Abstract:

In order to prevent the in-service failure of an item or a system, it is a common practice to employ an age replacement policy, in which a working item is replaced by a new one on its failure or at a prespecified time t, whichever occurs earlier. In this context, the mean time to failure (MTTF) function plays a prominent role in the study of reliability characteristics of systems under age replacement policy. Hence it is of profound importance to have statistical test procedures for comparing MTTF functions of two life distribution functions. On the other hand, it is of practical interest to detect whether lifetime data exhibits a possible departure from exponentiality toward notions of ageing where the MTTF function is monotonic. In the first part of the talk, a flexible two-sample nonparametric test, based on two independent samples, will be discussed for comparing mean time to failure (MTTF) functions of two life distributions. Finally, a consistent test of exponentiality against alternatives belonging to the decreasing mean time to failure class of life distributions will be presented.

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Given a strongly continuous orthogonal representation $(U_t)_{t\in \mathbb{R}}$ of $\mathbb{R}$ on a real Hilbert space$\mathcal{H}_{\mathbb{R}$, a decomposition $\mathcal{H}_{\mathbb{R}}:=\bigoplus\limits_{i\inN}\mathcal{H}_{\mathbb{R}}^{(i)}$ consisting of invariant subspaces of$(U_t)_{t\in \mathbb{R}}$ and an appropriate matrix $(q_{ij})_{N\times N}$ of real parameters, one can associate representations of the mixed commutation relations on a twisted Fock space. The associated von Neumann algebras are called mixed $q$-deformed Araki-Woods von Neumann algebras. These algebras turned out to be the non-tracial analogue of mixed $q$-Gaussian von Neumann algebras constructed by Bo\.{z}ejko-Speicher in 1992. In this talk we discuss the construction and structural properties of mixed $q$-deformed Araki-Woods von Neumann algebras.

Biography:

Rahul Kumar R is a research scholar working in the area of von Neumann algebras. He defended his thesis titled 'Mixed q-deformed Araki- Woods von Neumann algebras' in November 2022 at IIT Madras. He obtained his M.Sc. degree from CUSAT, Kerala. He was awarded Ph.D research fellowship from NBHM and UGC. As of now, he has co-authored four research articles of which three of them are either accepted/published in peer reviewed journals.

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Modular forms and their generalizations are one of the most central concepts in number theory. It took almost 200 years to cultivate the mathematics lying behind the classical (i.e. scalar) modular forms. All of the famous modular forms (e.g. Dedekind eta function) involves a multiplier, this multiplier is a 1-dimensional representation of the underlying group. This suggests that a natural generalization will be matrix valued multipliers, and their corresponding modular forms are called vector valued modular forms. These are much richer mathematically and more general than the (scalar) modular forms. In this talk, a story of vector valued modular forms for any genus zero Fuchsian group of the first kind will be told. The connection between vector-valued modular forms and Fuchsian differential equations will be explained.

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Let p be an odd prime, f be a p-ordinary newform of weight k and h be a normalized cuspidal p-ordinary Hecke eigenform of weight l < k. Let p be an Eisenstein prime for h i.e. the residual Galois representation of h at p is reducible. In this talk, we show that the p-adic L-function and the characteristic ideal of the p∞-Selmer group of the Rankin-Selberg convolution of f, h generate the same ideal modulo p in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for f⊗h holds modulo p. This is a joint work with Somnath Jha and Sudhanshu Shekhar.

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Mean Field Games (MFG) is a coupled system of equations consisting (i) backward Hamilton-Jacobi-Bellman equation and (ii) forward Fokker-Planck equation. These model a class of differential game problems with a large/ infinite number of agents. In this talk, I will discuss the numerical approximation of a class of MFG systems with nonlocal/fractional order diffusion. The problems include strongly degenerate diffusion and the solutions of such systems are usually interpreted by viscosity - very weak sense. Our approximations are based on semi-Lagrangian schemes. The prescribed approximations are monotone, stable and consistent. I will discuss the convergence analysis for both degenerate and nondegenerate cases. If time permits, I will give a brief overview about the new wellposedness result for fully nonlinear MFG as well.

Biography:

I am currently working as a postdoctoral fellow at University of Zagreb, Croatia. Prior to that, I was a postdoctoral researcher at Norwegian University of Science and Technology, Trondheim, Norway. I obtained my PhD from TIFR, Centre for applicable Mathematics, Bangalore, India under the supervision of Prof. Imran H. Biswas. I joined TIFR-CAM as an integrated PhD fellow in 2010. I finished my B.Sc. from St. Xavier's College Kolkata in 2007. My research interests include theory and numerical analysis of partial differential equations involving nonlocal operators.

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From the short Fields Medal citation:

"A very long-standing problem in mathematics is to find the densest way to pack identical spheres in a given dimension. It has been known for some time that the hexagonal packing of circles is the densest packing in 2 dimensions, while in 1998 Hales gave a computer assisted proof of the Kepler conjecture that the faced centered cubic lattice packing gives the densest packing in 3 dimensions. The densest packing wasn’t known in any other dimension until in 2016 Viazovska proved that the E8 lattice gave the densest packing in 8 dimensions and, very shortly afterwards, together with Cohn, Kumar, Miller and Radchenko, proved that the Leech lattice gave the densest packing in 24 dimensions. Viazovska’s approach built off work of Cohn and Elkies, who had used the Poisson summation formula to give upper bounds on the possible density of sphere packings in any dimension. Their work had suggested that in 8 and 24 dimensions there might exist a radial Schwartz function with very special properties (for instance it and its Fourier transform should vanish at the lengths of vectors in the respective lattice packings) which would give an upper bound equal to the lower bound coming from the known lattice packings. Viazovska invented a completely new method to produce such functions based on the theory of modular forms."

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In the first part of the talk, we will discuss the coefficient identification problems for partial differential equations. In the second part, we will discuss our results related to the inverse scattering problems. Finally, if time permits, we will also discuss very briefly about the obtained results related to integral geometry.

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GLMs are a powerful class of models applied ubiquitously in machine learning and signal processing applications. Learning these models often involves iteratively solving non-convex optimization problems. I will present an exact statistical analysis of learning in these models in a high dimensional setting. This framework is built on new developments in Random Matrix Theory, and does not rely on convexity. Using this framework, we can now analyze the effect of several design choices on the generalization error of the learned model. Example design choices include loss function, regularization, feature covariance, train-test mismatch.

Biography:

Parthe Pandit is a Simons postdoctoral fellow at the Halıcıoğlu Data Science Institute at UC San Diego. He obtained a PhD in ECE and MS in Statistics both from UCLA, and a dual degree in EE from IIT Bombay. He is a recipient of the Jack K Wolf student paper award at ISIT 2019. He has also been a research intern at Amazon and Citadel LLC.

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Extensions of valuations is a remarkably deep and open area of research which is significant from both algebraic and geometric points of view. The theory of ramification is developed to study extensions to algebraic extensions. However, a comprehensive theory to understand extensions of valuations to transcendental extensions is yet to be completely developed. In this talk, we will provide a gentle introduction to some of the classical concepts, and also mention some recent developments.

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Let E be a vector bundle on a smooth complex projective curve C of genus at least two. Fix an integer d>1. Let Q be the Quot scheme that parametrizes the torsion quotients of E of degree d. In this talk we will compute the sheaf cohomology of the tangent bundle of Q. In particular, we will study the space of first order infinitesimal deformations of Q. This is a joint work with Indranil Biswas and Ronnie Sebastian.

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Computational fluid dynamics constitutes a new approach to studying and developing the whole discipline of fluid dynamics. In this seminar, some basic concepts of flow in porous media and different governing equations for porous medium flow will be discussed. My interest lies in examining hydrodynamic stability analysis through linear and weakly nonlinear analysis. A brief discussion on a mathematical formulation for linear and weakly nonlinear stability analysis for a general nature flow will be considered. The linear stability analysis includes stability boundaries, energy spectrum at and around the critical point (bifurcation point), and disturbance pattern, which helps to understand the flow dynamics better. However, the weakly nonlinear stability analysis includes two major concepts; (i) growth of the most unstable linear wave and (ii) development of the Landau equation (or amplitude equation) to identify the regions of subcritical and supercritical bifurcations, energy spectrum, and pattern of disturbance flow. Apart from these, highlights on research achievement, undergoing research work in different directions, and future research plans will be discussed.

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Riemann-Roch theorems appear in various avatars in mathematics. For algebraic varieties it provides a functorial isomorphism between K-theory and Chow groups. In this talk we shall discuss various versions of Riemann-Roch theorems for algebraic stacks and matrix factorization categories. This talk in part is based on joint work with Dongwook Choa and Bumsig Kim.

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We define the logarithmic connection singular over a finite subset of a compact Riemann surface and its residues. We consider the moduli space of logarithmic connections with fixed residues over a compact Riemann surface. We show that there is a natural compactification of the moduli space of those logarithmic connections whose underlying vector bundle is stable. We compute the Picard group of this moduli space and show that it does not admit any non-constant algebraic function, but it admits non-constant holomorphic function.

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An E_0-semigroup over [0,\infty) is a semigroup $\{\alpha_t\}_{t\geq 0}$ of unital, normal *-endomorphisms of the algebra B(H) of bounded operators on a Hilbert space H such that for $A \in B(H)$ and $\xi, \eta \in H$, the map $[0,\infty) \ni t \to <\alpha_t(A)\xi, \eta> \in \mathbb{C}$ is continuous. Arveson associates with every E_0-semigroup over R_+ a product system and motivated by this observation, he introduced the notion of abstract product systems. Arveson established a one-to-one correspondence between isomorphism classes of product systems of Hilbert spaces and cocycle conjugacy classes of E_0-semigroups on B(H).

We have generalized this theory to E_0-semigroups over closed convex cone P in R^d, where $d\geq 2.$ In the multi-parameter definition of E_0-semigroups, we replace R_+ with a closed convex cone in R^d. We proved the following theorem.

Theorem (with Sundar) Product systems over $\Omega$ are in bijective correspondence with E_0-semigroups (up to cocycle conjugacy) over P, where $\Omega=Int(P)$. The above theorem is a very fundamental theorem that distinguishes one-parameter and multi-parameter E_0-semigroups.

Biography:

I am Murugan S P, a visiting fellow at Chennai Mathematical Institute. My native place is Sivakasi which belongs to the Virudhunagar district, Tamilnadu. My area of interest belongs to Operator Algebra. I carried out my thesis dissertation under the guidance of Prof. R. Srinivasan at Chennai Mathematical Institute. My thesis is entitled as follows: E_0-semigroups and Product Systems.

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It is well known that the real interpolation spaces can be represented in terms of analytic semigroup generated by a sectorial operator. In this talk, we shall obtain the representation of real interpolation spaces in terms of fractional resolvent families (or, the solution operators in fractional calculus). Utilizing this, we shall discuss several strict Hölder regularity results for a Caputo time-fractional abstract Cauchy problem.

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Labeling patients in electronic health records with respect to their statuses of having a disease or condition, i.e. case or control statuses, has increasingly relied on prediction models using high-dimensional variables derived from structured and unstructured electronic health recorddata. A major hurdle currently is a lack of valid statistical inference methods for the case probability. In this paper, considering high-dimensional sparse logistic regression models for prediction, we propose a novel biascorrected estimator for the case probability through the development of linearization and variance enhancement techniques. We establish asymptotic normality of the proposed estimator for any loading vector in high dimensions. We construct a confidence interval for the case probability and propose a hypothesis testing procedure for patient case-control labelling. We demonstrate the proposed method via extensive simulation studies.

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The theory of uniform algebra is a branch of mathematics in the interface between functional analysis and complex analysis. There was a general feeling that a uniform algebra A on X either is the algebra of all continuous complex-valued functions on X with the supremum norm or else there is a subset of the maximal ideal space of A that can be given the structure of a complex manifold on which the functions in A are holomorphic. However, it is well known that this feeling is not completely correct. We therefore consider weaker forms of analytic structure. Nonzero point derivation and nontrivial Gleason part are two such forms. However, the direct relation between these two notions is still unknown. Thus the question arises as to how these two notions are related. In this talk, we will explore whether the presence of one of these two notions imply the presence of the other.

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Population models with diverse type of catastrophes can be experienced in many real-life situations. In my talk, I will discuss about discrete-time population models which are prone to mild catastrophes such as binomial and geometric. As the population models involve different forms of arrival process of individuals as well as of catastrophes, I apply the supplementary variable technique to formulate the steady-state governing equations of the models. For further analysis, I use the roots method. The steady-state population size distributions at various epochs are obtained in terms of the roots of the associated characteristic equation. The methodology used throughout my work is analytically tractable and easily implementable. I will also talk about my contribution on population and queueing models. And finally, I will give a brief overview of my future research plan.

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The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We propose a class of tests to detect trend change in MTTF function. We develop test statistics utilising a measure of deviation based on a weighted integral approach. We derive the exact and asymptotic distributions of our test statistics exploiting L-statistic theory and also establish the consistency of the test as a consequence of our results. A Monte Carlo study is conducted to evaluate the performance of the proposed test. We also apply our test to some real life data sets for illustrative purposes. Further, the point at which the MTTF function changes trend has important implications in the context of cost optimization in such policies. So, we also develop a general methodology for change point estimation in this scenario and also establish the strong consistency of the proposed estimator. Finally, we have established some results to place in clear perspective the position of each of the well-known non-monotonic ageing classes.

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Nonlinear differential equations model many real-world physical phenomena encountered in scientific applications. Thus, the differential equations manifest themselves in numerous science branches and have a remarkable ability to predict the world around us. Generally, the evaluation of exact solutions to these problems poses a significant challenge. More precisely, the exact solutions are only available in a few cases. Thus, the construction of effective methods to solve them is crucial.

The main objective of the presentation is to present the computationally efficient iterative methods to solve nonlinear differential equations effectively. These methods are based on Newton's quasilinearization and the Picard iteration method. To demonstrate the efficiency, robustness, and applicability of the proposed methods, we consider various numerical examples, including real-life problems. The numerical simulations illustrate that the proposed methods are straightforward to implement and minimize the computational work compared with the existing methods. These methods are computationally efficient and overcome numerous shortcomings of the existing methods.

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The mixed convection in a duct filled with porous media is more concerned with the different industrial applications in micro-scale electronic equipment, macro-scale electrical transformers, solid-matrix heat exchangers, packed-bed reactors, energy-storage systems, etc. Knowledge of the heat-transfer characteristics and fluid flow mechanism, especially in the transition state, for duct-flow systems can guide in optimization of the thermal design and ensure a high degree of safety in the devices used in these applications. As a result, prior to initializing the duct flow through porous media in any appliance, it is essential to understand the flow dynamics and heat transfer mechanism in transition state under different geometrical situations. The understanding of the instability mechanism of annular flow through porous media may be of special interest because based on the gap between coaxial circular cylinders the annular flow provides a more general overview of the duct flow systems. As pointed out in the literature, in general, the transition from smooth laminar to disordered turbulent flow can involve a sequence of instabilities in which the system realizes progressively more complicated states or it can occur suddenly. In the former case, the complexity arises in well-defined steps in the form of sequence of bifurcations. Also, in the case of high permeable porous medium and reasonably high flow strength, the nonlinear interaction of superposed disturbance (fundamental mode) may have a significant role in the flow instability mechanism. Thus, to identify these bifurcations and nonlinear instability mechanisms, a nonlinear stability analysis of stably stratified flow is carried out in space of different controlling parameters.

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This presentation has two parts: first, the problem statement is examined, followed by a review of two common strategies for extending and modifying the Banach contraction principle, as well as various recent applications linked to the current state-of-the-art. Second, my main goal is to describe new research tools, techniques, and processes in the realm of fixed-point theorems, with an emphasis on their applicability in diverse fields of mathematical and technological sciences. My discussion will focus on explaining the study goals, expected outcomes, and methods that will be used to carry out and evaluate my research plan, which is centred on the application of contractions mappings to nonlinear systems.

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We consider making statistical inferences upon unknown parameters of some failure time distributions under different set ups namely, accelerated life testing, competing risks model and multicomponent stress-strength model. Various estimates of unknown quantities are obtained under censored data. Testing highly reliable experimental units under normal stress conditions may result in very few or no failures. Thus, many such products are usually tested at high stress levels so that a desired number of failures can be observed in some quick time. Furthermore, an experimental unit may fail due to multiple causes which compete with each other in product life cycle. These modes of failure are termed as competing risks. Sometimes identifying the actual causes of failures for all experimental units are not possible. In such situations, causes of failure are observed partially. In addition, reliability assessment of systems having multiple components is an important problem. In such set up a system properly functions provided strength of the system exceeds the stress experienced by it. Various inferences for unknown quantities can be obtained in many such fields of practical study based on some observed data. In general, life testing experiments are performed under time and budget limits. So it may not be possible to record failure times of all units put on the test. However, different censoring methods can be applied to collect data for further inference. In this work, we have mainly obtained various estimation results based on progressive censoring and generalized progressive hybrid censoring scheme by considering different probability models. For instance, estimation results for log-logistic distribution are obtained under progressive-stress accelerated life testing situations. Both classical and Bayesian estimates of model parameters are obtained. Future lifetimes of censored observations are predicted as well. Also, Kumaraswamy distributions is studied under partially observed competing risks models. The estimates for model parameters are obtained under generalized progressive hybrid censoring scheme. Multicomponent stress-strength reliability estimation is taken up under progressive censoring when stress and strength variables follow a family of inverted exponentiated distributions. Related results are derived under progressive censoring. The classical point estimates are obtained using maximum likelihood estimation method. Bayesian estimates are derived using Markov chain Monte Carlo, Lindley approximation and Tierney-Kadane methods. Asymptotic, bootstrap-percentile and bootstrap-t methods are used to construct classical intervals. Credible intervals are also discussed for comparison purposes. In each case, performance of various methods is assessed using simulation studies. Numerical examples are also discussed for illustration purposes.

Biography:

Dr. Amulya Kumar Mahto is an assistant professor at Kalinga Institute of Industrial technology. He received his PhD (Statistics) in April, 2021 and MTech (Mathematics and Computing) in May, 2015 from Indian Institute of Technology Patna and MSc. (Mathematics and Computing) in May, 2012 from Indian School of Mines, Dhanbad. His research interest include accelerated life testing, competing risks, multicomponent stress-strength reliability and has published 10 research papers in journal of repute such Journal of Applied Statistics, Journal of Statistical Computation and Simulation, Communication in Statistics: Theory and Methods, Quality and Reliability Engineering International, Journal of Statistical Theory and Practice, Annals of Data Science and OPSEARCH. Besides these many other research works are communicated for publication. His research interest also extends to transfer learning, an application of deep learning.

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We will discuss the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $\rho$ is determined by an equality of an $m$-power character $g \mapsto \operatorname{Tr}\left(\rho\left(g^{m}\right)\right)$ for some natural number $m$. Using this, we extend Faltings' finiteness criteria to determine the equivalence of two $\ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We will also discuss finiteness results for twist unramified representations. We will prove finiteness results for isomorphism classes of abelian varieties defined over a number field $K$, which have totally ramified reduction outside a finite set $S$ of places of $K$ and have good reduction at the places outside $S$ in some quadratic extension of $K$. This is a Joint work with Prof. C. S. Rajan.

Biography:

I have submitted my PhD thesis at Chennai Mathematical Institute in December 2021 under supervision of Prof Purusottam Rath and Prof C. S. Rajan. I have done 5 year Integrated MSc at UM-DAE Centre for Excellence in Basic Sciences, Mumbai (2009-2014). I was a INSPIRE PhD Fellow at UM-DAE Centre for Excellence in Basic Sciences, Mumbai (2009-2019) and transferred to Chennai Mathematical Institute in 2019.

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In 1961, Cornell proved that every finite abelian group G occurs as a subgroup of the class group of some cyclotomic field. There is no analogous result known for real cyclotomic fields. Class groups of real cyclotomic fields are very mysterious object. In this talk, we will prove that every finite abelian group G occurs as a subgroup of the class group of infinitely many real cyclotomic fields. This is a joint work with Prof. L. C. Washington and Prof. R. Schoof.

Biography:

I am a Ph.D student at Harish-Chandra Research Institute working in number theory. My primary interest is algebraic number theory and Elliptic curves. I am interested in understanding the structure of class groups of number fields and groups (Mordell-Weil group, Shafarevich-Tate group and Selmer group) attached to Elliptic curves.

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Exploring the mechanisms in tumor-immune dynamics is of utmost importance in oncology. In particular, a mathematical model of tumor-immune interactions can provide insight by analysing the nonlinear dynamics of tumor immune system. In this talk, I will discuss asymptomatic behaviours and long term dynamics of delayed and non-delayed tumor model under the viewpoint of dynamical system as well as tumor biology. Moreover, a dynamics of covid-19 model with comorbidity individual will also be discussed with epidemiological viewpoint.

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In this work, we study a weakly singular Volterra integrodifferential equation with Caputo type fractional derivative. First, we derive a sufficient condition for the existence and uniqueness of the solution of this problem based on the maximum norm. It is observed that the condition depends on the domain of definition of the problem. Thereafter, we show that this condition will be independent of the domain of definition based on an equivalent weighted maximum norm. In addition, we have also provided a procedure to extend the existence and uniqueness of the solution in its domain of definition by partitioning it. Next, we introduce an operator based parameterized method to generate an approximate solution of this problem. Convergence analysis of this approach is established here. Next, we optimize this solution based on least square method. For this, residual minimization is used to obtain the optimal values of the auxiliary parameter. In addition, we have also provided an error bound based on this technique. Comparison of the standard method and optimized method based on residual minimization signify the better accuracy of modified one.

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In this talk, I will discuss under what conditions we can be sure that a solution of first order ordinary differential equation exists. If solution exists, under what conditions can we be sure that there is a unique solution? The applicability of this result will also be discussed with a few examples.

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Applied mathematics has been playing a very important role in the field of science and engineering. Real world applications can be modelled by differential equations. Here, we study one practical application based on ordinary differential equation. Next, we study the existence and uniqueness of the solution of a first order ordinary differential equation. In addition, we discuss several numerical examples based on existence and uniqueness theorem.

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Turbulent transport of momentum, heat, and mass dominates many of the fluid flows investigated in physics, fluid mechanics, mathematics, engineering, and the environmental sciences. Intensive research on the dynamics of the wall turbulence in the boundary layer, pipe or channel, has been performed by many investigators. On the other hand, in hydraulics or river mechanics, the dynamics of turbulent open-channel water flow since it dominates the turbulent friction law, turbulent diffusion problem, sediment transport in the river field, and the flow characteristics near the hydraulic structures and so on. The detailed mathematical investigations of turbulence in open channel flows are still very insufficient. The ability to study the flow rate accurately using the Reynolds stress equations is very difficult. Since the Reynolds stress equations contain fluctuating components and fluctuating components are a highly irregular, random, complex, multi-scale, nonlinear, three-dimensional unsteady viscous phenomenon that occurs at high Reynolds number. The equations contain ten unknowns (namely, three components of mean velocity, the pressure, and six components of Reynolds stress) and hence the system is not closed and cannot be solved. Therefore, the main objective of this presentation is to investigate and model the Reynolds stress equations for turbulent flows.

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Many practical problems in science and engineering, when formulated mathematically, give rise to partial differential equations (PDE). In order to understand the physical behaviour of the mathematical model, it is necessary to have some knowledge about the mathematical character, properties, and the solution of the governing PDE. An equation which involves several independent variables (x, y, z, t,....), a dependent function u of these variables, and the partial derivatives of the dependent function u with respect to the independent variables, is called a partial differential equation. In this presentation, we shall consider second-order PDE involving two independent variables and find that the number of characteristic curves through a given point introduces a classification of the equation as one of three types: hyperbolic, parabolic, or elliptic. These three types are of fundamental importance in the theory of PDE. If time permits, we shall discuss some qualitative properties of second-order elliptic PDE such as maximum principal and mean value property etc.

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The Artin braid groups, also known as braid groups, are celebrated objects and appear in diverse areas of mathematics, and theoretical physics. One of the notable features of braid groups is their connection with knots and links. They also have explicit topological interpretation in the 3-space given by Artin in the 1920s. Doodles can be thought of as a planar analogue of knots. A class of right-angled Coxeter groups play the role of groups in the theory of doodles similar to the role that braid groups play in the theory of knots and links. Khovanov studied these groups and gave a topological interpretation to them as equivalence classes of configurations of intervals on the plane similar to the one known for braid groups. He called these groups twin groups. As these groups can be seen as planar analogues of braid groups, they are also known as planar braid groups. Since then, these groups are of interest to both topologists and algebraists. Recently these groups have also received attention from physicists who study them under the name of traid groups. Both braid groups and twin groups can also be seen as extensions of the symmetric groups. In this talk, we will introduce some structures related to twin groups, namely pure twin groups, virtual twin groups, and pure virtual twin groups. The focus of the talk will be on the results in their algebraic aspects. Towards the end of the talk, we will discuss a family of abstract groups, which arise as extensions (similar to the case of braid groups, and twin groups) of the symmetric groups.

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With the advent of continuous health monitoring with wearable devices, users now generate their unique streams of continuous data such as minute-level step counts or heartbeats. Summarizing these streams via scalar summaries often ignores the distributional nature of wearable data and almost unavoidably leads to the loss of critical information. We propose to capture the distributional nature of wearable data via user-specific quantile functions (QF) and use these QFs as predictors in scalar-on-quantile-function-regression (SOQFR). As an alternative approach, we also propose to represent QFs via user-specific L-moments, robust rank-based analogs of traditional moments, and use L-moments as predictors in SOQFR (SOQFR-L). These two approaches provide two mutually consistent interpretations: in terms of quantile levels by SOQFR and in terms of L-moments by SOQFR-L. We also demonstrate how to deal with multi-modal distributional data via Joint and Individual Variation Explained (JIVE) using L-moments. The proposed methods are illustrated in a study of association of digital gait biomarkers with cognitive function in Alzheimer's disease (AD). Our analysis shows that the proposed methods demonstrate higher predictive performance and attain much stronger associations with clinical cognitive scales compared to simple distributional summaries.

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Lectures on topics from the following will be covered. Random experiment, Sample space, Event, Sets and Set operations. Classical definition of probability, Relative frequency definition, Sigma Algebra, Axiomatic definition, Properties, Counting, Conditional probability, Bayes rule and independence of events.

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Global 3D display, Virtual Reality and Augmented Reality market is likely to experience considerable growth in coming years. Autostereoscopic or Naked-eye 3D displays are going to become a competitive alternative to stereoscopic 3D or standard 2D presentations. However, a number of central problems remain unsolved for supporting multi-user 3D displays that provide greater depths of field, wider fields of view, and continuous natural motion parallax, which invokes a strong feeling of immersion. Likewise, current VR/AR display technologies fall far short of truly recreating visual reality. Existing head-mounted displays cannot satisfy the natural accommodative perception of human eyes. This leads to visual discomfort after long hours of watching contents on such displays. Mixed reality devices today have a difficult time to display images up-close. This is a fundamental limit of current mixed reality displays that holding back entire industry.
Light field offers a solution today to inherent vergence-accommodation conflict (VAC) by reproducing apparent focal depth planes and stereo simultaneously. This removes the brainconfusing mismatch by enabling virtual objects to appear real at distances both near and far, and invoke natural coupling between eye vergence and accommodation in predicting the changes in perspective due to motion. However, prior automultiscopic displays work at the device level and use complex physical (mechanical, metallurgical, chemical, optical, etc.) means to reconstruct complete 4D light rays on the displays. Such approaches prove expensive and are hard to scale to acceptable standards due to large number of computational resources and bulky hardware. An alternate is to jointly optimize computation with sensing (optics) to economically alleviate the shortcomings of available 3D displays. This provides a solution with walk-around possibility in a reasonable field of view without any restrictions and sacrificing the resolution. Employing computation is critical to extend refocusing and directiondependent properties of light fields to portable glasses-free 3D displays and mixed reality HMDs. Still maintaining practical feasibility of processing large volume of light field data is hard on compact 3D displays and VR/AR wearables. The technology opens up new creative opportunities if a scene can be represented using a limited number of perspectives. Otherwise, it demands high complexity decoder. Addressing both ‘Depth of Field’ and ‘Field of View’ according to display size and characteristics is challenging. Therefore, it is imperative to take care of light field standardization requirements for naked-eye 3D displays and mixed reality platform.
This talk focuses on research problems that we have addressed in developing a viable lowcost, light field processing pipeline from capture to display. The proposed camera agnostic 3D pipeline leverages the advances in artificial intelligence/machine learning, geometrical optics, multilinear algebra, tensor optimization, and signal processing (Fourier analysis and compressive sampling) to reproduce high-fidelity light field on a compact 3D display. Harnessing the expressive power of deep neural networks with computational optics and mathematical models greatly reduces the data acquisition and processing cost for displaying full-parallax static or dynamic light field 3D images. The solutions work on top of existing display methodologies and are adaptable for different viewing conditions or multi-baseline geometries (e.g. from home theater projectors to mobile applications). This makes our systems scalable to support an apparent number of views from many directions (angles) simultaneously. Our target is to decouple light field content production from the display system and application scenario, which is currently not the case. In proposed schemes, we integrated computational optics with data-driven mathematical models and AI-enabled representation learning solutions that would address relevant issues associated with four basic display types that show the greatest promise of success: Compressive multi-layer, Integral Imaging, Eyesensing Super-stereoscopic, and Multi-view autostereoscopic 3D displays. Not just that, it also opens possibilities to address full range of computational tasks for accommodation-supporting HMDs using low-cost compressive (lensless/plenoptic) cameras or commodity RGB-D sensors. Further, the proposed mathematically valid solutions are generic and flexible to support light field interaction capabilities (post capture freedom) with Deep Optics, 3D vision technology and Haptic touch screens - a major step yet to be leveraged that bestow mobile opportunity for interactive full-parallax 3D displays. This talk will identify innovative solution modalities from different domains in order to have the best chance of deploying affordable light-field technologies with the main contenders for 3D displays and VR/AR HMDs.

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Bayesian learning and discriminant function analysis are fundamental statistical approaches to the problem of pattern classification. In this teaching talk, I shall cover the basics of Bayesian learning that enables to predict the error when we generalize the classification problem to novel patterns. I shall cover the topics that quantify the tradeoffs between various classification decisions using probability and the costs that accompany such decisions. Further, I shall cover multivariate technique for describing discriminant functions, and decision surfaces for pattern classifiers.

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Group signatures are significant primitive for anonymity, which allow group members to sign messages while hiding in the group, however, the signers remain accountable. Most of the existing schemes on group signature are relying on traditional cryptographic primitives, whereas rapid advancements in quantum computing suggest an originating threat to usual cryptographic primitives. This makes the necessity of quantum computer resistant cryptographic primitives. Multivariate public key cryptography (MPKC) is one of the promising options that may withstand quantum attacks. Its constructions are potential candidates for post-quantum (PQ) cryptography as they are very fast and require only modest computational resources. There are many existing secure and practical multivariate digital signatures. However, there is a deficiency of more advanced multivariate group signature scheme. The existing multivariate group signature has weaknesses interms of security and efficiency. We introduce a new multivariate group signature scheme employing a 5-pass identification protocol and multivariate signature scheme as its building blocks. The proposed signature scheme possesses unforgeability, user's anonymity, unlinkability, exculpability and traceability property. Unlike most of the existing post-quantum group signatures, the sizes of the signatures and the public parameters are not dependent on the number of group users in our construction. In particular, our construction is the first MPKC based group signature, where signature size and public parameter size are independent of the number of group users.

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The irreducibility of polynomials has a long history. In 1797, Gauss proved that the only irreducible polynomials with complex coefficients are linear polynomials. However, in view of Eisenstein Irreducibility Criterion proved in 1850, for each number $n\geq 1$, there are infinitely many irreducible polynomials of degree $n$ over rationals. In this talk, we shall discuss the development of this criterion through Newton polygons and see some recent results which provide us some explicit information regarding the degrees and number of irreducible factors of a polynomial. Further, we shall discuss some other applications of Newton polygons.

Abstract:

A commutator is an operator of the form AB - BA where A and B are operators on a complex Hilbert space. In finite-dimensional Hilbert space, commutators are characterized via the trace condition, i.e., an operator C is a commutator if and only if trace C = 0. In the more interesting case of an infinite-dimensional Hilbert space, the Pearcy-Topping question (1971) still remains a mystery: Is every compact operator a commutator of compact operators? A new perspective and some recent advances on this problem will be discussed in the talk.

Abstract:

In this presentation, I will talk about (a) PhD work, and (b) Post PhD work.

(a) PhD work:
PhD work is mainly aimed at development and implementation (extension of BBIE method) of a non-primitive boundary element method for (i) modelling two-dimensional flow of a viscous incompressible fluid through non-deformable porous medium using Brinkman equation, and (ii) modelling two-dimensional flow containing an interface between two porous medium having different permeability using Brinkman equation. Also, Discussing specific applications in porous media such as flow through a porous wavy channel and flow through a composite porous channel.

(b) Post PhD work:
Post PhD I have developed interest in the wide applications of Multi-domain Boundary Element Method (MBEM) and Dual Boundary Element Method (DBEM). This mainly consists of analysing the water wave scattering and radiation problems, liquid sloshing, energy extraction from Oscillating Water Columns (OWCs) and gravity wave interaction with floating elastic plates. These topics are very interesting due to the importance garnered for the socio-environment issues such as varying climate changes and increasing coastal activities. A couple of slides devoted to the developed codes and its validation.

Abstract:

In this talk, first I will present existence and uniqueness theorem (without proof) for first order problems involving
(a) Non-linear first order ODEs with initial condition
dy/dx=f(x,y), y(x₀)=y₀
(b) Linear first order ODEs with initial condition
dy/dx+P(x)y=Q(x), y(x₀)=y₀
Subsequently, I will explain the applicability of existence and uniqueness theorem by considering several examples.

Abstract:

We discuss the existence and regularity of periodic traveling wave solutions of a class of nonlocal equations with homogeneous symbols of order −r, where r > 1. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked periodic traveling wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol m(k) = k^ {−2}. Thereby we recover its unique highest 2π-periodic, peaked traveling wave solution, having the property of being exactly Lipschitz at the crest.

Abstract:

The electroosmotic flow of a non-Newtonian fluid (Power-law, Casson, Bingham and Hershel-Bulkley fluid) near a surface potential heterogeneity is studied numerically based on the Nernst–Planck model for ion transport. The objectives of this study are to highlight the limitations of the linear slip-model and the nonlinear Poisson–Boltzmann model at various flow conditions as well as to develop vortical flow to promote mixing of neutral solutes within the micro-channel. A power-law fluid, both shearthinning and shear-thickening, for the pseudoplastic behaviour of the non-Newtonian fluid or viscoplastic fluid with yield stress is adopted to describe the transport of electrolyte, which is coupled with the ion transport equations governed by the Nernst–Planck equations and the Poisson equation for electric field. The viscoplastic fluid is modeled as either Casson, Bingham or Hershel–Buckley fluid. A pressurecorrection based control volume approach has been adopted for the numerical computations of the governing equations. The nonlinear effects are found to be pronounced for a shear thinning liquid, whereas, the electroosmotic flow is dominated by the diffusion mechanisms for the shear thickening liquid. A maximum difference of 39% between the existing analytic solution based on the Debye– Huckel approximation and the present numerical model is found for a shear thinning power-law fluid. A vortex, which resembles a Lamb vortex, develops over the potential patch when the patch potential is of opposite sign to that of the homogeneous surface potential. Enhanced mixing of a neutral solute is also analyzed in this presentation. The yield stress reduces the electroosmotic flow however, promotes solute mixing.
We also consider the electroosmotic flow (EOF) of a viscoplastic fluid within a slit nanochannel modulated by periodically arranged uncharged slipping surfaces and no-slip charged surfaces embedded on the channel walls. The objective of this study is to achieve an enhanced EOF of a non-Newtonian yield stress fluid. The Herschel-Bulkley model is adopted to describe the transport of the non- Newtonian electrolyte, which is coupled with the ion transport equations governed by the Nernst-Planck equations and the Poisson equation for electric field. A pressure-correction-based control volume approach is adopted for the numerical computation of the governing nonlinear equations. We have derived an analytic solution for the power-law fluid when the periodic length is much higher than channel height with uncharged free-slip patches. An agreement of our numerical results under limiting conditions with this analytic model is encouraging. A significant EOF enhancement and current density in this modulated channel are achieved when the Debye length is in the order of the nanochannel height. Flow enhancement in the modulated channel is higher for the yield stress fluid compared with the power-law fluid. Unyielded region develops adjacent to the uncharged slipping patches, and this region expands as slip length is increased. The impact of the boundary slip is significant for the shear thinning fluid. The results indicate that the channel can be cation selective and nonselective based on the Debye layer thickness, flow behavior index, yield stress, and planform length of the slip stripes.

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The maximum principle states that a non-constant function satisfying Laplace equation cannot attain a maximum (or minimum) at an interior point of its domain. We shall formulate the result and its useful consequences

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Outline of Talk: The outline of my presentation contains mainly the following topics
1. First order partial differential equations and its characteristic
2. Classification of 2nd order partial differential equations and its characteristic

Abstract:

Advances in various _elds of modern studies have shown the limitations of traditional probabilistic models. The one such example is that of the Poisson process which fails to model the data tra_c of bursty nature, especially on mul-tiple time scales. The empirical studies have shown that the power law decay of inter-arrival times in the network connection session o_ers a better model than exponential decay. The quest to improve Poisson model led to the formulations of new processes such as non-homogeneous Poisson process, Cox point process, higher dimensional Poisson process, etc. The fractional generalizations of the Poisson process has drawn the attention of many researchers since the last decade. Recent works on fractional extensions of the Poisson process, commonly known as the fractional Poisson processes, lead to some interesting connections between the areas of fractional calculus, stochastic subordination and renewal theory. The state probabilities of such processes are governed by the systems of fractional differential equations which display a slowly decreasing memory. It seems a characteristic feature of all real systems. Here, we discuss some re- cently introduced generalized counting processes and their fractional variants. The system of di_erential equations that governs their state probabilities are discussed.

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In this talk, we will learn some basic concepts of conditional probability. The following topics will be discussed with examples: Multiplication Rule, Total Probability Theorem and Bayes' Rule.

Biography:

Dr. Kuldeep Kumar Kataria received his BSc (Hons) degree in Mathematics from St. Stephen's College, University of Delhi. He received his MSc degree in Mathematics from IIT Kanpur. In 2018, he received his PhD in Mathematics from IIT Bombay. Later, he joined IISc, Bangalore as a NBHM Post-Doctoral Fellow. He is currently working as an Assistant Professor in the Department of Mathematics at IIT Bhilai. The research interest of Dr. Kataria lies in the area of fractional stochastic processes and subordinated (time-changed) versions of certain counting pro- cesses. He deals with stable subordinators and space-time fractional versions of the Poisson process. In his PhD thesis, he has studied the applications of Ado- mian Decomposition Method to certain fractional stochastic processes. So far, Dr. Kataria has published 18 research articles in international journals of re- pute like Journal of Theoretical Probability, ALEA. Latin American Journal of Probability and Mathematical Statistics, Journal of Mathematical Analysis and Applications, Comptes Rendus Mathematique, Statistics and Probability Letters, Stochastic Analysis and Applications, etc. Also, he has published several expository articles in reputed mathematical magazine like American Mathematical Monthly, Mathematics Magazine, etc. For his research contributions he has been honoured with the Award of Excellence in Thesis Work for the year 2016- 2018 by Hon'ble Prime Minister of India at the 56th Convocation of the Institute (IIT Bombay).

Abstract:

The multiple gamma functions of Barnes introduced more than a century ago have been taken up during the last decades because they enter in several areas of modern mathematics. These functions are a generalization of Euler's gamma function. This talk is about multiple gamma functions □n and their applications via the important class of Pick functions. A Pick function is a function that is analytic in the upper half-plane with positive imaginary part and has a well understood integral representation. We derive a new class of Pick functions related to the multiple gamma function and obtain its integral representation. Furthermore, we discuss about completely monotonicity properties and some inequalities related to □n and their ratios. Inequalities for ratio of q-gamma functions are also obtained, which gives an alternative proof of Bohr-Mollerup theorem for q-gamma functions. More- over, we introduce new classes of logarithmically completely monotonic functions involving q-gamma function. Finally, asymptotic expansions for multiple gamma functions are derived with the formulae for determining the coe_cients. Using these asymptotic expansions, Pad_e approximants related to these asymptotic expansions are also obtained. Moreover, multiple gamma functions are approximated in terms of Pad_e approximant and continued fractions, namely S-fraction and J-fraction.

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A complex manifold is called hyperbolic if the Kobayashi pseudodistance associated with it is a distance. When this distance is complete, there are real-geodesics joining any two points of the manifold. Motivated by a property of the Gromov boundary of a Gromov hyperbolic space, we introduce a concept of visibility with respect to real-Kobayashi geodesics. Then the question arises under what conditions on complete hyperbolic complex manifolds the visibility property holds true? I will present a result that gives a complete answer to the question assuming that the Kobayashi distance is Gromov hyperbolic. To motivate the second theme, we shall begin with Cartan's Uniqueness Theorem (CUT) for bounded domains. Kobayashi generalized this result for hyperbolic complex manifolds. Then we shall introduce a family of matricial domains which are generalizations of the spectral unit ball studied in the literature extensively. These matricial domains are extremely nonhyperbolic andCUT is not true for such domains. However, associated with these domains are a class of domains that are hyperbolic. Using the hyperbolicity of these latter domains and their connection with the matricial domains, we shall present a structure theorem for holomorphic self-maps (as in the statement of CUT) of the matricial domains; namely, that such maps are spectrum-preserving.

Abstract:

Analytical study of linear and weakly/local-nonlinear stability analyses of Rayleigh-Bénard convection (RBC) in a chemically reactive two-component fluid system shall be presented by considering physically realistic as well as idealistic boundaries. Analytical expression for the eigenvalue as functions of the chemical potential and the relaxation parameter shall be discussed in the cases of stationary, oscillatory, and subcritical instabilities. The threshold value of relaxation parameter at which the transition from the subcritical to the critical motion takes place shall be depicted for different values of the chemical reaction rate. The higher-order generalized Lorenz model leads to the reduced-order cubic-quintic, Ginzburg-Landau equation (GLE) and using its solution, the heat transport is quantified in steady and unsteady convective regimes. The drawback of the cubic GLE and the need for the cubic-quintic GLE for studying the heat transfer in the case of subcritical regime shall be explained. The possibility of having pitchfork and inverted bifurcations at various values of the relaxation parameter shall be highlighted.

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Lecture Topic:
The teaching lecture shall cover the following topics:

• Overview of signals and the Fourier series
• Signals for which the Fourier series exists
• Condition for existence of Fourier series
• Limitation of Fourier series
• Need for Fourier transforms
• Applications of Fourier representations
• Need for wavelets

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In this talk, we present a proof of $C^{1, \alpha}$ regularity up to the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. The proof is achieved via compactness arguments combined with new boundary H\"{o}lder estimates for equations which are uniformly elliptic when the gradient is either small or large.

Biography:

Dr. Ram Baran Verma, currently working as an assistant professor in the department of Mathematics SRM University AP. He completed his Ph.D from IIT Gandhinagar in 2018. His research interest is to study the existence and regularity properties of solutions to nonlinear elliptic equations. Prior joining to SRM University AP, he was a post-doctorate at TIFR-CAM Bangalore. For more details about Dr. Verma you can visit https://srmap.edu.in/faculty/dr-ram-baran-verma

Abstract:

In this talk, I will be talking about the maximum principle for the Laplace equation and its consequences, like uniqueness of solutions to the Dirichlet problem, a prior uniform estimate and gradient estimate for solution of Laplace equation etc.

Biography:

Dr. Ram Baran Verma, currently working as an assistant professor in the department of Mathematics SRM University AP. He completed his Ph. D from IIT Gandhinagar in 2018. His research interest is to study the existence and regularity properties of solutions to nonlinear elliptic equations. Prior joining to SRM University AP, he was a post-doctorate at TIFR-CAM Bangalore. For more details about Dr. Verma you can visit https://srmap.edu.in/faculty/dr-ram-baran-verma

Abstract:

In this talk we will discuss my recent research works. We will consider basically three problems
1. First, I will talk about the controllability of the Landau-Lifshitz-Gilbert Equation (LLGE) in an interval in one spatial dimension with Neumann boundary conditions. The control force acting here is degenerate i.e., it acts through a fewer number of modes. We exploit Fourier series expansion of the solution. We use methods of Lie bracket generating property to establish the global controllability of finite-dimensional Galerkin approximations of LLGE. Then, we show L^2 approximate controllability of the full system.
2. Then, we discuss a trigonometric method for the stochastic Euler-Bernoulli beam equation. We consider a fully discrete approximation of the linear stochastic Euler-Bernoulli beam equation driven by additive noise. A standard finite element method is used for the spatial discretization and a stochastic trigonometric scheme for the temporal approximation.
3. Finally, we discuss my very recent work on a stochastic cross diffusion system. We have shown the existence of a global martingale solution of that system by exploiting its entropy structure. This is an ongoing work with Prof. Ansgar Jüngel at TU Wien, Austria.

Abstract:

We start discussing Gaussian elimination to solve system Ax=b. Then, we discuss LU, LDU^t, LL^t (Cholesky) factorization of a matrix A and show how these factorization methods are used to solve system Ax=b. We also discuss the algorithms and operation counts for each of these methods.

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The present paper deals with the Shear wave propagation in a multi-layered magnetoelastic anisotropic monoclinic medium with finite-difference modeling to comprehend the stability criteria, phase velocity, and group velocity. Utilizing Maxwell's fundamental theory of magnetoelasticity, the problem has been constructed. Stability analysis has been conducted based on the finite-difference technique to reduce the soaring error values and control its stability. Numerical evaluation as well as a graphical representation, have been employed to enlighten the effects of different values of courant number and magnetoelasticity on the phase and group velocities.

Abstract:

The teaching session will deal with the Existence and uniqueness of the solution of the initial value problem for a first-order ordinary differential equation.

Abstract:

Thermoelastic vibration of micro and nano-beam resonators is described by the Euler-Bernoulli beam theory in context with two-temperature generalized thermoelasticity theory. The amplitude of deflection and thermal moment in the case of simply supported and the isothermal beam are obtained by the Laplace transform method along with the finite Fourier sine transform method and the vibration frequency is examined by the normal mode analysis when the beam-ends are clamped and isothermal. The effects of the two-temperature parameter and micro and nano-beam thickness on the amplitude of deflection, thermal moment, and thermoelastic vibration frequency of the micro and nano-beam resonator have been studied and shown with the numerical results. A comparison of the results with the corresponding results of the two-temperature generalized thermoelasticity (2TLS) and the Lord-Shulman (LS) theories is also presented. Besides, the size dependency nature of the micro and nano-beam is analyzed by analytical and numerical results.

Biography::

Bachelor of Science in Mathematics (Honors) from Gaya College (Magadh University), Gaya, Bihar, India. (2010-2013) Master of Science in Mathematics from Central University Of South Bihar, Gaya, Bihar, India. (2013-2015) Doctor of Philosophy in Mathematics from Central University Of South Bihar, Gaya, Bihar, India. (2017-2021) Thesis Title-: Effects of Phase-Lag on Thermoelastic Damping in Micro and Nano Beam Resonators under Generalized Thermoelasticity Research Interest- Mathematical Physics; Solid Mechanics; Heat Transfer; Thermoelastic Damping in Resonators, Wave Propagation Life Member of "Indian Mathematical Society (IMS)". Best Research Award, International Research Awards on New Science Inventions (NESIN 2020), ScienceFather. 13 research publication in peer reviewed reputed international journals (1 SCOPUS Index, 12 SCI Index).
Link- https://orcid.org/0000-0002-4580-6513

Abstract:

The interaction between interstitial fluid flow and the deformation of the underlying porous structure gives rise to a variety of mechanisms of fluid-structure coupling. In the specific case of Biot poromechanics, this interaction occurs when the linearly elastic porous medium is saturated, and such problem is relevant to a large class of very diverse applications ranging from bone healing to, e.g., petroleum engineering, or sound isolation. We are also interested in the interface between elastic and poroelastic systems that are encountered in hydrocarbon production in deep subsurface reservoirs (a pay zone and the surrounding non-pay rock formation), or in the study of tooth and periodontal ligament interactions.
In this talk, we discuss the \textit{a posteriori} error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretisations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of degree $k$ for rotation vector, total pressure, and elastic pressure. Residual-based estimators are constructed, and upper and lower bounds (up to data oscillations) for all global estimators are rigorously derived. The methods are all robust with respect to the model parameters (in particular, the Lam\'e constants), they are valid in 2D and 3D, and also for arbitrary polynomial degree $k\geq 0$. The error behaviour predicted by the theoretical analysis is then demonstrated numerically on a set of computational examples including different geometries on which we perform adaptive mesh refinement guided by the \textit{a posteriori} error estimators.
Finally, we discuss some remarks related to other contributions.

Abstract:

We know how to obtain the general solution of the nth-order linear differential equation with constant coefficients. We have seen that in such cases the form of the complementary function may be readily determined. The general nth-order linear equation with variable coefficients is quite a different matter, however, and only in certain special cases can the complementary function be obtained explicity in closed form. One special case of considerable practical importance for which it is fortunate that this can be done is the so-called Cauchy-Euler equation.

Biography:

Bachelor of Science in Mathematics (Honors) from Gaya College (Magadh University), Gaya, Bihar, India. (2010-2013) Master of Science in Mathematics from Central University Of South Bihar, Gaya, Bihar, India. (2013-2015) Doctor of Philosophy in Mathematics from Central University Of South Bihar, Gaya, Bihar, India. (2017-2021) Thesis Title-: Effects of Phase-Lag on Thermoelastic Damping in Micro and Nano Beam Resonators under Generalized Thermoelasticity Research Interest- Mathematical Physics; Solid Mechanics; Heat Transfer; Thermoelastic Damping in Resonators, Wave Propagation Life Member of "Indian Mathematical Society (IMS)". Best Research Award, International Research Awards on New Science Inventions (NESIN 2020), ScienceFather. 13 research publication in peer reviewed reputed international journals (1 SCOPUS Index, 12 SCI Index).
Link- https://orcid.org/0000-0002-4580-6513

Abstract:

In this lecture we discuss numerical methods for solving a system of linear equations Ax = b, where the given matrix A is real, n × n, and nonsingular and b is a given real vector in Rn . We seek a solution x that is necessarily also a vector in Rn. It is well known that such problems arise frequently in any branch of science, engineering, economics, or finance. There is no single technique that is perfect for all cases. But the available numerical methods in the literature can generally be divided into two classes: direct methods and iterative methods. In this lecture, we will discuss methods of the first type. In the absence of roundoff error, we see that such methods would yield the exact solution within a finite number of steps. The basic direct method for solving linear systems of equations is Gaussian elimination. The Gaussian elimination method proceeds by successively eliminating the elements below the diagonal of the matrix of the linear system until the matrix becomes triangular, when the solution of the system is very easy. In the end part of the lecture, we discuss the LU decomposition method.

Abstract:

Before the scientific community, one of the most challenging problems is what sustains life or how living things originate from non-living things. A biological cell is assumed as the smallest unit of life. The multiple cellular processes are mainly controlled by tiny biological machines called motor proteins which move along macromolecular highways, namely microtubules, to transfer cargoes at distinct locations inside a cell. To get insight into the life-sustaining mechanism, it is necessary to understand the collective transport of motor proteins that fall into a specific category of the non-equilibrium system due to the presence of non-zero current governed by the continuous supply of energy. In recent decades, asymmetric simple exclusion process (TASEP), a Markov model which includes unidirectional particle hopping along a one-dimensional discrete lattice, has achieved the status of paradigm model to analyze stochastic non-equilibrium transport, including motor movement. The first part of the talk will be devoted to some beautiful results on the standard 1D TASEP model with fixed lattice size. Recent experimental observations suggest that the biological paths (microtubules) can polymerize or depolymerize under certain conditions, resulting in the growth or shrinkage of these paths. Modeling stochastic transport with dynamic microtubules will be explored in the second part of the talk, then validating theoretical results through experimental observations and extensively performed Monte Carlo simulations. Finally, challenges in modeling and analyzing intracellular transport will be discussed.

Biography:

I have completed BSc in Physics, Chemistry, Mathematics from CSJMU Kanpur (2012), followed by MSc in Industrial Mathematics & Informatics from IIT Roorkee (2014). I earned a Ph.D. on the topic "Mathematical Modelling of Driven Stochastic Transport Systems" from IIT Ropar (2019). After completing my Ph.D., I worked as Director's Fellow at IIT Ropar from Sep 2019 to May 2021. Since June 2021, I have been working as Assistant Professor at the Department of Mathematics, NIT Trichy, Tamil Nadu. I have published 9 research papers (all SCI) in international journals of repute like Physical Review E, Journal of Statistical Physics, Journal of Statistical Mechanics, etc.

Abstract:

During my teaching seminar, I will be discussing the basic theory of the system of homogeneous first order linear ordinary differential equations with constant coefficients.

Biography:

I have completed BSc in Physics, Chemistry, Mathematics from CSJMU Kanpur (2012), followed by MSc in Industrial Mathematics & Informatics from IIT Roorkee (2014). I earned a Ph.D. on the topic "Mathematical Modelling of Driven Stochastic Transport Systems" from IIT Ropar (2019). After completing my Ph.D., I worked as Director's Fellow at IIT Ropar from Sep 2019 to May 2021. Since June 2021, I have been working as Assistant Professor at the Department of Mathematics, NIT Trichy, Tamil Nadu. I have published 9 research papers (all SCI) in international journals of repute like Physical Review E, Journal of Statistical Physics, Journal of Statistical Mechanics, etc.

Abstract:

Determination of modular forms is one of the fundamental and interesting problems in number theory. It is known that if the Hecke eigenvalues of two newforms agree for all but finitely many primes, then both the forms are the same. In other words, the set of Hecke eigenvalues at primes determine the newform uniquely and this result is known as the multiplicity one theorem. In the case of Siegel cuspidal eigenform of degree two, the multiplicity one theorem has been proved only recently in 2018 by Schmidt. In this talk, after discussing a refinement of the multiplicity one theorem for newforms due to Rajan, we refine the result of Schmidt by showing that if the Hecke eigenvalues of two Siegel eigenforms agree at a set of primes of positive density, then they are the same (up to a constant). We also distinguish Siegel eigenforms from the signs of their Hecke eigenvalues. The main ingredient to prove these results are Galois representations attached to Siegel eigenforms, the Chebetarov density theorem and some analytic tools. 

Abstract:

In this talk, I will begin with a quick introduction to the PDE perspective of the Image Processing. Then, I will introduce two new image inpainting models using Modified Cahn-Hilliard equation and multi-well potential constructed from the histogram of a given image. The construction of the multi-well potential term which completely meets the challenge of capturing all the grey scale intensities of any image will be discussed. Later the existence- uniqueness of the proposed PDE model will be dealt with. Convexity Splitting method for time and Discrete Fourier spectral method in space has been used to discretize the PDE. Consistency, Stability, and convergence of the discretized models has been shown. Finally, we will present some results of our models and some applications will be shown.

Biography:

Dr. Abdul Halim is currently working as a Postdoc in the Department of Applied Mathematics, University of Twente, Netherland. Prior to that He was working as an Assistant Professor in Munger University, Bihar. He has completed his Ph.D. from the Department of Mathematics and Statistics, IIT Kanpur. His research area includes PDE based Image Processing, PDE eigenvalue Problems, Reduced Order Modeling.

Abstract:

In this talk, I will prove the existence and uniqueness theorem for first order initial value problem known as Picard’s Theorem using the Picard’s successive approximation on rectangle containing the initial point and discuss few examples.

Biography:

Dr. Abdul Halim is currently working as a Postdoc in the Department of Applied Mathematics, University of Twente, Netherland. Prior to that He was working as an Assistant Professor in Munger University, Bihar. He has completed his Ph.D. from the Department of Mathematics and Statistics, IIT Kanpur. His research area includes PDE based Image Processing, PDE eigenvalue Problems, Reduced Order Modeling.

Abstract:

Fundamental group of a topological space is an important topological invariant of a space. In algebraic geometry, there are many analogues of topological fundamental groups in the literature, which are generally group-schemes. We discuss the notions of étale fundamental group, Nori's fundamental group-scheme and the S-fundamental group-schemes of an algebraic variety, and relationships among them. Understanding these fundamental group-schemes for moduli spaces is an important problem to study. The Hilbert scheme of n points on a smooth projective surface X, denoted as $Hilb^n_X$ , is an important moduli space to study. In a recent joint work with Ronnie Sebastian, we established a relationship with the S-fundamental group-scheme and the Nori's fundamental group-scheme of $Hilb^n_X$ with that of X. After discussing this result, if time permits, we indicate some future research plans along this direction.

Abstract:

Flag varieties are natural generalizations of Grassmannians. There is a well-established connection between the geometry of flag varieties and representation theory of algebraic groups. Schubert varieties are subvarieties of a flag variety which parametrize families of linear subspaces of a vector space. In this talk, we consider the quotients of flag varieties and Schubert varieties for the action of a maximal torus. In the case of Grassmannian, we give a classification of smooth torus quotients of Schubert varieties. Finally, I will also give a few words about the Gromov-width of Bott-Samelson varieties, these are desingularizations of Schubert varieties.

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It is well known that the Orlicz spaces are natural generalizations of the Lebesgue spaces. We will discuss the Banach algebra structure of the Orlicz spaces associated to vector measures over compact groups. We will also discuss the Fourier transform associated to vector measures over compact groups.

Abstract:

Zadeh introduced the notion of fuzzy subsets. The notions introduced by Atanassov (1983) and Atanassov&Gargov (1989) in defining intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets are interesting and useful in modelling real-life problems. An intuitionistic fuzzy set plays a vital role in decision making, data analysis, and artificial intelligence. The information received from a source is represented by an information system involving quantitative, qualitative, and incomplete information. Such incomplete information is fed into the intelligent system for enhancing accuracy using Trapezoidal Intuitionistic Fuzzy Numbers (TrIFN). The ranking of fuzzy numbers plays an important role in Decisionmaking problems and many other applications. The ranking of Fuzzy Numbers have started in the early eighties in the last century, different researchers have proposed different methods for the Ranking of Fuzzy numbers. But none of them yields a total ordering on the class of fuzzy numbers and intuitionistic fuzzy numbers. The main aim of this lecture is to introduce a ranking (ordering) principle for comparing arbitrary intuitionistic fuzzy numbers (TrIFNs) and discuss its application in decision-making.

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In the present research work, we have analysed the movement of solute in a homogeneous aquifer. Firstly, the fractional-order (1+1)-dimensional solute transport model is considered which is simply an advection-diffusion equation having source/sink term with given initial condition and first type source boundary conditions. To describe the movement of solute concerning the column length, the considered problem is solved by using shifted Legendre collocation method. Secondly, the shifted Legendre collocation method is extended to obtain the solution of the (2+1)-dimensional fractional-order nonlinear advection-reaction-diffusion model. The main feature of the present contribution is the graphical exhibitions of the effects of advection term, reaction term, and fractional-order parameters on the solution profile. To authenticate the effectiveness of the method, a drive has been taken to compare the obtained results with the existing analytical results of the integer-order form of the considered model through error analysis. The striking feature of the work is the damping effect of the field variable on the solution profile when the system approaches fractional-order from the integer-order for specified values of the parameters of the system which greatly describes the physical phenomenon where the rate of transportation is much faster than the usual one.

Biography:

I, Anup Singh, Assistant Professor (Guest faculty) in the Department of Mathematics, Mahatma Gandhi Central University, Motihari, Bihar. I have completed my Ph.D. in the Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, under the supervision of Prof. Subir Das on June 08, 2020. The title of the thesis is "Study and analysis of advection-reaction-diffusion equations in porous media." I did B.Sc. (Hons.) and M.Sc. at Banaras Hindu University (BHU), Varanasi. I have published 06 research articles in the reputed SCI/SCIE journals.

Abstract:

Tensors (also known as multidimensional arrays or n-way arrays) have significantly impacted many areas of science and engineering in recent years. However, operations with tensors are still a major challenge and preoccupation for the scientific community. Therefore, it is appropriate to develop an infrastructure that supports reasoning about tensor computations. In this talk, we will discuss tensor singular value decomposition based on a new notion of tensor-tensor multiplication. Then I shall present tensor singular value decomposition applications to a color image compression and computation of the tensor Moore-Penrose inverse. Further, we will discuss a color image deblurring problem and the solution of the Poisson problem in a tensor-structure domain.

Abstract:

I will discuss about the nonlocal operators, in particular, the fractional Laplacian, and investigate the positivity properties of nonlocal Schr\"odinger type operators, driven by the fractional Laplacian by developing a criterion that links the positivity of the spectrum of such operators with the existence of certain positive supersolutions, thereby establishing necessary and sufficient conditions for the existence of a configuration of poles that ensures the positivity of the corresponding Schr\"odinger operator.

Biography:

I am currently visiting Prof. Mousomi Bhakta, IISER Pune since July, 2021. Prior to this, I was a postdoctoral fellow at Montan University, Leoben, Austria and Masaryk University, Brno, Czech Republic between 2018 and 2021. I defended my doctoral thesis on 18th July, 2018 on the title" Nonlocal elliptic equations: existence and multiplicity results" from IISER Pune under the supervision of Dr. Mousomi Bhakta.

Abstract:

In this talk, uniqueness and existence results for first-order ordinary differential equations will be studied. Several examples are given to illustrate the applicability of this result.

Biography:

I am currently visiting Prof. Mousomi Bhakta, IISER Pune since July, 2021. Prior to this, I was a postdoctoral fellow at Montan University, Leoben, Austria and Masaryk University, Brno, Czech Republic between 2018 and 2021. I defended my doctoral thesis on 18th July, 2018 on the title" Nonlocal elliptic equations: existence and multiplicity results" from IISER Pune under the supervision of Dr. Mousomi Bhakta.

Abstract:

Systems of linear algebraic equations arise in many different problems, such as traffic flow, electrical networks, balancing chemical equations, evaluating the integrals, etc. In this talk, we will discuss the solution of systems of linear algebraic equations; the existence and uniqueness of the solution. Then we will discuss LU and Cholesky factorizations for solving special types of linear systems.

Abstract:

In this talk, we begin by translating the problem of classification of Hopf algebras of a fixed dimension over an algebraically closed field into an algebro-geometric question. This naturally leads us to the classical invariant theory of the space of mixed tensors. We obtain a finite set of polynomial invariants for this space and use this to construct a set of invariants which separate closed orbits of the variety of Hopf algebras of a fixed dimension, under the action of a general linear group. We show that this gives a finite set of complete invariants for the isomorphism classes of Hopf algebras of 'small dimensions'.

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It is known that every irreducible subfactor $N\subset M$ with finite Jones index has only finitely many intermediate subfactors. We provide an explicit bound for the cardinality of this set. To achieve this, in a joint work with Das, Liu and Ren , we have introduced and investigated a notion of angle between a pair of intermediate subfactors. As a consequence, we answer a question of R. Longo published in 2003. More recently, in a joint work with Gupta we generalize the above result in the purely $C^*$-algebra setting exploiting a notion of generalized Pimsner-Popa basis.

Abstract:

Various porous medium flows like enhanced oil recovery, CO2 sequestration are of significant importance to various engineers, geologists and chemists. But a mathematical insight into these is equally important to obtain certain parameters before performing the experiments and to validate the experimental findings with mathematical theories. We discuss hydrodynamic instabilities ubiquitous in such flows through mathematical modelling and performing linear stability analysis as well as non-linear simulations to understand the dynamics.

Biography:

Dr. Vandita Sharma obtained her Ph.D from department of Mathematics, Indian Institute of Technology Ropar. She has expertise in mathematical modeling and scientific computing, computational fluid dynamics, hydrodynamic stability. In particular, she has worked on modeling, computation and stability analysis of reaction-diffusion systems. She is a Gold medallist in M.Sc Mathematics and has won various competitive awards like SIAM student travel award, ICTAM travel award and ICIAM financial support, to name a few, during her PhD.

Abstract:

We will present the results related to the eigenvalue statistics of random Schrodinger operators with various types of random potentials. Mostly, we will study the limiting behaviour of the sequence of point processes (or random measures) associated with the spectrum of finite volume restriction of the Schrodinger operators. We will also talk about the smoothness of the integrated density of states (IDS) and multiplicity of the singular spectrum for general Anderson type Hamiltonian.

Abstract:

We will discuss a system of linear equations Ax =b and try to obtain a solution when this system is actually inconsistent. We shall discuss a couple of ways for finding the solution, building upon the disadvantages of one method and proceeding to the next method.

Biography:

Dr. Vandita Sharma obtained her Ph.D from department of Mathematics, Indian Institute of Technology Ropar. She has expertise in mathematical modeling and scientific computing, computational fluid dynamics, hydrodynamic stability. In particular, she has worked on modeling, computation and stability analysis of reaction-diffusion systems. She is a Gold medallist in M.Sc Mathematics and has won various competitive awards like SIAM student travel award, ICTAM travel award and ICIAM financial support, to name a few, during her PhD.

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In 1970, Davenport and Schmidt studied a Diophantine property of pairs of real numbers; it concerned those pairs for which the classical Dirichlet theorem can be improved. They showed that the set of Dirichlet-improvable pairs, while small in the sense of having zero Lebesgue measure, has full Hausdorff dimension. We study a similar Dirichlet-improvable property, where the approximations are made using an arbitrary norm rather than the supremum norm, and show the same result. To this end this, we recast the Dirichlet-improvable property into a dynamical property of certain orbits in the space of unimodular lattices, and prove a Hajos-Minkowski type result in the geometry of numbers.

Biography:

I graduated from Brandeis University in 2020 with a thesis in the area of metric Diophantine approximation. This area of research comes under number theory and studies the approximation properties of real numbers by rational ones. I am also interested in the related areas of ergodic theory, Lie groups and geometry of numbers.

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Bordification of spaces obtained by adjoining a boundary at infinity has created a lot of interest in recent times. One such well known object is the Gromov boundary of hyperbolic metric spaces. In this talk, we will give some recent developments made in bordification of spaces.

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Let A and B be two elliptic curves defined over the field of rational numbers with equivalent residual Galois representations at an odd prime p. We compare the parity of p-Selmer rank and the root number of A and B over a number field K. As a consequence, in many cases, we show that the p parity conjecture over K holds for A if and only if it holds for B.

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Recent developments in the p-adic Langlands program allow us to revisit classical conjectures in Iwasawa Theory which help in understanding the arithmetic of Galois representations arising from elliptic curves, modular forms and automorphic forms. This talk will cover various aspects of my research domain. We will present results in Iwasawa Theory and p-adic Hodge Theory and combine them to frame the signed Iwasawa main conjectures for non-ordinary automorphic representations. Alongside we will also discuss Greenberg’s p-rationality conjecture and use it to construct Galois representations with big open image in reductive groups. Furthermore, we will discuss results in p-adic functional analysis and show the existence of rigid analytic vectors in the crystalline representations arising from the p-adic Langlands program. These works are a culmination of several projects with various collaborators.

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Moser-Trudinger inequality (1971) is connected to the embedding results of Sobolev spaces (W^{1,p}) for certain critical exponent (p). Classical Moser-Trudinger inequality was generalised to a singular form by Adimurthi-Sandeep in 2006. In this talk, I will discuss the issue of the existence of extremal functions for such inequality. This talk is based on a joint work with G.Csato and V. H. Nguyen.

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The approximate analytical solutions for the system of fractional order Van der Pol equations with independent initial profiles are investigated. The influence of two main physical parameters such as angular frequency (a) and the amplitude (ω) are included for the study. The effect of the physical parameters on phase portrait and the time-history curves for various values of fractional orders are plotted and discussed. It is found that the variations of in-phase and out-of phase periodic solutions and convergence rate strongly depend on the initial conditions. Based on the HAM method, the convergence rate, accuracy, and efficiency of the governing equations are demonstrated, which exhibit meaningful structures and advantages in science and engineering.

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The second order Partial differential equations (PDEs) are extremely important in both mathematics and physics. This talk provides a review on the nomenclature of the geometrical figures, if B2 − 4AC > 0 the partial differential equation is said to be hyperbolic; if B2 − 4AC = 0 the equation is parabolic, and if B2 − 4AC < 0 the equation is elliptic. Also, illustrate these on real life situational examples and its solution behaviour lucidly.

Biography:

I have studied all my education starting from 1st class to Ph. D in Government Schools/ Social Welfare Residential Hostels and Government Universities. I have completed my Ph.D in Mathematics from the Department of Mathematics, Central University of Pondicherry, Pondicherry India under the guidance of Prof. Rajeswari Seshadri.(Ph.D. from IISc Bangalore) in the areas of Solutions for nonlinear differential equations. Also, I have completed my one year Post Doctoral Fellow (PDF) research work in Applied Mathematics/Mechanics from School of Naval Architecture, Ocean and Civil Engineering(1st Rank in World by GLOBAL RANKING OF WORLD OF ACADEMIC SUBJECTS-2018), Shanghai Jiao Tong University, Shanghai in China under the guidance of senior Scientist and Distinguished Professor. Shi Jun Liao (HAM method Founder and 2016 Most Highly Cited Researchers in Global Mathematics).
I have "5 years" of teaching experience and “1 year” Post Doctoral fellow experience and Published 28(SCI/SCOPUS/UGC) international journal articles/conference proceedings and have communicated 4 more papers. I also presented 5 papers and gave 8 lectures at various conferences and seminars. I have participated in more than 74 Conferences / Seminars / Workshops and presented papers in some of them. I have received prestigious fellowship “Rajiv Gandhi National Fellowship(JRF & SRF) from UGC, New Delhi, Govt of India during my M.Phil and Ph.D. I have received renowned the SJTU Post Doctoral Fellowship from Shanghai Jiao Tong University, Shanghai, China in 2016 and Prof. Meenakshi Sundaram Memorial Best paper award in Applied Mathematics by APSMS in 2017. Also awarded QITCS Post Doctoral Scientist fellow from Shandong University, Govt of China-2020(Not availing due to Covid). Recently, I have received CONFIRMATION LETTER FOR HIGH LEVEL FOREIGN TALENTS from CHINA (i.e. prestigious “China Talent Visa(R)” i.e, Foreign High-Level Talents Fellow Researcher among all 100 International fellows from Government of China) for 2 years free VISA services for work as a Postdoctoral Scientist from 6th April, 2020 to 6th April, 2022 at Qingdao Institute for Theoretical and Computational Sciences, Shandong University, Qingdao, China under the supervision of Director, Prof. Wenjian Liu.

• Secured 1st rank in All India Entrance (PU CET) for M.Phill. in Mathematics at Pondichetty University ( A Central University), Pondicherry in 2009.
• Secured 1st rank, All India Selections for Ph.D. in Mathematics at University of Hyderabad in 2011.
• Secured 1st rank in All Andhra Pradesh state Entrance (KU CET) for M. Sc in Mathematics, Kakatiya University (State University), Warangal in 2005.

My research area is on application of approximate

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Dynamical systems is the branch of mathematics which investigates how a system evolves with time. In my talk I will discuss two broad sects of this subject One-Dimensional Dynamical Systems and Complex Dynamical Systems in the spirit of "Developing efficient machinery to make conclusions about the dynamics of a given map f from reduced information". In the former part, we will accomplish the above stated goal by studying the forcing relation among over-rotation numbers of points under a map f and concocting algorithms to compute out its over-rotation interval I(f) = [r(f), 1/2]. It is known that this single number r(f) epitomizes the limiting dynamical behaviour of points of the interval which in turn portrays the dynamical complexity of the map f. In the later discipline we will accomplish the stated goal by developing efficient topological models for the parameter space of complex polynomials. This will allow us to partition the whole parameter space into parts depending upon dynamics. Then if we just knew the location of a polynomial in the parameter space we can immediately unveil its dynamics.

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The Theory of Ordinary Differential Equations is playing an ever more important role in provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classicaltechniques of mathematics. Examining stability of solutions is one of the central problems in the Theory of Ordinary Differential Equations. In my talk I will present an introduction to this topic focussing on Systems of Linear Differential Equations. Important components of this disquisition entails : determining stable , unstable and centre subspaces of a given linear system , classification of equilibrium points of a linear system and use of Liapunov function in ascertaining stability of a linear system. We will elaborate upon the particular case of Linear systems in the plane.

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Individuals in an ecosystem may exhibit different behavioural strategies for their long term survival. Mathematical modelling of different ecological phenomenon by systems of non-linear ordinary differential equations with non-negative initial conditions can give us important insights of ecological dynamics. The systems may be autonomous or non-autonomous and depending upon different scenarios particular forms can be chosen. A linear stability analysis of an autonomous system tells us about the stability of the equilibrium points of the systems and provide understanding about the bifurcation scenarios. For a non-autonomous system that is when the species are seasonally forced (parameters of the systems are explicitly dependent on time), coincidence degree theorem is useful to check the existence of periodic solutions of the systems. Apart from deterministic system, the effects of environmental fluctuations on a system can be analysed by considering the stochastic set up. It is observed that in the study of erythropoiesis mathematically, the influence of stem cell dynamics on the subsequent process of erythrocyte production has been less explored in the recent studies. Combine study of these processes under the framework of multi-scale modelling may provide important insights of the collective cell dynamics.

Biography:

Jyotirmoy Roy had completed his doctoral degree form IIEST, Shibpur in the year 2021. His research interests include mathematical modelling and numerical analysis of different ecological, eco-epidemiological and biological phenomenon. He had published ten articles in different journals and conferences.

Abstract:

Continued and remarkable empirical successes of increasingly complicated machine learning models such as neural networks without a sound theoretical understanding of success and failure conditions can leave a practitioner blind-sided and vulnerable, especially in critical applications such as self-driving cars and medical diagnosis. As such, there has been an enhanced interest in recent times in research on building interpretable models as well as interpreting model predictions. In this talk, I will discuss various theoretical and practical aspects of interpretability in machine learning along both these directions through the lenses of feature attribution and example-based learning. In the first part of the talk, I will present novel theoretical results to bridge the gap in theory and practice for interpretable dimensionality reduction aka feature selection. Specifically, I will show that feature selection satisfies a weaker form of submodularity. Because of this connection, for any function, one can provide constant factor approximation guarantees that are solely dependent on the condition number of the function. Moreover, I will discuss that the cost of interpretability accrued because of selecting features as opposed to principal components is not as high as was previously thought to be.
In the second part of the talk, I will discuss the development of a probabilistic framework for example-based machine learning to address ``which training data points are responsible for making given test predictions?“. This framework generalizes the classical influence functions. I will also present an application of this framework to understanding the transfer of adversarially trained neural network models.

Short Biography:

Dr. Rajiv Khanna is currently a Visiting Faculty Researcher at Google Research, and an incoming Assistant Professor at the Department of Computer Science at Purdue University. Previously, he was a postdoc at the Department of Statistics at UC Berkeley and was also associated with the Foundations of Data Analytics Institute (FODA) at UC Berkeley. Before that, he was a Research Fellow in the program of Foundations of Data Science at the Simons Institute for the Theory of Computing, also at UC Berkeley.
His research is focussed on elucidating mechanisms of success/failure conditions of machine learning through optimization, learning theory and interpretability. His work on beyond worst-case analysis on the Column Subset Selection won the best paper award at NeurIPS 2020. He earned his PhD in Electrical and Computer Engineering at UT Austin.

Abstract:

In this talk, after briefly describing my research areas, I focus on the topic of importance sampling (IS). The standard IS method uses samples from a single proposal distribution and assigns weights to them, according to the ratio of the target and proposal pdfs. This naive IS estimator, generally does not work well in multiple target examples as the weights can take arbitrarily large values making the estimator highly unstable. In such situations, alternative generalized IS estimators involving samples from multiple proposal distributions are preferred. Just like the standard IS, the success of these multiple IS estimators crucially depends on the choice of the proposal distributions. For selecting these proposal distributions, we propose three methods based on a geometric space filling coverage criterion, a minimax variance approach, and a maximum entropy approach, respectively. The proposed methods for selecting proposal densities are illustrated using some examples.

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Maximum likelihood estimation is probably the most common method of estimation in use in frequentist statistics. After discussing the basic idea of the maximum likelihood estimator (MLE), we will define MLE and go over several examples. Finally, we will discuss some nice properties of the MLE.

Short Biography:

Dr. Roy is an Associate Professor in the Department of Statistics at Iowa State University. Currently, he is serving two statistics journals - Journal of Computational and Graphical Statistics and Sankhya, Series B as an Associate Editor. His research interests are Markov chain Monte Carlo, importance sampling, high dimensional data analysis, model selection and Bayes and empirical Bayes methods.

Abstract:

D & C paradigm is commonly used in designing efficient algorithms. Here, the problem is divided into subproblems, each subproblem is solved recursively, and the solutions to the subproblems are combined to get the overall solution. In this talk, I will discuss this design strategy with the example of efficiently finding the closest pair of points. Here, given n points in a plane, the goal is to find a pair of points that have the smallest Euclidean distance between them. I will show that the brute force approach here leads to a time complexity of O(n^2), while a D & C strategy leads to time complexity of O(n log^2 n). This complexity can be reduced further. Depending upon the availability of time, I would also summarize the use of D & C strategy for the problem of efficiently multiplying two matrices.

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Algorithmic scalability is a challenge in many complex systems and machine learning applications. Inexact computing is one solution to this scalability challenge. With the focus on the problem of efficiently reducing large-scale dynamical systems, we propose the use of approximate (and hence, faster) linear solves in a set of model reducing algorithms. Theoretically, we prove that under mild conditions (easily satisfiable), these algorithms are backward stable with respect to the error introduced by inexact linear solves. Experimentally, we demonstrate that while reducing an industrial disk-brake model of size 1.2 million, one of our approximate algorithms leads to relative savings of up to 64% in the total computation time (when compared with the corresponding exact algorithm). In absolute terms, this leads to a saving of 5 days.
Another solution to this scalability challenge is sampling, which leads to a reduction in the problem size. With the focus on the problem of efficiently grouping plant species, we propose a class of probabilistically-sampled spectrally-clustered algorithms. Using experiments on multiple Soybean datasets comprising of thousands of species, we demonstrate the superiority of our approach over the current standard in-terms of both the cluster quality (up to 45% better) as well as the time complexity (an order-of-magnitude lesser).

Short Biography:

With Master’s and Ph.D. degrees in Mathematics and Computer Science (from Virginia Tech), Dr. Kapil Ahuja has a strong interdisciplinary focus. After graduating from VT, he received his postdoctoral training from the Max Planck Institute in Magdeburg. Since then, he has established his independent research program in Mathematics of Data Science and Simulation (MODSS) at IIT Indore, where he is currently working as an Associate Professor. In the recent past, he has also held visiting professor positions at TU Braunschweig, TU Dresden, and Sandia National Labs.
Dr. Ahuja’s core research interests are in Machine Learning, Optimization, Game Theory, and Numerical Methods. Dr. Ahuja's research output includes external funding worth more than half-a-million USD and over forty high-quality publications. While achieving this, he has graduated 4 PhD students with 1 more to graduate soon. On the teaching end, Dr. Ahuja has received the best teacher award four times at IIT Indore, and administratively, recently, he headed International Affairs at IIT Indore as its founding Dean.

Abstract:

Finitely many agents have preferences on a finite set of alternatives, singlepeaked with respect to a connected graph with these alternatives as vertices. A probabilistic rule assigns to each preference profile a probability distribution over the alternatives. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles at which all peaks are on leaves of the tree and, thus, extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leaves. Finally, the two results are combined to obtain a general characterization for every connected graph by using its block tree representation.

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In this talk, I will define real-valued random variables on a probability space and talk about the probability distribution and distribution function of such a random variable. Then I will go through the details of some well-known random variables such as Binomial, Poisson, Normal, etc., and their different moments. At the end of this talk, I will touch on some inequalities and various modes of convergence.

Short Biography:

Dr. Soumyarup Sahdukhan has obtained his Ph.D. in Economics (28th August 2020) from the Economic Research Unit of ISI Kolkata. During his Ph.D., he has worked in the areas of mechanism design, random social choice theory, and division problem, and has published in top journals such as Mathematics of Operations Research, Journal of Economic Theory, Economic Theory, and Journal of Mathematical Economics. He did his B.Sc. in Statistics from Narendrapur Ramakrishna Mission (with 85% marks) and M.Stat from ISI (with distinction). Currently, he is working as a CV Raman post-doctoral fellow in the Department of Computer Science and Automation of IISc Bangalore. Previously, he has worked as a visiting scientist in the Applied Statistics Unit of ISI Kolkata under the supervision of Professor Mridul Nandi.

Abstract:

In this talk we examine the relationship between maxi–min, Bayes and Nash designs for some hypothesis testing problems. In particular we consider the problem of sample allocation in the standard analysis of variance framework and show that the maxi–min design is also a Bayes solution with respect to the least favourable prior, as well as a solution to a game theoretic problem, which we refer to as a Nash design. In addition, an extension to tests for order is provided.

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In this talk a simple linear regression model is considered. An ordinary least square technique is carried out for the estimation of the model parameters. Elementary properties of the estimators are discussed. To check the significance of the parameters, relevant hypotheses and associated statistical tests are formulated. A data example is given to illustrate the methodology.

Short Biography:

Dr. Satya Prakash Singh is currently working as an Assistant Professor at the Department of Mathematics, IIT Hyderabad. He did M.Sc. (Mathematics) from IIT Delhi and M.Sc. (Statistics) from IIT Kanpur. He obtained his Ph.D. from IIT Bombay. He did his Postdoc at the University of Haifa, Israel. His research interests are optimal experiments for cluster and crossover studies and designing optimal experiments under order restrictions.

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Short Biography:

Dr. Neeraj Bhauryal is currently a postdoctoral fellow at TIFR-CAM, Bangalore. He obtained his Masters and PhD in Mathematics from TIFR-CAM Bangalore. His current research interests include Nonlinear Partial Differential Equations, Conservation Laws and Stochastic Analysis.

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Short Biography:

Dr. Paramita Pramanick is currently working as a Research Associate in the Department of Mathematics, Indian Institute of Science, Bangalore. Dr. Pramanick completed her B.Sc. and M.Sc. from Jadavpur University, and her Ph.D. from Indian Institute of Science, Bangalore in 2021. Her research focuses on Functional Analysis, Operator Theory: more speci fically, operator theory on Hilbert space of analytic functions, reproducing kernel Hilbert modules, homogeneous operators, subnormal operators, hyponormal operators and trace inequalities.

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Hierarchical Bayesian modeling of high-dimensional datasets is a very active area of research in geostatistics and spatial extremes. In this work, we estimate extreme sea surface temperature (SST) hotspots, i.e., high threshold exceedance regions, for the Red Sea, a vital region of high biodiversity. We analyze high-resolution satellite-derived SST data comprising daily measurements at 16703 grid cells across the Red Sea over the period 1985–2015. We propose a semiparametric Bayesian spatial mixed-effects linear model with a flexible mean structure to capture spatially-varying trend and seasonality, while the residual spatial variability is modeled through a Dirichlet process mixture (DPM) of low-rank spatial Student’s t-processes (LTPs). By specifying cluster-specific parameters for each LTP mixture component, the bulk of the SST residuals influence tail inference and hotspot estimation only moderately. Our proposed model has a nonstationary mean, covariance, and tail dependence, and posterior inference can be drawn efficiently through Gibbs sampling. In our application, we show that the proposed method outperforms some natural parametric and semiparametric alternatives. Moreover, we show how hotspots can be identified and we estimate extreme SST hotspots for the whole Red Sea, projected until the year 2100, based on the Representative Concentration Pathway 4.5 and 8.5. The estimated 95% credible region for joint high threshold exceedances includes large areas covering major coral reefs in the southern Red Sea. Some future research directions include parametric and nonparametric Bayesian approaches for spatial geostatistics and extremes, Bayesian hotspot estimation, and extreme event attribution.

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The last three decades have seen an explosion of interest in spatial and spatiotemporal problems. The increased availability of inexpensive and high-speed computing facilities worked as an indispensable tool for fitting realistic spatial statistical models and drawing inferences, and currently, the applications of spatial statistics span a large number of scientific disciplines. Spatial data are generally divided into three categories: point-referenced data, areal unit data, and spatial point patterns. We discuss examples of different types of spatial data and possible scientific questions. Gaussian processes are at the heart of this area of statistics, and hence we review some necessary multivariate statistical inference techniques. We discuss a brief overview of linear models, which provides some necessary insights about spatial mean modeling. The variogram is a common exploratory technique for understanding spatial dependence, and we cover the main concepts. Some spatial covariance models and some simple examples of kriging are discussed. Finally, we also cover some areal data modeling approaches. All the concepts are illustrated with R programming and real data examples.

Short Biography:

Arnab Hazra is currently a postdoctoral fellow of statistics at the CEMSE Division at King Abdullah University of Science and Technology, Saudi Arabia. He obtained his Bachelor of Statistics and Master of Statistics degrees from the Indian Statistical Institute, Kolkata, and his Ph.D. in Statistics from North Carolina State University, Raleigh, US, in 2018. His current research interests include spatial statistics, extreme value analysis, heavy-tailed processes, high-dimensional data analysis, hierarchical Bayesian modeling, and environmental statistics.

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Short Biography:

Dr. Kapil Kant completed his B.Sc. in Mathematics from University of Allahabad in 2013. After that, he obtained his M.Sc. and Ph.D. in Mathematics from Indian Institute of Technology, Kharagpur in 2015 and 2020, respectively. His broad area of research is numerical analysis and scientific computing.

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Short Biography:

Subhabrata (Subho) Majumdar is a Senior Inventive Scientist in the Data Science and AI Research group of AT&T Labs. His research interests have two focus areas - (1) statistical machine learning - specifically predictive modelling and complex high-dimensional inference, and (2) trustworthy machine learning methods, with emphasis on human-centric qualities such as social good, robustness, fairness, privacy protection, and causality.
Subho has a PhD in Statistics from the School of Statistics, University of Minnesota under the guidance of Prof. Anshu Chatterjee. His thesis was on developing inferential methods based on statistical depth functions, focusing on robust dimension reduction and variable selection. Before joining AT&T, Subho was a postdoctoral researcher at the University of Florida Informatics Institute under Prof. George Michailidis. Subho has extensive experience in applied statistical research, with past and present collaborations spanning diverse areas like statistical chemistry, public health, behavioral genetics, and climate science. He recently co-founded the Trustworthy ML Initiative (TrustML), to bring together the community of researchers and practitioners working in that field, and lower barriers to entry for newcomers. Link to his webpage: https://shubhobm.github.io/.

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Short Biography:

Prof. Juan A. Cuesta-Albertos (Segovia, Spain, 1955) has a degree in Mathematics, specializing in Statistics and Operations Research, from the University of Valladolid, Spain. He completed his doctoral thesis at the same university, in Mathematical Sciences, under the direction of Prof. Dr. D. Miguel Martín Díaz. He taught at the Colegio Universitario de Burgos until the 1981/82 academic year, and from that moment until the present day, at the University of Cantabria, where he is a Full Professor of Statistics.
He is the author of about 85 research papers on topics such as the laws of large numbers and centralization measures in abstract spaces, bootstrap, robustness, functional data analysis, etc. Apart from continuing his research work, he has recently been paying attention to the dissemination of statistics. He was the Chief Editor of the journal TEST from 2002 to 2004, and is currently an Associate Editor of JASA. Please find more details in this link: https://personales.unican.es/cuestaj/index.html.

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Short Biography:

Dr. Mallesham K completed his MSc in Mathematics from IIT Madras in 2011, and joined Harish-Chandra Research Institute (HRI), Allahabad, as PhD student. He received his PhD degree from HRI in September, 2018. Currently, he is a visiting scientist in the Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata. His research area of interest lies in Number theory, especially in Analytic number theory.

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Short Biography:

Dr. Ramiz Reza completed his B.Sc. from Ramakrishna Mission Vidyamandira, Belur (University of Calcutta) in 2009. After that, he obtained his Integrated Ph.D. degree (both M.S. and Ph.D.) from Indian Institute of Science, Bangalore in 2017. He started his post-doctoral research as a SERB-NPDF fellow in IISER Kolkata in September 2017. In February 2018, he moved to Chalmers University of Technology, Gothenburg, Sweden to continue his post-doctoral studies as a SERB-OPDF fellow.
Currently, he is an institute postdoctoral fellow at Indian Institute of Technology, Kanpur (since May 2019).

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Short Biography:

Dr. Kumar completed his graduation in Mathematics from C.C.S. University Meerut in 2012, his post-graduation in Mathematics from IIT Guwahati in 2014, and Ph.D. in Mathematics from NISER, Bhubaneswar under the supervision of Dr. Anil Kumar Karn in 2020. He is currently working as an adhoc faculty in School of Sciences (Mathematics), NIT, Andhra Pradesh.

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Short Biography:

Dr. Panda is currently a Post Doctoral Fellow at the School of Mathematical Sciences, NISER, Bhubaneswar. He obtained his B.Sc. from Gangadhar Meher Autonomous College, Sambalpur; M.Sc. from IIT Kanpur and Ph.D. from IIT Guwahati (all in Mathematics). His main research interest is algebraic combinatorics.

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Short Biography:

Dr. Adhikari is currently a Viterbi and Zeff Post Doctoral Fellow at the Department of Electrical Engineering, Technion, Israel. Prior to that, he was a Post Doctoral Fellow at Indian Statistical Institute, Kolkata. He obtained both degrees M.S. and Ph.D. (in Mathematics) from Indian Institute of Science, Bangalore. He completed his Bachelor degree (Mathematics honours) from Ramakrishna Mission Residential College, Narendrapur (under Calcutta University).
His research interests mainly lie in probability theory and analysis, specifically, random matrix, determinantal point processes, large deviation, Steins method for normal approximation, and potential theory, free probability, stochastic geometry and random graphs.

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such as medical imaging and geophysical imaging. First half of the talk will be a brief overview of our work related to introduced transforms and the second half will be focused on a recent work on V-line tomography for vector fields in

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Short Biography:

Dr. Adhikari is currently a Viterbi and Zeff Post Doctoral Fellow at the Department of Electrical Engineering, Technion, Israel. Prior to that, he was a Post Doctoral Fellow at Indian Statistical Institute, Kolkata. He obtained both degrees M.S. and Ph.D. (in Mathematics) from Indian Institute of Science, Bangalore. He completed his Bachelor degree (Mathematics honours) from Ramakrishna Mission Residential College, Narendrapur (under Calcutta University). His research interests mainly lie in probability theory and analysis, specifically, random matrix, determinantal point processes, large deviation, Steins method for normal approximation, and potential theory, free probability, stochastic geometry and random graphs.

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The aim of this talk is to prove a very important result of analysis named the Fundamental Theorem of Calculus (FTC) which basically relates the concepts of integration and differentiation. To achieve this

Short Biography:

Dr. Mishra received his PhD in 2017 under the guidance of Dr. Venky P. Krishnan from TIFR Centre for Applicable Mathematics, Bangalore, India. His primary research interests are in the field of inverse problems related to integral geometry, partial differential equations, microlocal analysis and medical imaging. Currently, he is a postdoctoral scholar in the Department of Mathematics at the University of Texas at Arlington (UTA), Texas. Before joining UTA, he was a postdoctoral scholar at the University of California, Santa Cruz from August 2017 to July 2019.

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We have established limit theorems for the sums of dependent Bernoulli random variables. Here each successive random variable depends on previous few random variables. A previous k-sum dependent model is considered. This model is a combination of the previous all sum and the previous k-sum dependent models. The law of large numbers, the central limit theorem and the law of iterated logarithm for the sums of random variables following this model are established. A new approach using martingale differences is developed to prove these results.

Short Biography:

Currently, Dr. Singh is working at Redpine Signals India as a Research Engineer, where he is exploring the mathematical concept of Quantum Machine Learning Algorithms. He has completed his Ph.D. in Probability Theory under the supervision of Prof. Somesh Kumar in June, 2020. During his Ph.D., he also served as a Teaching Assistant for NPTEL courses for three years.

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We have established limit theorems for the sums of dependent Bernoulli random variables. Here each successive random variable depends on previous few random variables. A previous k-sum dependent model is considered. This model is a combination of the previous all sum and the previous k-sum dependent models. The law of large numbers, the central limit theorem and the law of iterated logarithm for the sums of random variables following this model are established. A new approach using martingale differences is developed to prove these results.

Short Biography:

Currently, Dr. Singh is working at Redpine Signals India as a Research Engineer, where he is exploring the mathematical concept of Quantum Machine Learning Algorithms. He has completed his Ph.D. in Probability Theory under the supervision of Prof. Somesh Kumar in June, 2020. During his Ph.D., he also served as a Teaching Assistant for NPTEL courses for three years.

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The aim of this talk is to prove a very important result of analysis named the Fundamental Theorem of Calculus (FTC) which basically relates the concepts of integration and differentiation. To achieve this

Short Biography:

Dr. Mishra received his PhD in 2017 under the guidance of Dr. Venky P. Krishnan from TIFR Centre for Applicable Mathematics, Bangalore, India. His primary research interests are in the field of inverse problems related to integral geometry, partial differential equations, microlocal analysis and medical imaging. Currently, he is a postdoctoral scholar in the Department of Mathematics at the University of Texas at Arlington (UTA), Texas. Before joining UTA, he was a postdoctoral scholar at the University of California, Santa Cruz from August 2017 to July 2019.

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Short Biography:

Dr. Adhikari is currently a Viterbi and Zeff Post Doctoral Fellow at the Department of Electrical Engineering, Technion, Israel. Prior to that, he was a Post Doctoral Fellow at Indian Statistical Institute, Kolkata. He obtained both degrees M.S. and Ph.D. (in Mathematics) from Indian Institute of Science, Bangalore. He completed his Bachelor degree (Mathematics honours) from Ramakrishna Mission Residential College, Narendrapur (under Calcutta University).
His research interests mainly lie in probability theory and analysis, specifically, random matrix, determinantal point processes, large deviation, Steins method for normal approximation, and potential theory, free probability, stochastic geometry and random graphs.

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such as medical imaging and geophysical imaging. First half of the talk will be a brief overview of our work related to introduced transforms and the second half will be focused on a recent work on V-line tomography for vector fields in

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Short Biography:

Dr. Adhikari is currently a Viterbi and Zeff Post Doctoral Fellow at the Department of Electrical Engineering, Technion, Israel. Prior to that, he was a Post Doctoral Fellow at Indian Statistical Institute, Kolkata. He obtained both degrees M.S. and Ph.D. (in Mathematics) from Indian Institute of Science, Bangalore. He completed his Bachelor degree (Mathematics honours) from Ramakrishna Mission Residential College, Narendrapur (under Calcutta University).
His research interests mainly lie in probability theory and analysis, specifically, random matrix, determinantal point processes, large deviation, Steins method for normal approximation, and potential theory, free probability, stochastic geometry and random graphs.

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Short Biography:

Dr. Panda is currently a Post Doctoral Fellow at the School of Mathematical Sciences, NISER, Bhubaneswar. He obtained his B.Sc. from Gangadhar Meher Autonomous College, Sambalpur; M.Sc. from IIT Kanpur and Ph.D. from IIT Guwahati (all in Mathematics). His main research interest is algebraic combinatorics.

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Short Biography:

Dr. Kumar completed his graduation in Mathematics from C.C.S. University Meerut in 2012, his post-graduation in Mathematics from IIT Guwahati in 2014, and Ph.D. in Mathematics from NISER, Bhubaneswar under the supervision of Dr. Anil Kumar Karn in 2020. He is currently working as an adhoc faculty in School of Sciences (Mathematics), NIT, Andhra Pradesh.

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Short Biography:

Dr. Ramiz Reza completed his B.Sc. from Ramakrishna Mission Vidyamandira, Belur (University of Calcutta) in 2009. After that, he obtained his Integrated Ph.D. degree (both M.S. and Ph.D.) from Indian Institute of Science, Bangalore in 2017. He started his post-doctoral research as a SERB-NPDF fellow in IISER Kolkata in September 2017. In February 2018, he moved to Chalmers University of Technology, Gothenburg, Sweden to continue his post-doctoral studies as a SERB-OPDF fellow.
Currently, he is an institute postdoctoral fellow at Indian Institute of Technology, Kanpur (since May 2019).

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Short Biography:

Dr. Mallesham K completed his MSc in Mathematics from IIT Madras in 2011, and joined Harish-Chandra Research Institute (HRI), Allahabad, as PhD student. He received his PhD degree from HRI in September, 2018. Currently, he is a visiting scientist in the Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata. His research area of interest lies in Number theory, especially in Analytic number theory.

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Short Biography:

Prof. Juan A. Cuesta-Albertos (Segovia, Spain, 1955) has a degree in Mathematics, specializing in Statistics and Operations Research, from the University of Valladolid, Spain. He completed his doctoral thesis at the same university, in Mathematical Sciences, under the direction of Prof. Dr. D. Miguel Martín Díaz. He taught at the Colegio Universitario de Burgos until the 1981/82 academic year, and from that moment until the present day, at the University of Cantabria, where he is a Full Professor of Statistics.
He is the author of about 85 research papers on topics such as the laws of large numbers and centralization measures in abstract spaces, bootstrap, robustness, functional data analysis, etc. Apart from continuing his research work, he has recently been paying attention to the dissemination of statistics. He was the Chief Editor of the journal TEST from 2002 to 2004, and is currently an Associate Editor of JASA. Please find more details in this link: https://personales.unican.es/cuestaj/index.html.

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Short Biography:

Subhabrata (Subho) Majumdar is a Senior Inventive Scientist in the Data Science and AI Research group of AT&T Labs. His research interests have two focus areas - (1) statistical machine learning - specifically predictive modelling and complex high-dimensional inference, and (2) trustworthy machine learning methods, with emphasis on human-centric qualities such as social good, robustness, fairness, privacy protection, and causality.
Subho has a PhD in Statistics from the School of Statistics, University of Minnesota under the guidance of Prof. Anshu Chatterjee. His thesis was on developing inferential methods based on statistical depth functions, focusing on robust dimension reduction and variable selection. Before joining AT&T, Subho was a postdoctoral researcher at the University of Florida Informatics Institute under Prof. George Michailidis. Subho has extensive experience in applied statistical research, with past and present collaborations spanning diverse areas like statistical chemistry, public health, behavioral genetics, and climate science. He recently co-founded the Trustworthy ML Initiative (TrustML), to bring together the community of researchers and practitioners working in that field, and lower barriers to entry for newcomers. Link to his webpage: https://shubhobm.github.io/.

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Applications of Singular perturbation and its related boundary layer phenomena are very common in today's literature. Presence of a small parameter in the differential equation changes the behavior of the solution rapidly. Uniform meshes are inadequate for the convergence of numerical solution. The aim of the present talk is to consider the adaptive mesh generation for singularly perturbed differential equations based on moving mesh strategy. I shall start this talk with a small introduction on singular perturbation. The analytical and computational difficulties on the existed methods will be discussed. The concept of moving mesh strategy will be explained. For a posteriori based convergence analysis, a system of nonlinear singularly perturbed problems will be considered. In addition, the difference between a priori and a posteriori generated meshes and the effectivity of a posteriori mesh on the present research will be discussed. The parameter independent a priori based convergence analysis for a parabolic convection diffusion problem will be presented. In addition, an approach.

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In this talk, definitions of different types of cross-sectional dependence and the relations between them would be discussed. We then look into parameter estimation and their asymptotic properties.

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I will discuss on the two sections of Numerical Analysis- numerical root finding methods and interpolation techniques. I will start with the introduction on these two topics and the restriction of the popular methods. I will also discuss about interpolation and their applications.

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In this series of two talks, we will first define long memory processes and discuss practical and theoretical examples of such processes. In addition we shall discuss the uniform reduction principle for such processes and some of its implications. This principle says that for long memory moving average processes the suitably standardized empirical process converges weakly to a degenerate Gaussian process. This is completely unlike what happens in the independent or weakly dependent case, where the suitably standardized empirical process converges weakly to Brownian bridge. In the second talk we shall discuss the problem of fitting a known d.f. or density

to the marginal error distribution of a stationary long memory moving-average process when its mean is known and unknown. When the mean is unknown and estimated by the sample mean, the first-order difference between the residual empirical and null distribution functions is asymptotically degenerate at zero. Hence, it cannot be used to fit a distribution up to an unknown mean. We show that by using a suitable class of estimators of the mean, this first order degeneracy does not occur. We also present some large sample properties of the tests based on an integrated squared-difference between kernel-type error density estimators and the expected value of the error density estimator. The asymptotic null distributions of suitably standardized test statistics are shown to be chi-square with one degree of freedom in both cases of known and unknown mean. This is totally unlike the i.i.d. errors set-up where such statistics are known to be asymptotically normally distributed.

An interested person may find the following two references helpful.

Giraitis, L., Koul, H.L. and Surgailis, D. (2012). Large sample inference for long memory processes. Imperial College Press and World Scientific.

Koul, H.L., Mimoto, N. and Surgailis, D. (2013). Goodness-of-fit tests for long memory moving average marginal density. Metrika, 76, 205-224.

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In this series of two talks, we will first define long memory processes and discuss practical and theoretical examples of such processes. In addition we shall discuss the uniform reduction principle for such processes and some of its implications. This principle says that for long memory moving average processes the suitably standardized empirical process converges weakly to a degenerate Gaussian process. This is completely unlike what happens in the independent or weakly dependent case, where the suitably standardized empirical process converges weakly to Brownian bridge. In the second talk we shall discuss the problem of fitting a known d.f. or density

to the marginal error distribution of a stationary long memory moving-average process when its mean is known and unknown. When the mean is unknown and estimated by the sample mean, the first-order difference between the residual empirical and null distribution functions is asymptotically degenerate at zero. Hence, it cannot be used to fit a distribution up to an unknown mean. We show that by using a suitable class of estimators of the mean, this first order degeneracy does not occur. We also present some large sample properties of the tests based on an integrated squared-difference between kernel-type error density estimators and the expected value of the error density estimator. The asymptotic null distributions of suitably standardized test statistics are shown to be chi-square with one degree of freedom in both cases of known and unknown mean. This is totally unlike the i.i.d. errors set-up where such statistics are known to be asymptotically normally distributed.

An interested person may find the following two references helpful.

Giraitis, L., Koul, H.L. and Surgailis, D. (2012). Large sample inference for long memory processes. Imperial College Press and World Scientific.

Koul, H.L., Mimoto, N. and Surgailis, D. (2013). Goodness-of-fit tests for long memory moving average marginal density. Metrika, 76, 205-224.

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Lie algebras are infinitesimal counterpart of Lie groups. In fact, there is a one-to-one correspondence between simply-connected Lie groups and Lie algebras (Lie III theorem).

In this talk, we will introduce 'multiplicative Poisson structures' on Lie groups (called Poisson-Lie groups) and 'Lie bialgebra' structures on Lie algebras. Finally, we show that there is a one-to-one correspondence between Poisson-Lie groups and Lie bialgebras. If time permits, I will mention some generalizations of the above correspondence.

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While studying the dynamics of polynomial automorphisms in C2, it turns out that the class of Hénon maps, which exhibits extremely rich dynamical behaviour, is the single most important class to study. An extensive research has been done in this direction by many authors over the past thirty years. In this talk, we shall see a `rigidity' property of Hénon maps which essentially replicates a classical `rigidity' theorem of Julia sets of polynomial maps in the complex plane. In particular, we shall give an explicit description of the automorphisms in C2 which preserve the Julia sets of a given Hénon map. If time permits, we shall see a few more results regarding `rigidity' property of some special classes of automorphisms in higher dimension.

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Natural language processing (NLP) has become a hot topic of research since the big five (GAFAM) are investing heavily into the development of this field. NLP tasks are wide-ranging, e.g. machine translation, auto completion, spam detection, argument mining, text classification, sentiment analysis, chat bots and named entity recognition. In any case, the text has to be converted somehow into numbers to use the text e.g. in classification problems.

The talk will be split into two parts. In the first part, the problem is introduced and the necessary preprocessing steps as well as simple methods for the representation of documents are described. These methods allow, e.g., to include text documents as features in modern machine learning methods used for classification and regression (supervised learning). The second part introduces the so-called neural language models which are unsupervised methods to “learn” a language. This results in embedding words and documents using so-called word or document vectors (e.g. word2vec). A disadvantage is that these vectors are not context sensitive. Therefore, we also give an outlook to the newest, usually very large, language models (based on deep learning methods) which are often freely available and can therefore be used by everybody for own purposes.

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Current tools for multivariate density estimation struggle when the density is concentrated near a nonlinear subspace or manifold. Most approaches require choice of a kernel,with the multivariate Gaussian by far the most commonly used. Although heavy-tailed and skewed extensions have been proposed, such kernels cannot capture curvature in the support of the data. This leads to poor performance unless the sample size is very large relative to the dimension of the data. This article proposes a novel generalization of the Gaussian distribution, which includes an additional curvature parameter. We refer to the proposed class as Fisher-Gaussian (FG) kernels, since they arise by sampling from a von Mises-Fisher density on the sphere and adding Gaussian noise. The FG density has an analytic form,and is amenable to straightforward implementation within Bayesian mixture models using Markov chain Monte Carlo. We provide theory on large support, and illustrate gains relative to competitors in simulated and real data applications.

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T11 Target structure (T11TS), a membrane glycoprotein isolated from sheep erythrocytes, reverses the immune suppressed state of brain tumor induced animals by boosting the functional status of the immune cells. This study aims at aiding in the design of more efficacious brain tumor therapies with T11 target structure. In my talk, I will discuss about the dynamics of brain-tumor immune interaction though a system of coupled non-linear ordinary differential equations. The system undergoes sensitivity analysis to identify the most sensitive parameters. In the model analysis, I obtained the criteria for the threshold level of T11TS for which the system will be tumor-free. Computer simulations were used for model verification and validation.

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Computationally determinant and permanent of matrices are the two extreme problems. The determinant is in P whereas the permanent is a #P- complete problem, which means that the permanent cannot be computed in polynomial time unless P = NP. In this talk, we will discuss how the determinant (permanent) of a matrix can be computed in terms of the determinant (permanent) of blocks in the corresponding digraph. Under some conditions on the number of cut-vertices and block sizes the computation beats the asymptotic complexities of the state of art methods. Next, as an application of the computation of determinant using blocks, we will discuss a characterization of nonsingular block graphs which was an open problem, proposed in 2013 by Bapat and Roy.

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I will discuss a discrete scattering problem which can be reduced to a 2x2 matrix factorization of Wiener-Hopf on an annulus (including the unit circle) in complex plane. An advancement made last year will be also presented that effectively solves this factorization problem but in an intricate manner. The particular factorization problem is still open from point of view of explicit factors and corresponds to an analogous problem on an infinite strip (including the real line) in complex plane (also open since several decades).

Short Biography:

Dr. Arkaprava Roy is currently a postdoctoral associate at Duke University, currently working with Dr. David Dunson. I have completed my Ph.D. in statistics in April 2018 from North Carolina State University (NCSU), under the supervision of Dr. Subhashis Ghosal and Dr. Ana-Maria Staicu after a Bachelor and Masters degree in statistics from Indian Statistical Institute, Kolkata. My focus is on data science and in developing innovative statistical modeling frameworks and corresponding inference methodology motivated by complex applications. I will be joining University of Florida, Biostatistics department in June, 2020 as an Assistant Professor.

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Dr. Ravitheja Vangala did his B.Math. from ISI Bangalore, M.Sc. in Mathematics from CMI Chennai, and Ph.D.in Mathematics from TIFR Mumbai. He works in number theory. His current research interests are reductions of local Galois representations, modular representation theory, p-adic L-functions and Iwasawa theory.

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In 1961, James and Stein introduced an estimator of the mean of a multivariate normal distribution that achieves a smaller mean squared error than the maximum likelihood estimator in dimensions three and higher. This was in fact a surprising result by Stein. I will discuss the proof of this result. After that I will discuss some other estimators which are competitors of the James - Stein estimator and if the time permits the James-Stein estimator as an Empirical Bayes Estimator will also be discussed.

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In this long talk, motivated by the need to deal with free boundary or interface problems of practical engineering applications, a variety of numerical methods will be presented. Apart from isogeometric methods (and its variants), we provide a primer to various numerical methods such as meshfree methods, cutFEM methods, and collocation methods based on Taylor series expansions. We review traditional methods and recent ones which appeared in
The last decade for a variety of applications.

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In this talk, the first a posteriori error-driven adaptive finite element approach for real-time surgical simulation will be presented, and the method will be demonstrated on needle insertion problems. For simulating soft tissue deformation, the refinement strategy relies upon a hexahedron-based finite element method, combined with a posteriori error estimation driven local h-refinement. The local and global error levels in the mechanical fields (e.g., displacement or stresses) are controlled during the simulation. After showing the convergence of the algorithm on academic examples, its practical usability will be demonstrated on a percutaneous procedure involving needle insertion in a liver and brain. The brain shift phenomena is taken in to account which occurs when a craniotomy is performed. Through academic and practical examples it will be demonstrated that our adaptive approach facilitates real-time simulations. Moreover, this work provides a first step to discriminate between discretization error and modeling error by providing a robust quantification of discretization error during simulations. The proposed methodology has direct implications in increasing the accuracy, and controlling the computational expense of the simulation of percutaneous procedures such as biopsy, brachytherapy, regional anaesthesia, or cryotherapy. Moreover, the proposed approach can be helpful in the development of robotic surgeries because the simulation taking place in the control loop of a robot needs to be accurate, and to occur in real time. The talk will conclude with some discussion on future outlook towards personalised medicines.

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In this talk, we will discuss about the boundedness of the conformal composition operators on Besov spaces defined on domains in the Euclidean plane. We will see how the regularity of domains effect the boundedness of the operators. The related open problems will also be discussed.

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In binary classification, results from multiple diagnostic tests are often combined in many ways like logistic regression, linear discriminant analysis to improve diagnostic accuracy. In recent time, combining methods like direct maximization of the area under the ROC curve (AUC) has received significant interest among researchers in the field of medical science. In this article, we develop a combining method that maximizes a smoothing approximation of the hyper-volume under manifolds (HUM), an extended notion of AUC when disease outcomes are multi-categorical with ordinal in nature. The proposed method is distribution-free as it does not assume any distribution of the biomarkers. Consistency and asymptotic normality of the proposed method are established. The method is illustrated using simulated data sets as well as two real medical data sets.

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Recent years have witnessed tremendous activity at the intersection of statistics, optimization and machine learning. The consequent bi-directional flow of ideas has significantly improved our understanding of the computational aspects of different statistical learning tasks. Our talk presents new results for two fundamental tasks at the intersection of statistics and optimization namely, (i) efficiently generating random samples given partial knowledge of a probability distribution, and (ii) learning parameters of an unknown distribution given samples from it. In the first part of the talk, we present non-asymptotic convergence guarantees of several popular Monte Carlo Markov Chain (MCMC) algorithms including Langevin algorithms and Hamiltonian Monte Carlo (HMC). We also underline beautiful connections between optimization and sampling, which leads to a design of faster sampling algorithms. In the second part of the talk, we present nonasymptotic results for parameter estimation of mixture models given samples from the distribution.
In particular, we provide algorithmic and statistical guarantees for the Expectation-Maximization (EM) algorithm when the number of components is incorrectly specified by the user.

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In this talk, I shall talk about my recently finished joint work with Simeng Wang and Xumin Wang, which is the first progress on the pointwise convergence of noncommutative Fourier series, solving an open problem since Junge-Xu's remarkable ergodic maximal inequality in noncommutative analysis. Going back harmonic analysis on Euclidean space, one of our results suggests a new class of maximal inequalities which is quite interesting but challenging and deserves to be investigated.

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Majorization is a concept from linear algebra that is used to compare disorderness in physics, computer science, economics and statistics. Recently, Gour et al (2018) extended matrix majorization to the quantum mechanical setting to accommodate ordering of quantum states.In this talk, I will discuss a generalization of their concept of quantum majorization to the infinite dimensional setting. The entropic characterization of quantum majorization will be presented using operator space tensor products and duality . This is based on joint work with Li Gao, Satish Pandey and Sarah Plosker.

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Let $X_1,X_2, X_3$ be Banach spaces of measurable functions in $L^p(\mathbb R)$ and let $m(\xi,\eta)$ be a locally integrable function in $\R^2$. We say that $m$ belongs to $BM(X_1,X_2,X_3)$ if \[ B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(\xi) \hat{g}(\eta)m(\xi,\eta)e^{2\pi i <\xi+\eta, x>}d\xi d\eta, \] defined for $f$ and $g$ with compactly supported Fourier transform, extends to a bounded bilinear operator from $X_1 \times X_2$ to $X_3$.
In this talk we investigate some properties of the class $BM(X_1,X_2,X_3)$ for general spaces which are invariant under translation, modulation and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus in the case $m(\xi,\eta)=M(\xi-\eta)$ and find conditions for these classes to contain non zero multipliers in terms of the Boyd indices for the spaces.

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The Vortex Filament Equation (VFE) is a model for the dynamics of a vortex _lament in a 3-D inviscid incompressible uid, and due to its geometric properties and simplicity, it has received a lot of attention recently. Given an arc-length parameterized curve X(s; _) in R3, the equation describes its evolution as

Where ^ is the usual cross product, s arc-length parameter, t time and subscripts denote the partial derivatives.
In this talk, we consider the initial datum X(s; 0) as a regular polygonal curve in the Euclidean geometry; through algebraic and numerical results, it will be shown that X(s; t) is a skew polygon at any rational time. Hence, we will see that the evolution can be related to the \Talbot e_ect" in optics. We also comment on the trajectory of one point, i.e., X(0; t) which appears to be a multifractal and resembles to the so-called Riemann's non-differentiable function.

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This talk will provide an overview of some of the main ideas in the theory of spacings, i.e. the gaps between successive observations. After reviewing some basic properties of spacings, their use in testing statistical hypotheses and in estimating parameters will be discussed. Two-sample tests based on “spacings-frequencies” and their relationship to locally most powerful rank tests will be explored, as are some possible extensions to observations in higher dimensions.

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This talk provides an easy introduction to the novel area of statistics, where the observations are “directions”. It is introduced by raising various scientific issues where the empirical evidence comes in the form of measuring directions, and how such measurements answer the question at hand. Such examples arise in many natural sciences like geology and biology where one may be directly measuring directions in two or three dimensions, as well as in several other seemingly unrelated and unexpected situations. New descriptive measures as well as statistical models are needed for studying such data, and the basic question one needs to answer before doing any inference in this context, is to ask if the data exhibit any preferred direction i.e. test for isotropy.

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The theory of semihypergroups and hypergroups allows a detailed study of measure algebras that can be expressed in terms of a convolution of measures on the underlying spaces. In particular, the class of semihypergroups contains many important examples of coset and orbit spaces in locally compact groups, which do not have enough structure to be a semigroup or a hypergroup. The lack of an extensive prior research since its inception in 1972 and the signi_cant examples it contains in coset theory, orbit spaces and homogeneous spaces, opens up a number of intriguing new paths of research on semihypergroups.
In our talk, we will give a brief overview on how some well-known algebraic and analytic concepts and language of classical semigroup and group theory can be translated for semihypergroups, and investigate where the theory deviates from the classical theory of semigroups. In particular, we will discuss ideals and homomorphisms, spaces of almost periodic and weakly almost periodic functions and free-product structures in the category of semihypergroups.

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This talk will provide a brief exposition to variable selection and sparse recovery using the L_1 penalty. Primarily, I will talk about the primal-dual witness approach and geometric insights into sparse recovery.

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Prof. Chatterjee will discuss basic facts about high dimensional model selection in regression setup. It is currently a burning area of research in statistics. He will state and explain some recent non-asymptotic results.

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Prof. Chatterjee will discuss basic facts about high dimensional model selection in regression setup. It is currently a burning area of research in statistics. He will state and explain some recent non-asymptotic results.

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We consider the problem of analysis of variance, where the sample observations are random functions, and propose a test based on spatial signs. An asymptotic implementation and a bootstrap implementation of this test are developed, and their properties are investigated. We compare the performance of our test with that of several mean based tests of ANOVA for functional data in the literature, and found that Our test not only outperforms the mean based tests in several non-Gaussian models with heavy tails, but in some Gaussian models also, it exhibits better performance than the mean based tests.

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In this talk I will give a brief introduction to G_2 geometry and gauge theory on G_2 manifolds. I will discuss the deformation theory of instantons on nearly G_2 manifolds. We will study the deformation space by identifying it as the kernel of a Dirac operator and will use this identification to specify some cases where the deformation space is trivial.

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We shall discuss how one can estimate values of various Diophantine exponents for standard linear actions of some $2 \times 2$ matrix groups provided we have continued fraction algorithms whose denominator sizes rise exponentially. Thereafter, we illustrate this with few cases which we have been able to resolve satisfactorily. This builds upon previous works of Bugeaud, Dani, Laurent and Nogueira, among many othersThe talk will be partly based on joint work with Yann Bugeaud and Zhenliang Zhang.

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In this talk, I will first give some historical account of work on cure models. I will then describe some of the important cure rate models and then present a flexible cure rate model that encompasses some of the special cases and describe both direct maximum likelihood estimation and an efficient EM algorithm. I will then discuss some model discrimination results. Finally, I will describe some generalizations including proportional hazards model, proportional odds model, and some destructive cure models. All the models will be illustrated with some melanoma data sets.

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A celebrated theorem of Margulis characterizes arithmetic lattices in terms of density of their commensurators. A question going back to Shalom asks the analogous question for thin subgroups. We shall report on work during the last decade or so and conclude with a recent development. In recent work with Thomas Koberda, we were able to show that for a large class of normal subgroups of rank one arithmetic lattices, the commensurator is discrete.

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We study first passage percolation (FPP) in a Gromov-hyperbolic group G with boundary equipped with the Patterson-Sullivan measure. We associate an i.i.d. collection of random passage times to each edge of a Cayley graph of G, and investigate classical questions about asymptotics of first passage time as well as the geometry of geodesics in the FPP metric. Under suitable conditions on the passage time distribution, we show that the 'velocity' exists in almost every direction, and is almost surely constant by ergodicity of the G-action on the boundary.For every point on the boundary, we also show almost sure coalescence of any two geodesic rays directed towards the point. Finally, we show that the variance of the first passage time grows linearly with word distance along word geodesic rays in every fixed boundary direction.
This is joint work with Riddhipratim Basu.

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Warped cone is a geometric object associated with a measure preserving isometric action of a finitely generated group on a compact manifold. It encodes the geometry of the manifold, geometry of the group (Cayley graph) and the dynamics of the group. This geometric object has been introduced by J. Roe in the context of Coarse Baum-Connes conjecture (CBC conjecture). Warped cones associated with the action of amenable groups give examples of CBC conjecture and some expander graphs can be constructed from the warped cones associated with the action of Property (T) group. On the other hand, Measured Equivalence (ME) is an equivalence relation between two countable groups introduced by M. Gromov as a measure-theoretic analogue of quasi-isometry. If the ‘cocyles’ associated with a measured equivalence relation are bounded, the relation is called Uniform Measured Equivalence. In this lecture, we prove that if two warped cones are quasi-isometric, then the associated groups are Uniform Measured Equivalent. As an application, we will talk about different ME-invariants which distinguish two warped cones up to quasi-isometry. This is a work in progress.

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I will give a brief introduction to A^1-homotopy theory and describe some applications to algebraic geometry. The presentation will be non-technical and will be based on a lot of examples.

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Consider k ($\geq 2$) populations characterized by k probability distributions that differ only in the numerical value of a parameter (say mean). In the analysis of variance if the hypothesis of homogeneity of populations is rejected then a natural question to ask is that which of the k populations is the best population, where a population is considered to be better than the other if the numerical value of some function of the parameter associated with it is larger (or smaller) than the corresponding value for the other population. Ranking and selection problems provide a satisfactory solution to this problem. Although the ranking and selection problems have been extensively studied in the literature since 1950 there are still some unresolved problems in the area.

In this talk we will discuss one such unresolved problem and discuss a partial solution to this problem. For the ease of presentation, the problem will be discussed through the example of gamma populations.

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The E_2-term of the Adams spectral sequence may be identified with certain derived functors, and this also holds for other Bousfield-Kan types spectral sequence.

In this talk, we'll explain how the higher terms of such spectral sequences are determined by truncations of relative derived functors, defined in terms of certain spectrally enriched functor called mapping algebras.

This is ongoing joint work with David Blanc.

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In 1812, F. Gauss introduced classical hypergeometric series to the Royal Society of Sciences at Gotingen. Since then, this kind of special functions have been well studied by mathematicians and established many significant contributions in different branches of mathematics. In the meantime, based on an analogy between the character sum expansion of a complex valued function over finite field and the power series expansion of an analytic function, J. Greene developed an analogue for classical hypergeometric series over finite field. This function is known as Gaussian hypergeometric function. Gaussian hypergeometric functions were introduced to have a parallel study with classical hypergeometric series. However, Gaussian hypergeometric functions have certain limitations. To overcome the limitations, D. McCarthy introduced an analogue for classical hypergeometric series in the p-adic setting. In this talk, we will discuss Gauusian hypergeometric functions and hypergeometric functions in the p-adic setting. Hypergeometric functions have been applied to different areas of mathematics but the two areas of most interest to us are their relations with the traces of Hecke operators and Kloosterman sums. In the first part of the talk, we will discuss certain connections of hypergeometric functions with the traces of Hecke operators. In the 2nd part we will review Kloosterman sums and their relations with hypergeometric functions.

Abstract:

In a large variety of fluid systems, flow properties are a function of space and time, and their variation can have a dramatic effect on the flow instability. The knowledge of instability behaviour of such flow is essential for mathematical modeling, design and application of compact tools to ensure desired mechanical, optical properties and barriers of the products. Moreover, many intriguing and important fluid dynamic phenomena in nature and engineering tools are associated with complex spatio-temporal patterns (e.g. water waves, clouds, sprays, blood flow and turbulent flows in various industries etc.). The study of hydrodynamic stability is an easier way to understand the spatio-temporal behaviour of complex flow systems. This talk includes the discussion on modal linear stability analysis, analytical and numerical techniques for solving stratified multilayer problems. I will explain how the instability characteristics of bounded as well as semi-bounded viscosity-stratified flows alter effectively by scalar diffusion and boundary slip. I will also discuss the derivation of generalized Orr-Sommerfeld equation and Reynolds-Orr energy equation for stratified miscible flow with Navier-slip boundary condition.

Abstract:

We are often interested in problems in which we look for a solution y(x) of a differential equation so that y(x) satisfies a prescribed condition—that is, condition imposed on the unknown y(x) or its derivatives. In this talk, I want to discuss about the existence and uniqueness of solution for such kind of equation. I will define an initial value problem (IVP) and seek general existence theorem for real solution. I will also explain how a unique solution of the initial value problem, can be obtained by an approximation process and check the necessary/sufficient conditions for the unique solution.

Abstract:

In this talk I will give an exposition on two remarkable results of Sudipta (one jointly with P. Bandyopadhyay and the other with D. Narayana) on the geometry of subspace of finite co-dimension in space of continuous functions on a compact set K. These results which are nearly a decade old had a great impact on the study of the structure of such subspaces in other classes of Banach spaces.

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No

Abstract:

The aim of this talk is to explain the behaviour of some conformal metrics and invariants near a smooth boundary point of a domain in the complex plane. We will be interested in the invariants associated to the Carathéodory metric such as its higher-order curvatures that were introduced by Burbea and the Hurwitz metric.

The basic technical step in all these is the method of scaling the domain near a smooth boundary point. To estimate the higher-order curvatures using scaling, we generalize an old theorem of Suita on the real analyticity of the Carathéodory metric on planar domains and in the process, we show convergence of the Szegő and Garabedian kernels as well.

We then talk about the Hurwitz metric that was introduced by D. Minda. Its construction is similar to the Kobayashi metric but the essential difference lies in the class of holomorphic maps that are considered in its definition. We show that this metric is continuous and also strengthen Minda’s theorem about its comparability with the quasi-hyperbolic metric by estimating the constants in a more natural manner.

Abstract:

In Langlands program certain invariant linear functionals on irreducible representations of algebraic groups over locally compact _elds (ex: GL2(R), GL2 (Qp)) play a central role. The simplest version of these linear functionals are called Whittaker functionals. Whittaker functionals are deeply related to invariant harmonic analysis on such groups, with arithmetic and geometry. For instance, for the group GL2, the existence of Whittaker linear functional imply the multiplicity one results in the theory of automorphic representations. The situation of multiplicity one fails for almost all other linear groups and this failure is captured by Langlands formalism of dual group. I will explain my results in the case of unitary groups in three variables.

Finally, if time permits, we shall see some weak estimates on the generalized upper and lower curvatures of the Hurwitz metric.

Abstract:

This is joint work with Marc Technau (University of Graz). We generalize a classical result by R.C. Vaughan on Diophantine approximation restricted to fractions with prime denominator to imaginary quadratic number field of class number one. Our treatment is based on Harman's sieve method in the number field setting. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

Abstract:

A collective motion of cells which responds to an attractant gradient is known as “chemotaxis”. Chemotaxis-convection-diffusion is a particular type of bio convection. Due to its significant role in medical, industrial, and geophysical areas, research effort has been performed to understand the dynamics of the bacterial motility in suspension, studies through analytical, experimental, and numerical attempts previously were only for a flat free-surface of a suspension of chemotaxis bacteria in a shallow/deep chamber. We consider now a three-dimensional chemotaxis-convection-diffusion flow system with a deformed free surface. The influences of the aggregated chemotactic cells on the deformed free surface of a shallow chamber are studied analytically. The aim of our research work is to explore the nature of the instability in the system by performing a detailed linear stability analysis of steady-state oxygen and cell concentration distributions. A weakly nonlinear stability analysis has been carried out as well to determine the relative stability of the pattern formation at the onset of instability where Rayleigh number R_(α_T ) is the nonlinear control parameter of the system. The system becomes dominated by nonlinear convection terms beyond a critical R_(α_T ) , which also depends on the critical wavenumber k and Nusselt number N_(u_T ) as well as the other parameters. We have investigated the issue of how the critical R_(α_T ) in this system varies with three different sets of parameters. The Lorenz model is derived under the assumption of Bossinesq approximation. Using the method of multiscales, a Ginzburg-Landau equation is derived from the Lorenz model, the solution of which helps to quantify the energy transport through the Nusselt number N_(u_T )

Abstract:

In 1980s Goldman introduced various Lie algebra structures on the free vector space generated by the free homotopy classes of closed curves in any orientable surface F. Naturally the universal enveloping algebra and the symmetric algebra of these Lie algebras admit a Poisson algebra structure. In this talk I will define and discuss some properties of these Poisson algebras. I will explain their connections with symplectic structure of moduli space and the skein algebras of F\times [0,1] . I will also discuss how to compute center of these Poisson algebras using geometric group theory. I will mention some open problems related to these objects.

Abstract:

Differential equations have fundamental importance in engineering mathematics because many physical laws and relations can be expressed mathematically in the form of differential equations. The mathematical problems of science and engineering fields can appear as a differential equations. For example, the problem of satellite motion, current flow in an electric circuit, population growth, radioactive decay, temperature control etc. lead to differential equations. Each of the above problems are characterized by some laws which involve the rate of change of one or more quantities, with respect to the other quantities. The laws characterizing these problems when expressed mathematically, become equations involving derivatives and such equations are called differential equations. It is important to study the methods of ordinary differential equations to solve these problems. Differential equations that depends on a single variable is called as ordinary differential equations. Simplest method to be discussed are ODEs of the first order because they involve only the first derivative of the unknown function. Some first-order ODEs examples will be solved and plot their solution curve.

Abstract:

It is well known from a result of Milnor on the topology of isolated singularities that the Milnor number is a topological invariant in the complex case. We will show that the Milnor number in the real case is not a bi-Lipschitz invariant. We will produce a one-parameter deformation of a singularity which is bi-Lipschitz trivial but the Milnor number is different for two different values of the parameter variable.

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Abstract:

Define a configuration in R^n to be a family C of finite subsets of the R^n which is closed under dilations. We say that X avoids C if no set in C is contained in X. We will discuss some problems of the following form: Given a configuration C in R^n and a subset X of R^n, can we find a "large" subset Y of X such that Y avoids C? "Y is a large subset of X" will be interpreted both measure theoretically (Y has the same Lebesgue outer measure as X) and topologically (Y is everywhere non meager in X).

Abstract:

An algorithm is a step by step procedure to solve a problem. Every algorithm has inherited parallelism in it. But all algorithms cannot be parallelized. To explore the inherited parallelism of an algorithm, few basic steps play a vital role. Interconnection network is one of such steps. Few algorithm’s performance is suited to a specific kind of interconnection network. The present talk will explore such interconnection networks and will also discuss about the problems which have better performance over which network.

Abstract:

In this talk, we discuss a semi-discrete finite difference scheme for a conservation laws driven by noise. Thanks to BV estimates, we show a compact sequence of approximate solutions, generated by the finite difference scheme, converges to the unique entropy solution of the underlying problem, as the spatial mesh size goes to zero. Moreover, we show that the expected value of the L^1 difference between the approximate solution and the unique entropy solution converges at an expected rate.

Abstract:

Several types of spatiotemporal patterns for ecological communities are ubiquitous in natural habitat such as the vegetation patterns in semiarid region and the patchy spatial distribution in plankton ecosystem. It has been believed that the generation of spatiotemporal pattern in ecological communities is a result of the combination effect of the local and external factors. There exist several external factors, such as chemical-physical limitations and environmental heterogeneity, which take part in shaping the dynamics of ecological communities. However, there exist numerous empirical evidences which suggest that one should not point at the environmental heterogeneity as sole reason behind the emergence of spatiotemporal patterns. On the other hand, studies indicate that internal factors can lead to spatiotemporal pattern formation in a completely homogeneous environment and this gives rise to the well-known theory of the self-organized pattern formation. The theoretical study on self-organized pattern formations in ecological communities has been accomplished by analyzing the reaction-diffusion systems which take into account the random movements of the concerned species. In this presentation, we will talk about a complete self-organized pattern formation scenario for a spatiotemporal prey-predator model with strong Allee effect in prey growth, Holling type-II functional response and density dependent death rate of predator. The effect of the half-saturation constant on the emergence of different types of stationary patterns and the persistence enhancing effect of the spatial component will be discussed. Also, we will present different types of invasive patterns and their implications on control and management strategies for invasive species. Further, we will discuss about the destabilizing effect of gestation delay on spatial distributions of the concerned species. Overall, a comparative study between the dynamics of spatial and corresponding non-spatial systems will be presented.1

Abstract:

Sign changes of Fourier coefficients of cusp forms are central problem in number theory and many results have been obtained in this direction. In particular, it is interesting to study the sign changes at subsequences. In this talk, we will consider the problem of sign changes of Fourier coefficients of cusp form at sum of two squares and will give a quantitative result in this direction.

Abstract:

In this talk we will describe the Safety Analysis of the Indian Oceanic Airspace project, which Indian Statistical Institute is conducting jointly with the Airports Authority of India (AAI), starting from the year 2011. The talk will concentrate on the understanding of the goal of the project, typical data structure and the current "state of the art". We will also discuss some of the statistical challenges related to such an analysis.

Abstract:

The Selmer group of an elliptic curve over a number field encodes several arithmetic data of the curve providing a p-adic approach to the Birch and Swinnerton Dyer, connecting it with the p-adic Lfunction via the Iwasawa main conjecture. Under suitable extensions of the number field, the dual Selmer becomes a module over the Iwasawa algebra of a certain compact p-adic Lie group over Z_p (the ring of padic integers), which is nothing but a completed group algebra. The structure theorem of GL(2) Iwasawa theory by Coates, Schneider and Sujatha (C-S-S) then connects the dual Selmer with the “reflexive ideals” in the Iwasawa algebra. We will give an explicit ring-theoretic presentation, by generators and relations, of such Iwasawa algebras and sketch its implications to the structure theorem of C-S-S. Furthermore, such an explicit presentation of Iwasawa algebras can be obtained for a much wider class of p-adic Lie groups viz. pro- p uniform groups and the pro-pIwahori of GL(n,Z_p). If we have time, alongside Iwasawa theoretic results, we will state results (joint with Christophe Cornut) constructing Galois representations with big image in reductive groups and thus prove the Inverse Galois problem for p-adic Lie extensions using the notion of “p-rational” number fields.

Abstract:

When examining dependence in spatial data, it can be helpful to formally assess spatial covariance structures that may not be parametrically specified or fully model-based. That is, one may wish to test for general features regarding spatial covariance without presupposing any particular, or potentially restrictive, assumptions about the joint data distribution. Current methods for testing spatial covariance are often intended for specialized inference scenarios, usually with spatial lattice data. We propose instead a general method for estimation and testing of spatial covariance structure, which is valid for a variety of inference problems (including nonparametric hypotheses) and applies to a large class of spatial sampling designs with irregular data locations. In this setting, spatial statistics have limiting distributions with complex standard errors depending on the intensity of spatial sampling, the distribution of sampling locations, and the process dependence. The proposed method has the advantage of providing valid inference in the frequency domain without estimation of such standard errors, which are often intractable, and without particular distributional assumptions about the data (e.g., Gaussianity). To illustrate, we develop the method for formally testing isotropy and separability in spatial covariance and consider confidence regions for spatial parameters in variogram model fitting. A broad result is also presented to justify the method for application to other potential problems and general scenarios with testing spatial covariance. The approach uses spatial test statistics, based on an extended version of empirical likelihood, having simple chi-square limits for calibrating tests. We demonstrate the proposed method through several numerical studies.

Abstract:

We will begin with a quick review of quantum channels and briefly present the celebrated Choi-Kraus representation theorem. These constitute the key concepts and results that are required to move forward and define a special class of quantum channels called "entanglement breaking channels" --- the quantum channels that admit a Choi-Kraus representation consisting of rank-one Choi-Kraus operators.

We then introduce the entanglement breaking rank of an entanglement breaking channel and define it to be the least number of rank-one Choi-Kraus operators required in its Choi-Kraus representation. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. In particular, we show that the problem of computing the entanglement breaking rank of the quantum channel, that sends X in M_d to 1/(d+1) [X + Tr(X)I_d] in M_d, is equivalent to the existence problem of SIC POVM in dimension d. This is a joint work with Vern Paulsen, Jitendra Prakash, and Mizanur Rahman.

Abstract:

In this talk, we will discuss the problem whether two regression functions modelling the relation between a response and covariate in two samples differ by a shift in the horizontal and vertical axis. We consider a nonparametric situation assuming only smoothness for the regression functions. A graphical tool based on the derivatives of the regression functions and their inverses is proposed to answer this question and studied in several examples. We also formalize this question in a corresponding hypothesis and develop a statistical test. The asymptotic properties of the corresponding test statistic are investigated under the null hypothesis and under local alternatives and the finite sample properties of the new test are investigated by means of a small simulation study and real data example. This is a joint work with Holger Dette (University of Ruhr, Germany) and Weichi Wu (Tsinghua University, China).

Abstract:

In this talk, we will discuss estimation and prediction inferences for the generalized half normal and Kumaraswamy distributions under different censoring schemes. In particular we will discuss hybrid Type-I censoring, Type II progressive hybrid censoring and adaptive Type-II progressive censoring schemes. Likelihood inference and Bayesian inference will be addressed under different censoring schemes. Further we will discuss about prediction estimates and associated intervals of censored/ future observations using both frequentist and Bayesian approaches. We will numerically compare different estimations and prediction estimates. We will illustrate different methods using various real data sets. Finally we will discuss some ongoing research problem on step stress life testing model.

Abstract:

Lang, Jorgenson, and Kramer have successfully employed techniques coming from theory of heat kernel, to study and estimate various arithmetic invariants. Inspired by their ideas, we describe an approach to study estimates of cusp forms, using theory of heat kernels.

Abstract:

Let G be a connected reductive group over a finite field f of order q. When q is small, we make further assumptions on G. Then we determine precisely when G(f) admits irreducible, cuspidal representations that are self- dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive p-adic groups. This is a joint work with Jeffrey Adler.

Abstract:

Given a smooth function f : [0,1) --> \R, and scalars u_j, v_j in (0,1), I will compute the Taylor (Maclaurin) series of the function F(t) := \det A(t), where A(t) is the 2x2 matrix

f( t u_1 v_1 ) f( t u_1 v_2 )
f( t u_2 v_1 ) f( t u_2 v_2 )

C. Loewner computed the first two of these Maclaurin coefficients, in the thesis of his student R.A. Horn (Trans. AMS 1969). This was in connection with entrywise functions preserving positivity on matrices of a fixed dimension -- the case of all dimensions following from earlier work of Schur (Crelle 1911) and his student Schoenberg (Duke 1942).

It turns out that an "algebraic" family of symmetric functions is hiding inside this "analysis". We will see how this family emerges when one computes the second-order (and each subsequent higher-order) Maclaurin coefficient above. This family of functions was introduced by Cauchy (1800s) and studied by Schur in his thesis (1901). As an application, I will generalize a determinant formula named after Cauchy to arbitrary power series (over any c ommutative ring); the above is the special case f(x) = 1/(1-x) and t=1.

Abstract:

I want to explain the oldest unsolved major problem in Mathematics (called the congruent number problem). It can be traced back at least to the 10th century but it is possibly much older. It turns out to be a beautiful example of the modern theory of the arithmetic of elliptic curves, but it is more accurate to say that this theory grew out of the study of this problem.

In the 17th century, Fermat gave a wonderful proof of the first special case of this problem. It also led Fermat to his so called Last Theorem (now solved by Andrew Wiles). But the original congruent number problem remains unsolved, despite the fact that conjecturally there is a very simple and beautiful answer to it.

Abstract:

Over the last few decades, models based on fluid flow and deformation in porous media are getting a lot of attention because these models have wide range of applications in science and engineering. Specifically, Biot’s consolidation model has many applications which cover the range from geoscience to medicine. It is the aim of this talk to present the a posteriori error analysis for locking free mixed finite element method of Biot's consolidation model. We discuss three novel a posteriori error estimators and show that all three a posteriori error estimators are reliable, efficient and robust. Finally, numerical results are presented to validate the theoretical results.

Abstract:

Estimation of animal abundance and distribution over large regions remains a central challenge in statistical ecology. In our first study, we use a Bayesian smoothing technique based on a conditionally autoregressive (CAR) prior distribution and Bayesian regression to integrate data from reliable but expensive surveys conducted at smaller scales with cost-effective but less reliable data generated from surveys at wider scales to address this problem. We also investigate whether the random effects which represent the spatial association due to the CAR structure have any confounding effect on the fixed effects of the regression coefficients. Next, we develop a novel Bayesian spatially explicit capture-recapture (SECR) model that disentangles the latent ecological process of animal arrival within a detection region from the process of recording this arrival by a given set of detectors. We integrate this into an advanced version of a recent SECR model by Royle (2015) involving partially identified individuals. The above is a joint work with Prof. Mohan Delampady.

Abstract:

This is an informal talk meant for aspiring researchers in Mathematics and related fields such as Theoretical Physics. The purpose of this talk is to convey a sense of what it takes, on a day to day basis, to succeed in research. I would like to lay bare the inner attitude of a researcher. It is based partly on my own experience, but mostly on what I have learnt while collaborating, and generally hobnobbing, with some truly great mathematicians of our times. The talk will be totally non-technical and I hope to make it very interactive.

Abstract:

The representation theory of string algebras started with the work of Gelfand and Ponomarev on Harish-Chandra representations of the Lorentz group. The methods they introduced were adapted and extended to cover the representation theory of a broad class of algebras. The term "string" algebra refers to the combinatorial description of their indecomposable finitely generated representions. I will recall that description, and that of the morphisms between representations, outline the broad picture of the category of finitely generated modules in terms of Auslander-Reiten components and say something about the infinite-dimensional representation theory.

Abstract:

When _lms thin the viscous resistance of the con_ning surfaces controls key aspects of their dynamics. However, those surfaces do more than provide the typical boundary conditions. For example, if one of the boundaries is elastic, there are nonlocal interactions that lead to interesting and challenging mathematical problems and important applications. One unique setting occurs when the free surfaces of most solids approach the bulk melting temperature from below, and they are wet by the melt phase. Mathematically, the uid dynamics of this so-called \premelting" falls under the rubric of a class of higher-order di_usion equations governing the dynamics of viscous current down an incline, viscous gravity currents between a rigid surface and a deformable elastic sheet, wetting/dewetting, and a spate of other thin-_lm settings, to name a few. However, theunderlying forces driving the ow are uniquely associated with intermolecular forces. We discuss aclass of experimentally tested and testable premelting dynamics ows. Finally, through its inuence on the viscosity, the con_nemente_ect implicitly introduces a new universal length scale into the volume ux. Thus, there are a host of thin _lm problems, from droplet breakup to wetting/dewetting dynamics, whose properties (similarity solutions, regularization, and compact support) will changeunder the action of the con_nemente_ect. Therefore, our study suggests revisiting the mathematical structure and experimental implications of a wide range of problems within the framework of the con_nemente_ect.

"The game of Hex gives a constructive proof of the Brouwer's fixed point theorem”
on Feb 21, 2019 at 4:00 PM in FB 567, MATHS Dept. Seminar room by Prof. TES Raghavan from University of Illinois at Chicago

Abstract:

Abstract:

Abstract:

We will discuss results concerning conjugacy-invariant norms on free groups, in particular ones thatare homogeneous and quasi-homogeneous.

On values of the Riemann zeta function at odd positive integers. on January 25, 2019 at 2:30 PM in FB 567, MATHS Dept Seminar room by Prof. Atul Dixit from Gandhinagar

Abstract:

KMS states are generalization of what is known as Gibbs state for matrices to in_nitedimen-sionalC^∗-algebras. Given an inverse temperature, unlike the Gibbs state for matrix algebra,KMS statesfor general in_nite dimensional C^∗algebras are far from unique. We discuss how theclassical and quantum symmetry of graphs act as symmetry objects for the corresponding graphC^∗algebras and the invariance of KMS states on graph C^∗algebras under such symmetries. Inparticular, we shall give an example of a graph admitting in_nitely many KMS states at criticalinverse temperature all of which are invariant under classical symmetry. But interestingly, thatgraph admits unique quantum symmetry invariant KMS state.

Absract:

A topological dynamical system is a pair (X,T) where T is a homeomorphism of a compact space X. A measure preserving action is a triple (Y, \mu, S) where Y is a standard Borel space, \mu is a probability measure on X and S is a measurable automorphism of Y which preserves the measure \mu. We say that (X,T) is universal if it can embed any measure preserving action (under some suitable restrictions).Krieger’s generator theorem shows that if X is A^Z (bi-infinite sequences in elements of A) and T is the transformation on X which shifts its elements one unit to the left then (X,T) is universal. Along with Tom Meyerovitch, we establish very general conditions under which Z^d (where now we have dcommuting transformations on X)-dynamical systems are universal. These conditions are general enough to prove that 1) A self-homeomorphism with non-uniform specification on a compact metric space (answering a question by Quas and Soo and recovering recent results by David Burguet)2) A generic (in the sense of dense G_\delta) self-homeomorphism of the 2-torus preserving Lebesgue measure (extending result by Lind and Thouvenot to infinite entropy)3) Proper colourings of the Z^d lattice with more than two colours and the domino tilings of the Z^2 lattice (answering a question by Şahin and Robinson) are universal. Our results also extend to the almost Borel category giving partial answers to some questions by Gao and Jackson.The talk will not assume background in ergodic theory and dynamical systems.

Abstract:

The standard isoperimetric inequality states that among all sets with a given fixedvolume(or area in dimension 2) the ball has the smallest perimeter. That is, written here in dimension2, the following infimum is attained by the ball  I will give an elementary overview on this inequality, such as techniques developed for theproof (symmetrization, variational method, Fourier series method in 2 dimensions), higherdimensional version, connections to partial differential equations and some generalizations. Iwill keep the talk basic, requiring no specialized knowledge in differential geometry and partial differential equations.

Abstract:

The dynamical setting of a non-autonomous system is related to the Bedford conjecture. Further,Short C^k's arises as basins of attraction of a fixed point in the non-autonomous setting. In this talk, I will discuss the relation between non-autonomous holomorphic dynamics, Short C^k's and their importance in the context of the Levi-problem. Also, I will give some construction of Short C^k's with pathological properties and finally discuss some related results and questions.

Abstract:

In multivariate linear regression, one of the major challenges is to model dependent responses from correlated predictors in a compact and interpretable manner. One promising approach is to model the coefficient matrix as a jointly sparse and low­-rank matrix. Learning such a decomposition is, however, very challenging due to the simultaneous presence of orthogonality and sparsity constraints.Here, we introduce a divide and conquer strategy to infer such a coefficient matrix from data. In the divide step, we decompose the coefficient matrix into a sum of unit-­rank matrices whose left and right singular vectors are sparse. In the conquer step, each unit­-rank matrix is estimated either by a sequential(greedy) approach or an exclusive extraction approach. We show that the proposed algorithm is guaranteed to converge to an accumulation point and provide an efficient implementation in the R package secure. We demonstrate the efficacy of the procedure in simulation studies and two applications in genetics.

Abstract:

Over the last few decades, models based on fluid flow and deformation in porous media are getting a lot of attention because these models have wide range of applications in science and engineering. Specifically, Biot’s consolidation model has many applications which cover the range from geoscience to medicine. It is the aim of this talk to present the a posteriori error analysis for locking free mixed finite element method of Biot's consolidation model. We discuss three novel a posteriori error estimators and show that all three a posteriori error estimators are reliable, efficient and robust. Finally, numerical results are presented to validate the theoretical results.

Abstract:

The problem of estimating quantiles has received considerable attention by several researchers in the recent past from classical as well as decision theoretic point of view. It is noted that, exponential quantiles are widely used in the study of reliability, life testing and survival analysis, and related fields. Suppose there are k(≥2) normal populations with a common mean and possibly different variances. The problem of estimation of quantiles of the first population is considered with respect to a quadratic loss function. The concept of invariance is introduced to the problem. As a result, sufficient conditions are derived to improve affine and location equivariant estimators. Consequently, some complete class results have been obtained. Next, the problem of estimating ordered quantiles of two exponential distributions has been considered assuming equality on location parameters. Using order restriction on the quantiles, several new estimators have been proposed including the restricted maximum likelihood estimator. Using quadratic loss function, a sufficient condition has been obtained to improveequivariant estimators under the order restriction. Consequently, some improved estimators have been obtained. All the proposed estimators have been compared numerically. It has been seen that the percentage of risk reduction after using order restriction is quite significant.

 Lack-of-fit of a parametric AR(1) model in the presence of measurement error.On January 14, 2019 at 11:00 AM in FB567, MATHS Dept Seminar room by Prof. Hira Koul from Department of Statistics & Probability, Michigan State University

Abstract:

Recently, power law (also known as Zipf's law) has been turned out to be very natural in modelling and inferring about networks, finance, environment studies, rare events. In this seminar, Hill's estimator (best known to estimate index of regular variation) will be discussed with examples. We shall talk about the advantages and disadvantages of using Hill's estimator once population deviates from pure power law. We shall also raise some issues which are yet to be resolved.

Abstract:

We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n-th generation converges weakly to a Cox cluster process. This is a joint work with Rajat Subhra Hazra and Parthanil Roy.

Abstract:

We will start by defining a G_2 and a nearly G_2 structure on a seven dimensional manifold M. After surveying known results about G_2 manifolds and explaining the importance of such manifolds, we will talk about hypersurfaces of a G_2 manifold, which inherits an SU(3) structure from the ambient G_2 structure. We will give a necessary and sufficient condition for a hypersurface of a nearly G_2 manifold to be nearly Kahler.After that, we will focus our attention on minimal hypersurfaces (zero mean curvature) and will discuss some new results in that direction.

Abstract:

Sequential multiple assignment randomized trials (SMART) are used to construct data-driven optimal treatment strategies for patients based on their treatment and covariate histories in di_erent branches of medical and behavioral sciences where a sequence of treatments are given to the patients; such sequential treatment strategies are often called dynamic treatment regimes (DTR). In the existing literature, the ma- jority of the analysis methodologies for SMART studies assume a continuous primary outcome. However, ordinal outcomes are also quite common in clinical practice; for example, the quality of life is often measured in an ordinal scale (e.g., poor, moderate, good). In this work, _rst, we develop the notion of dynamic generalized odds-ratio (dGOR) to compare two dynamic treatment regimes embedded in a 2-stage SMART with an ordinal outcome. We propose a likelihood-based approach to estimate dGOR from SMART data. Next, we discuss some combinatorial properties of dGOR and derive the asymptotic properties of its estimate. We discuss some alternative ways to estimate dGOR using concordant-discordant pairs and multi-sample U-statistic. Then, we extend the proposed methodology to a K-stage SMART. Furthermore, we propose a basic policy search algorithm that uses dGOR to _nd an optimal DTR within a _nite class. A simulation study shows the performance of the estimated dGOR in terms of the estimated power corresponding to the derived sample size. We analyze data from Sequenced Treatment Alternatives to Relieve Depression (STAR*D), a multi-stage randomized clinical trial for treating major depression, to illustrate the proposed methodology. A freely available online tool using R statistical software is provided to make the proposed methodology accessible to other researchers and practitioners.

Abstract:Quantile regression is a more robust, comprehensive, and flexible method of statistical analysis than the commonly used mean regression methods. Quantile function completely characterizes a distribution. More and more data is getting generated in the form of multivariate, functional, and multivariate functional data and the quantile analysis gets more challenging as the data complexity increases. We propose a set of quantile analysis methods for multivariate data and multivariate functional data. In plant science, the study of salinity tolerance is crucial to improving plant growth and productivity and we apply our methods to barley field data for a salinity tolerance analysis. We apply our methods to radiosonde wind data to illustrate our proposed quantile analysis methods for visualization, outlier detection, and prediction. We model multivariate functional data for flexible quantile contour estimation and prediction.The estimated contours are flexible in the sense that they can characterize non-Gaussian and non-convex marginal distributions. We aim to perform spatial prediction for non-Gaussian processes.

Abstract:

Abstract:

Arens irregularity of a Banach algebra is due to elements in its Banach dual which are not weakly almost periodic. The usual algebras in harmonic analysis (such as the group algebra or the Fourier algebra, when known) turned out to be all Arens irregular (even in an extreme way, as we shall see in the talk). To start, more than sixty years ago, Richard Arens himself proved that I_1 is irregular, and was followed by Mahlon Day proving the same result for many discrete groups including the abelian ones. A long exciting story followed after. The talk is an attempt to tell you the story and to explain the combinatorial reason (on the group or its dual) creating such irregularity.

Abstract:These will be an informal talk on the use of Carleman estimates for formally determined inverse problems for hyperbolic PDEs.

Abstract: In this talk, we will discuss the existence theory of distributional solutions solving.

 On a bounded domain Ω. The nonlinear structure A(x, t, Δu) is modelled after the standard parabolic p-Laplace operator. In order to do this, we develop suitable techniques to obtain a prior estimates between the solution and the boundary data. As a consequence of these estimates, a suitable compactness argument can be developed to obtain the existence result. An interesting ingredient in the proof is the careful use of the boundedness of a new type of Maximal function defined on negative Sobolev spaces. The a priori estimates proved discussed in this talk are new even for the heat equation on bounded domains.

Abstract:

Bounded sequences in the Sobolev space of non-compact manifold $M$ converge weakly in $L^p$ only after subtraction of countably many terms supported in the neighborhood of infinity. These terms are defined by functions on the manifolds-at-infinity of $M$, defined by a gluing procedure, and the sum of their respective Sobolev energies is dominated by the energy of the original sequence. Existence of minimizers in isoperimetric problems involving Sobolev norms is therefore dependent on comparison between Sobolev constants for $M$ and its manifolds-at-infinity. This is a joint work with Leszek Skrzypczak.

Abstract:

In this introductory talk, we describe an extension of the probabilistic model of diffusion that was introduced by K.Itô using his method of `stochastic differential equations'. In our model, which is based on ideas from stochastic partial differential equations, the evolving system of (solute) particles is modeled by a tempered distribution. Our method can also be viewed as an extension of the ‘method of characteristics’ in partial differential equations.

Abstract:

Since Lovasz introduced the idea of associating complexes with graphs to solve problems in Graph Theory, the properties that these complexes enjoy have been a topic of study. In this talk, I will be discussing the complexes associated with some classes of graphs, including complete graphs, Kneser and Stable Kneser graphs. The connectedness of some of the complexes associated with the graphs provide good lower bounds for the chromatic number of these graphs.

Abstract:Geometric invariant theory (GIT) provides a construction of quotients for projective algebraic varieties equipped with an action of a reductive algebraic group. Since its foundation by Mumford, GIT plays an important role in the construction of moduli (or parameter) spaces. More recently, its methods have been successfully applied to problems of representation theory, operator theory, computer vision and complexity theory. In the first part of the talk, I will give a very brief introduction to GIT and in the second part I will talk about some geometric properties of the torus and finite group quotients of Flag varieties.

Abstract:A branching random walk is a system of growing particles that starts from one particle at the origin with each particle branching and moving independently of the others after unit time. In this talk, we shall discuss how the tails of progeny and displacement distributions determine the extremal properties of branching random walks. In particular, we have been able to verify two related conjectures of Eric Brunet and Bernard Derrida in many cases that were open before.This talk is based on a joint works with Ayan Bhattacharya (PhD thesis work at Indian Statistical Institute (ISI), Kolkata, presently at Centrum Wiskunde & Informatica, Amsterdam), Rajat Subhra Hazra (ISI Kolkata), Souvik Ray (M. Stat dissertation work at ISI Kolkata, presently at Stanford University) and Philippe Soulier (University of Paris Nanterre).

Abstract:

During the last four decades, it has been realized that some problems in number theory and, in particular, in Diophantine approximation, can be solved using techniques from the theory of homogeneous dynamics. We undertake this as the theme of the talk. We shall first give a broad overview of the subject and demonstrate how some Diophantine problems can be reformulated in terms of orbit properties under certain flows in the space of lattices, following the ideas of G. A. Margulis, S. G. Dani, D. Y. Kleinbock and G. A. Margulis, E.Lindenstrauss. Subsequently, we will discuss the joint work of the speaker with Victor Beresnevich, Anish Ghosh and Sanju Velani on Inhomogeneous dual approximation on affine subspaces.

Abstract:

Randomized experiments have long been considered to be a gold standard for causal inference. The classical analysis of randomized experiments was developed under simplifying assumptions such as homogeneous treatment effects and no treatment interference leading to theoretical guarantees about the estimators of causal effects. In modern settings where experiments are commonly run on online networks (such as Facebook) or when studying naturally networked phenomena (such as vaccine efficacy) standard randomization schemes do not exhibit the same theoretical properties. To address these issues we develop a randomization scheme that is able to take into account violations of the no interference and no homophily assumptions. Under this scheme, we demonstrate the existence of unbiased estimators with bounded variance. We also provide a simplified and much more computationally tractable randomized design which leads to asymptotically consistent estimators of direct treatment effects under both dense and sparse network regimes.

Abstract:

Step-stress life testing is a popular experimental strategy, which ensures an efficient estimation of parameters from lifetime distributions in a relatively shorter period of time. In our analysis, we have used two different stress models, viz., the Cumulative Exposure and Khamis-Higgins models, with respect to the one and two-parameter exponential distributions, and the two-parameter Weibull distributions, respectively, under various censoring schemes. Both Bayesian and frequentist (wherever possible) approaches have been applied for the estimation of parameters and construction of their confidence/credible intervals. Under Bayesian analysis, estimation with order restriction on the mean lifetimes of units has been considered as well. In yet another attempt to analyze lifetime observations, we construct optimal variable acceptance sampling plans, an instrument to test the quality of manufactured items with a crucial role in the acceptance or rejection of the lot. Assuming the one-parameter exponential lifetime distribution, in presence of Type-I and Type-I hybrid censoring, we propose decision-theoretic approach based plans with a new estimator of the scale parameter. The optimal plans are obtained by minimizing the Bayes’ risk under a well-defined loss function.

Abstract:

The Cauchy dual subnormality problem asks when the Cauchy dual operator of an m-isometry is subnormal. This problem can be considered as the non-commutative analog of the fact that the reciprocal of a Bernstein function is completely monotone. We discuss this problem, its connection with a Hausdorff moment problem, its role in a problem posed by N. Salinas dating back to 1988, and some instances in which this problem can be solved. This is a joint work with A. Anand, Z. Jablonski, and J. Stochel.

Abstract:

One of the central questions in representation theory of finite groups is to describe the irreducible characters of finite groups of Lie type, namely matrix groups over finite fields. The theory of character sheaves, initiated by Lusztig, is a geometric approach to this problem. I will give a brief overview of this theory.

Abstract:

Consider a non-parametric regression model y = μ(x) + ε, where y is the response variable, x is the scalar covariate, ε is the error, and μ is the unknown non-parametric regression function. For this model, we propose a new graphical device to check whether v-th (v ≥ 1) derivative of the regression function μ is positive or not, which includes checking for monotonicity and convexity as special cases. An example is also presented that demonstrates the practical utility of the graphical device. Moreover, this graphical device is employed to formulate a class of test statistics to test the aforementioned assertion. The asymptotic distribution of the test statistics are derived, and the tests are implemented on various simulated and real data.

Abstract:

Mathematical modelling of complex ecological interactions is a central goal of research in mathematical ecology. A wide variety of mathematical models are proposed and relevant dynamical analysis is performed to understand the complex interaction among various trophic levels. Existing models are modified as well to remove their ecological discrepancies. Main objective of this talk is to provide a overview of current research trend in mathematical ecology and how dynamical complexities among interacting populations can be captured and analyzed with the help of mathematical models involving ODEs, PDEs, DDEs, SDEs and their combinations.

Abstract:

In this talk, we study the configuration of systoles (minimum length geodesics) on closed hyperbolic surfaces. The set of all systoles forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface, we call these admissible.There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first result characterises admissibility.It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, to the classical result that there are only two minimal non-planar graphs).

Abstract:

Lag windows are commonly used in the time series, steady state simulation, and Markov chain Monte Carlo (MCMC) literature to estimate the long range variances of ergodic averages. We propose a new lugsail lag window specifically designed for improved finite sample performance. We use this lag window for batch means and spectral variance estimators in MCMC simulations to obtain strongly consistent estimators that are biased from above in finite samples and asymptotically unbiased. This quality is particularly useful when calculating effective sample size and using sequential stopping rules where they help avoid premature termination.Further, we calculate the bias and variance of lugsail estimators and demonstrate that there is little loss compared to other estimators. We also show mean square consistency of these estimators under weak conditions. Our results hold for processes that satisfy a strong invariance principle, providing a wide range of practical applications of the lag windows outside of MCMC. Finally, we study the finite sample properties of lugsail estimators in various examples.

Abstract:

Intractable integrals appear in many areas in statistics; for example, generalized linear mixed

models and Bayesian statistics. Inference in these models relies heavily on the estimation of said integrals. In this talk, I present Monte Carlo methods for estimating intractable integrals. I introduce the accept-reject sampler and demonstrate its use on an example. Although useful, the accept-reject sampler is not effective for estimating high-dimensional integrals. To this end, I present Markov Chain Monte Carlo (MCMC) methods, like the Metropolis-Hastings sampler, which allow estimation of high-dimensional integrals. I discuss important theoretical properties of MCMC methods and some statistical challenges in its practical implementation.

Abstract:

Intractable integrals appear in many areas in statistics; for example, generalized linear mixed

models and Bayesian statistics. Inference in these models relies heavily on the estimation of said integrals. In this talk, I present Monte Carlo methods for estimating intractable integrals. I introduce the accept-reject sampler and demonstrate its use on an example. Although useful, the accept-reject sampler is not effective for estimating high-dimensional integrals. To this end, I present Markov Chain Monte Carlo (MCMC) methods, like the Metropolis-Hastings sampler, which allow estimation of high-dimensional integrals. I discuss important theoretical properties of MCMC methods and some statistical challenges in its practical implementation.

Abstract:

We talk about the mod 2 cohomology ring of the Grassmannian $\widetilde{G}_{n,3}$ of oriented 3-planes in $\mathbb{R}^n$. We first state the previously known results. Then we discuss the degrees of the indecomposable elements in the cohomology ring. We have an almost complete description of the cohomology ring. This description provides lower and upper bounds on the cup length of $\widetilde{G}_{n,3}$. This talk is based on my work with Somnath Basu.

Abstract:

How can we determine whether a mean-square continuous stochastic process is finite-dimensional, and if so, what its precise dimension is? And how can we do so at a given level of confidence? This question is central to a great deal of methods for functional data, which require low-dimensional representations whether by functional PCA or other methods. The difficulty is that the determination is to be made on the basis of iid replications of the process observed discretely and with measurement error contamination. This adds a ridge to the empirical covariance, obfuscating the underlying dimension. We build a matrix-completion inspired test statistic that circumvents this issue by measuring the best possible least square fit of the empirical covariance’s off-diagonal elements, optimised over covariances of given finite rank. For a fixed grid of sufficient size, we determine the statistic’s asymptotic null distribution as the number of replications grows. We then use it to construct a bootstrap implementation of a stepwise testing procedure controlling the family-wise error rate corresponding to the collection of hypothesis formalising the question at hand. Under minimal regularity assumptions we prove that the procedure is consistent and that its bootstrap implementation is valid. The procedure involves no tuning parameters or pre-smoothing, is indifferent to the omoskedasticity or lack of it in the measurement errors, and does not assume a low-noise regime. An extensive study of the procedure’s finite-sample accuracy demonstrates remarkable performance in both real and simulated data.

This talk is based on an ongoing work with Victor Panaretos (EPFL, Switzerland).

Abstract:

Virtual element methods(VEM) is a recently developed technology considered as generalization of FEM hav- ing firm mathematical foundations, simplicity in implementation, efficiency and accuracy in computations. Unlike FEM which allows element like triangle and quadrilateral only, VEM permits very general shaped polygons including smoothed voronoi, random voronoi, distorted polygons, nonconvex elements. Basis func- tions in VEM are constructed virtually and can be computed from the informations provided by degrees of freedom(DoF) associated with the VEM space.

Moreover, basis functions are solution of some PDEs which determine the dimension of VEM space. Furthermore, we have two projection operators on VEM space; orthogonal L2projection operator Π0k,K and elliptic projection operator Π∇k,K.Both operators are definedlocally element-wish on K ∈ Th , where Th and K denote mesh partition, polygon respectively and project basis function to computable polynomial subspace sitting inside VEM space. Basically, in abstract VEMformulation, we split the bilinear form into two parts polynomial part and non-polynomial part or stabi- lization part. Polynomial part can be computed directly from degrees of freedom and non-polynomial part can be approximated from DoF ensuring same scaling as polynomial part. However, the above mentioned framework will not work in case of non-linear problems. The primary reason is that term involving nonlinearfunction e.g. (f (u)∇u • ∇v)K can not be split into polynomial and non-polynomial parts. Hence discreteform will not be computable from DoF. In view of this difficulty, we introduce a graceful idea of employingorthogonal projection operator Π0k,Kin order to discretise nonlinear term. Exploiting this technique, weencounter semi-linear parabolic and hyperbolic problems ensuring optimal order of convergence in L2 and H 1 norms. However, we assert that this technique can be employed to discretize general nonlinear type of problem.

Abstract:

Main objects of study in model theory are definable subsets of structures. For example, the definable sets in the field of complex numbers are precisely the (boolean combinations of) varieties. The (model-theoretic) Grothendieck ring of a structure aims to classify definable sets up to definable bijections. The Grothendieck ring of varieties is central to the study of motivic integration, but its computation is a wide open problem in the area. The problem of classification is simplified to a large extent if the theory of the structure admits some form of elimination of quantifiers, i.e., the complexity of the formulas describing the definable sets is in control.

In this ``double talk'', we will begin by describing the construction of the Grothendieck ring and by giving a survey of the known Grothendieck rings. Then we will present the results and techniques used to compute the Grothendieck ring in the case of dense linear orders (joint with A. Jain) and atomless boolean algebras. On our way we will also state the ``implicit function theorem’’ for boolean varieties.

Abstract:

We begin by presenting a spectral characterization theorem that settles Chevreau's problem of characterizing the class of absolutely norming operators --- operators that attain their norm on every closed subspace. We next extend the concept of absolutely norming operators to several particular (symmetric) norms and characterize these sets. In particular, we single out three (families of) norms on B(H,K): the ``Ky Fan k-norm(s)", ``the weighted Ky Fan (\pi, k)-norm(s)", and the ``(p, k)-singular norm(s)", and thereafter define and characterize the set of absolutely norming operators with respect to each of these three norms.

We then restrict our attention to the algebra B(H) of operators on a separable infinite-dimensional Hilbert space H and use the theory of symmetrically normed ideals to extend the concept of norming and absolutely norming from the usual operator norm to arbitrary symmetric norms on B(H). In addition, we exhibit the analysis of these concepts and present a constructive method to produce symmetric norm(s) on B(H) with respect to each of which the identity operator does not attain its norm.

Finally, we introduce the notion of "universally symmetric norming operators" and "universally absolutely symmetric norming operators" and characterize these classes. These refer to the operators that are, respectively, norming and absolutely norming, with respect to every symmetric norm on B(H).

In effect, we show that an operator in B(H) is universally symmetric norming if and only if it is universally absolutely symmetric norming, which in turn is possible if and only if it is compact. In particular, this result provides an alternative characterization theorem for compact operators on a separable Hilbert space.

Abstract:

In this talk, we address the question of identifying commutant and reflexivity of the multiplication d-tuple M_z on a reproducing kernel Hilbert space H of E-valued holomorphic functions on Ω, where E is a separable Hilbert space and Ω is a bounded domain in C^d admitting bounded approximation by polynomials. In case E is a finite dimensional cyclic subspace for M_z, under some natural conditions on the B(E)-valued kernel associated with H , the commutant of M_z is shown to be the algebra H^{∞}_ B(E)(Ω) of bounded holomorphic B(E)-valued functions on Ω, provided M_z satisfies the matrix-valued von Neumann’s inequality. Also, we show that a multiplication d-tuple M_z on H satisfying the von Neumann’s inequality is reflexive. The talk is based on joint work with Sameer Chavan and Shailesh Trivedi.

Abstract:

Although we can trace back the study of epidemics to the work of Daniel Bernoulli nearly two and a half centuries ago, the fact remains that key modeling advances followed the work of three individuals (two physicians) involved in the amelioration of the impact of disease at the population level a century or so ago: Sir Ronald Ross (1911) and Kermack and McKendrick (1927). Ross' interests were in the transmission dynamics and control of malaria while Kermack and McKendrick's work was directly tied in to the study of the dynamics of communicable diseases. In this presentation, I will deal primarily with the study of the dynamics of influenza type A, a communicable disease that does not present a “fixed” target. The study of the short-term dynamics of influenza, single epidemic outbreaks, makes use of extensions/modifications of the models first introduced by Kermack and McKendrick while the study of its long-term dynamics requires the introduction of modeling modifications that account for the continuous emergence of novel influenza variants: strains or subtypes. Here, I will briefly review recent work on the dynamics of influenza A/H1N1, making use of single outbreak models that account for the movement of people in the transmission process over various regions within Mexico. This research has been carried in collaboration with a large number of researchers over a couple of decades.

From a theoretical perspective, I will observe that over the past 100 years modeling epidemic processes have been based primarily on the use of the mass action law. What have we learned from this approach and what are the limitations? In this lecture, I will revisit old and “new” modeling approaches in the context of the dynamics of vector borne, sexually-transmitted and communicable diseases.

Abstract:

Let F(x; y) that belongs to Z[x; y] be a homogeneous and irreducible with degree 3. Consider F(x; y) = h for some fixed nonzero integer h. In 1909, Thue proved that this has only finitely many integral solutions. These eponymous equations have several applications. Much effort has been made to obtain upper bounds for the number of solutions of Thue equations which are independent of the size of the coefficients of F. Siegel conjectured that the number of solutions could be bounded only in terms of h and the number of non-zero coefficients of F. This was settled in the affirmative by Mueller and Schmidt. However, their bound doesn’t have the desired shape. In this talk, we present some instances when their result can be improved. This is joint work with N. Saradha.

Abstract:

Suppose p(t)=X_0+X_1t+...+X_n t^n is a polynomial where each X_i is randomly chosen to be +1 or -1. How many real roots does the polynomial have, on average? Turns out that the answer is of order \log(n). More generally, given a subset of the complex plane, how many roots are in the given subset (on average, say)? Turns out that the roots are almost all close to the unit circle, and distributed roughly uniformly in angle. We survey basic results answering these questions. The talk is aimed to be accessible to advanced undergraduate and graduate students.

Abstract:

Let F(x; y) that belongs to Z[x; y] be a homogeneous and irreducible with degree 3. Consider F(x; y) = h for some fixed nonzero integer h. In 1909, Thue proved that this has only finitely many integral solutions. These eponymous equations have several applications. Much effort has been made to obtain upper bounds for the number of solutions of Thue equations which are independent of the size of the coefficients of F. Siegel conjectured that the number of solutions could be bounded only in terms of h and the number of non-zero coefficients of F. This was settled in the affirmative by Mueller and Schmidt. However, their bound doesn’t have the desired shape. In this talk, we present some instances when their result can be improved. This is joint work with N. Saradha.

Abstract:

Penalized regression techniques are widely used in practice to perform variable selection (also known as model selection). Variable selection is important to drop the covariates from the regression model, which are irrelevant in explaining the response variable. When the number of covariates is large compared to the sample size, variable selection is indeed the most important requirement of the penalized method. Fan and Li(2001) introduced the Oracle Property as a measure of how good a penalized method is. A penalized method is said to have the oracle property provided it works as well as if the correct sub-model were known (like the Oracle who knows everything beforehand). We categorize different penalized regression methods with respect to oracle property and show that bootstrap works for each category. Moreover, we show that in most of the situations, the inference based on bootstrap is much more accurate than the oracle based inference.

Abstract:

The Adaptive Lasso (Alasso) was proposed by Zou (2006) as a modification of the Lasso for the purpose of simultaneous variable selection and estimation of the parameters in a linear regression model. Zou (2006) established that the Alasso estimator is variable-selection consistent as well as asymptotically Normal in the indices corresponding to the nonzero regression coefficients in certain fixed-dimensional settings. Minnier et al. (2011) proposed a perturbation bootstrap method and established its distributional consistency for the Alasso estimator in the fixed-dimensional setting. In this paper, however, we show that this (naive) perturbation bootstrap fails to achieve the desired second order correctness [i.e. with uniform error rate o(n^{-1/2})] in approximating the distribution of the Alasso estimator. We propose a modification to the perturbation bootstrap objective function and show that a suitably studentized version of our modified perturbation bootstrap Alasso estimator achieves second-order correctness even when the dimension of the model is allowed to grow to infinity with the sample size. As a consequence, inferences based on the modified perturbation bootstrap is more accurate than the inferences based on the oracle Normal approximation. Simulation results also justifies our method in finite samples.

Abstract:

Let g be a simple finite dimensional Lie algebra. Let A be the Laurent polynomial algebra in n + 1 commuting variables. Then g ⊗ A is naturally a Lie algebra. We now consider the universal central extension g ⊗ A ⊕ ΩA/dA. Then we add derivations of A, Der(A) and consider τ = g ⊗ A ⊕ ΩA/dA ⊕ Der(A). τ is called full toroidal Lie algebra. In this talk, we will explain the classification of irreducible integrable modules for the full toroidal Lie algebra. In the first half of the lecture, we will recall some general facts of toroidal Lie algebras and then we will go on into the technical part.

Abstract:

We briefly discuss the concept of quantum symmetry and mention how it fits into the realm of noncommutative geometry. We take a particular noncommutative topological space coming from connected, directed graph which is called graph C*-algebra and introduce a notion of quantum symmetry of such noncommutative space. A few concrete examples of such quantum symmetry will also be discussed.

Abstract:

Let Gq be the q-deformation of a simply connected simple compact Lie group G of type A, C or D and Oq(G) be the algebra of regular functions on Gq. In this talk, we show that the Gelfand-Kirillov dimension of Oq(G) is equal to the dimension of the underlying real manifold G. If time allows then we will discuss some applications of this result.

Abstract:

Cryptographic protocols base their security on the hardness of mathematical problems. Discrete Logarithm Problem (DLP) is one of them. It is known to be computationally hard in the groups of cryptographic interest. Most important among them are the multiplicative subgroup of finite fields, group of points on an elliptic curve and the group of divisor classes of degree 0 divisors (Jacobian) on a hyperelliptic curve.

Abstract:

Let g be a simple finite dimensional Lie algebra. Let A be the Laurent polynomial algebra in n + 1 commuting variables. Then g ⊗ A is naturally a Lie algebra. We now consider the universal central extension g ⊗ A ⊕ ΩA/dA. Then we add derivations of A, Der(A) and consider τ = g ⊗ A ⊕ ΩA/dA ⊕ Der(A). τ is called full toroidal Lie algebra. In this talk, we will explain the classification of irreducible integrable modules for the full toroidal Lie algebra. In the first half of the lecture, we will recall some general facts of toroidal Lie algebras and then we will go on into the technical part.
In this talk, I will discuss some of the index calculus algorithms for solving the discrete logarithm problem on these groups. More specifically, I will discuss the tower number field sieve algorithm (TNFS) for solving the discrete logarithm problem in the medium to large characteristic finite fields.

Abstract:

Multivariate two-sample testing is a very classical problem in statistics, and several methods are available for it. But, in the current era of big data and high-dimensional data, most of the existing methods fail to perform well, and they cannot even be used when the dimension of the data exceeds the sample size. In this talk, I will propose and investigate some methods based on inter-point distances, which can be conveniently used for data of arbitrary dimensions. I will discuss the merits and demerits of these methods using theoretical as well as numerical results.

Abstract:

Let F be a non-Archimedean local field F with ring of integers O and a fi nite residue fi eld k of odd characteristic. In contrast to the well-understood representation theory of the fi nite groups of Lie type GL_n(k) or of the locally compact groups GL_n(F), representations of groups GL_n(O) are considerably less understood. For example, the uniqueness of Whittaker model is well known for the complex representations of both GL_n(k) and GL_n(F) but is not known for GL_n(O).

Abstract:

In this talk we will see two possible generalizations, due to A. Connes and Frohlich et al., of the de-Rham calculus on manifolds to the noncommutative geometric context. Computations of both these will be highlighted for a class of examples provided by the quantum double suspension, which helps to compare these two generalizations in a very precise sense.

Abstract:

In this presentation, we analyze a semi-discrete finite difference scheme for a stochastic balance laws driven by multiplicative L´evy noise. By using BV estimate of approximate solutions, generated by finite difference Scheme, and Young measure technique in stochastic setting, we show that the approximate solutions converges to the unique BV entropy solution of the underlying problem. Moreover, we show that the expected value of the L^1-difference between approximate solutions and the unique entropy solution converges at a rate O(√∆x), where ∆x being a spatial mesh size.

Abstract:

We will give an overview of some techniques involved in computing the mod p reduction of p-adic Galois representations associated to certain cusp forms of GL(2).

Abstract:

We study the congruences between Galois representations and their base-change along a p-adic Lie extension. We formulate a Main conjecture arising out of these Galois representations which explains how the congruences are related to values of L-functions. This formulation requires the Galois representations to satisfy some conditions and we provide some examples where the conditions can be verified.

Abstract:

The Bellman and Isaac equations appear as the dynamic programming equations for stochastic control and differential games. This talk is concerned with the error estimates for monotone numerical approximations with the viscosity solutions of such equations. I will be focusing on both local and non-local scenario. I will discuss the most general results for equations of order less than one (non-local) and first order (local) equations. For second order equations, convex and non-convex cases (partial results!) will be treated separately. I will explain why the methods are quite different for convex and non-convex cases. To the end, I will discuss the recent developments on error estimates for non-local Isaac equations of order greater than one.

Abstract:

Transport phenomenon is of fundamental as well as practical importance in a wide spectrum of problems of different length and time scales, viz., enhanced oil recovery (EOR), carbon-capture and storage (CCS), contaminant transport in subsurface aquifers, and chromatographic separation. These transport processes in porous media feature different hydrodynamic instabilities [1, 2]. Viscous

Abstract:

This talk will present a class of tests for fitting a parametric model to the regression function in the presence of Berkson measurement error in the covariates without specifying the measurement error density but when validation data is available. The availability of validation data makes it possible to estimate the calibrated regression function non-parametrically. The proposed tests are based on a class of minimized integrated square distances between a nonparametric estimate of the calibrated regression function and the parametric null model being fitted. The asymptotic distributions of these tests under the null hypothesis and against certain alternatives are established. Surprisingly, these asymptotic distributions are the same as in the case of known measurement error density. In comparison, the asymptotic distributions of the corresponding minimum distance estimators of the null model parameters are affected by the estimation of the calibrated regression function. A simulation study shows desirable performance of a member of the proposed class of estimators and tests. This is co-authored with Pei Geng.

Abstract:

Residual empirical processes are known to play a central role in the development of statistical inference in numerous additive models. This talk will discuss some history and some recent advances in the Asymptotic uniform linearity of parametric and nonparametric residual empirical processes. We shall also discuss their usefulness in developing asymptotically distribution free goodness-of-fit tests for fitting an error distribution functions in nonparametric ARCH(1) models. Part of this talk is based on some joint work with Xiaoqing Zhu.

Abstract:

Passive transport models are equations of advection-diffusion type. In most of the applications involving passive transport, the advective fields are of greater magnitude compared to molecular diffusion.
This talk attempts to present a novel theory developed by myself, Thomas Holding (Imperial) and Jeffrey Rauch (Michigan) to address these strong advection problems. Loosely speaking, our strategy is to recast the advection-diffusion equation in moving coordinates dictated by the flow associated with the advective field. Crucial to our analysis is the introduction of a fast time variable and the introduction of some new notions of weak convergence along flows in Lp spaces. We also use ideas from the theory of “homogenization structures” developed by Gabriel Nguetseng.
Our asymptotic results show the following dichotomy:

• If the Jacobian matrix associated with the flow satisfies certain structural conditions (loosely speaking, boundedness in the fast time variable) then the strong advection limit is a non-degenerate diffusion when seen along flows.

• On the other hand, when the Jacobian matrix associated with the flow fails to satisfy the aforementioned structural conditions, then the strong advection limit is a parabolic problem with a constraint. Here we show the appearance of an initial layer where there is an enhanced dissipation along flows.

Our results have close links to

• the Freidlin-Wentzell theory on perturbations of dynamical systems.

• the theory of Relaxation enhancing Lipschitz flows.

This talk will illustrate the theoretical results via various interesting examples. We address some well-known advective fields such as the Euclidean motions, the Taylor-Green cellular flows, the cat’s eye flows and some special class of the Arnold-Beltrami-Childress (ABC) flows. We will also comment on certain examples of hyperbolic or Anosov flows. Some of the results to be presented in this talk can be found in the following

Publication:

T. Holding, H. Hutridurga, J. Rauch. Convergence along mean flows, SIAM J Math.
Anal., Volume 49, Issue 1, pp. 222–271 (2017).

Abstract:

In the problem of selecting a linear model to approximate the true unknown regression model, some necessary and/or sufficient conditions will be discussed for the asymptotic validity of various model selection procedures including Akaike’s AIC, Mallows’ Cp, Schwarz’ BIC, generalized AIC, etc. We shall see that these selection procedures can be classified into three distinct classes according to their asymptotic behaviour. Under some fairly weak conditions, the selection procedures in one class are asymptotically valid if there exists fixed dimensional correct models; while the selection procedures in another class are asymptotically valid if no fixed dimensional correct model exists. On the other hand, the procedures in the third class are compromises of the procedures in the first two classes.

Abstract:

We consider the problem of computationally-efficient prediction from high-dimensional and highly correlated predictors in challenging settings where accurate variable selection is effectively impossible. Direct application of penalization or Bayesian methods implemented with Markov chain Monte Carlo can be computationally daunting and unstable. Hence, some type of dimensionality reduction prior to statistical analysis is in order. Common solutions include application of screening algorithms to reduce the regressors, or dimension reduction using projections of the design matrix. The former approach can be highly sensitive to threshold choice in finite samples, while the later can have poor performance in very high-dimensional settings. We propose a Targeted Random Projection (TARP) approach that combines positive aspects of both the strategies to boost performance. In particular, we propose to use information from independent screening to order the inclusion probabilities of the features in the projection matrix used for dimension reduction, leading to data-informed sparsity. Theoretical results on the predictive accuracy of TARP is discussed in detail along with the rate of computational complexity. Simulated data examples, and real data applications are given to illustrate gains relative to a variety of competitors.

Abstract:

Let G be a simple algebraic group over the field of complex numbers and B be a Borel subgroup of G. Let X_w be a Schubert variety in the flag variety G/B corresponding to an element w of the Weyl group of G, and let Z_w be the Bott–Samelson variety, a natural desingularization of X_w. In this talk we discuss the classification of the "reduced expressions of w" such that Z_w is Fano or weak Fano.

Abstract:

GATE is a big exam used for PG admissions by academic institutes as well as hiring by PSUs. In 2015, more than 8 lakh people appeared for GATE, all subjects combined.

GATE uses formula scoring with negative marking for multiple choice questions or MCQs, e.g., 1, 0, and -1/3 marks for correct, omitted, and wrong answer, respectively. Some questions have 2 marks, with -2/3 for wrong answers. Some have numerical answers, with no negative marking. The number of distinct scores possible is small (below 400), and the number of candidates is large (lakhs).

A modern statistical approach to evaluating MCQ exams uses item response theory (IRT; also called latent trait models). In this approach, each question has some parameters, called “difficulty” or “discrimination” etc., written abstractly as vector a, and each student has an ability or talent attribute, written abstractly as a scalar theta. The probability that a given question (with vector a) will be answered correctly is taken to be some specified function f(theta,a).

The definition of a, and choice of f, are modeling decisions. Later calculations, though complex, are routine. The aim is to estimate a for every question and theta for every candidate.

Two common IRT models are the Rasch model and the 2-parameter logistic (2PL) model, which I will describe. Since our question outcomes are not dichotomous (right/wrong) but polytomous (right/ omitted/wrong), we use the generalized partial credit model (GPCM), which I will describe. GPCM results are poor. The estimated abilities have low correlations with formula scores; these correlations vary across disciplines; and there are also clear conceptual problems in applying GPCM to GATE.

I will then present our new two-step IRT model, where the candidate first decides whether or not to attempt the question, and then (if attempting) gets it right or wrong. The corresponding mathematical model is simple, and aligned with how we believe GATE works. Results are better. The correlation with formula scores is higher, and near-constant across disciplines.

The policy implications of our model are positive. We now have a two-dimensional score of each candidate’s performance. The formula score represents an overall knowledge score, which may appeal to industry. The IRT ability estimate represents an academic potential estimate, which may appeal to academic institutes. If admission and hiring processes are no longer based on the same measure, both may benefit. A minor extra advantage is the possibility of awarding rationally derived ranks to individual candidates, with very few clashes.

As part of the talk, I will also discuss the estimation methods and numerical implementation. However, the emphasis will be on model statement, results, and policy implications. I hope that most of the talk will be accessible to all stakeholders in GATE.

 Abstract:

Henry-Parusinski proved that the Lipschitz right classification of function germs admits continuous moduli. This allows us to introduce the notion of Lipschitz simple germs and list all such germs. We will present the method of the classification in this talk and mention some open problems.

 Abstract:

In this talk, the likelihood construction is explained under different censoring schemes. Further, the techniques for estimation of the unknown parameters of the survival model are discussed under these censoring schemes.

 Abstract:

In this talk, the change point problem in hazard rate is considered. The Lindley hazard change point model is discussed with its application to model bone marrow transplant data. Further, a general hazard regression change point model is discussed with exponential and Weibull distribution as special cases.

 Abstract:

A {\em simplicial cell complex} K of dimension d is a poset isomorphic to the face poset of a d-dimensional simplicial CW-complex X. If a topological space M is homeomorphic to X, then K is said to be a {\em pseudotriangulation} of M. In 1974, Pezzana proved that every connected closed PL d-manifold admits a (d+1)-vertex pseudotriangulation. For such a pseudotriangulation of a PL d-manifold one can associate a (d+1)-regular colored graph, called a crystallization of the manifold.
Actually, crystallization is a graph-theoretical tool to study topological and combinatorial properties of PL manifolds. In this talk, I shall define crystallization and show some applications on PL d-Manifolds for d=2, 3 and 4.

In dimension 2, I shall show a proof of the classification of closed surfaces using crystallization. This concept has some important higher dimensional analogs, especially in dimensions 3 and 4. In dimensions 3 and 4, I shall give lower bounds for facets in a pseudotriangulation of a PL manifolds. Also, I shall talk on the regular genus (a higher dimensional analog of genus) of PL d-manifolds. Then I shall show the importance of the regular genus in dimension 4. Additivity of regular genus has been proved for a huge class of PL 4-manifolds. We have some observations on the regular genus, which is related to the 4-dimensional Smooth Poincare Conjecture.

 Abstract:

Rayleigh-Bénard convection is a classical extended dissipative system which shows a plethora of bifurcations and patterns. In this talk, I'll present the results of our investigation on bifurcations and patterns near the onset of Rayleigh-Bénard convection of low-Prandtl number fluids. Investigation is done by performing direct numerical simulations (DNS) of the governing equations. Low dimensional modeling of the system using the DNS data is also done to understand the origin of different flow patterns. Our investigation reveals a rich variety of bifurcation structures involving pitchfork, Hopf, homoclinic and Neimar-Sacker bifurcations.

 Abstract:

In this talk, we will discuss some recent results on the existence and uniqueness of strong solutions of certain classes of stochastic PDEs in the space of Tempered distributions. We show that these solutions can be constructed from the solutions of "related" finite dimensional stochastic differential equations driven by the same Brownian motion. We will also discuss a criterion, called the Monotonicity inequality, which implies the uniqueness of strong solutions.

 Abstract:

The investigation of solute dispersion is most interesting topic of research owing to its outspread applications in various fields such as biomedical engineering, physiological fluid dynamics, etc. The aim of the present study is to know the different physiological processes involved in the solute dispersion in blood flow by assuming the relevant non-Newtonian fluid models. The axial solute dispersion process in steady/unsteady non-Newtonian fluid flow in a straight tube is analyzed in the presence and absence of absorption at the tube wall. The pulsatile nature of the blood is considered for unsteady flow. Owing to non-Newtonian nature of blood at low shear rate in small vessels, non-Newtonian Casson, Herschel-Bulkley, Carreau and Carreau-Yasuda fluid models which are most relevant for blood flow analysis are considered. The three transport coefficients i.e., exchange, convection and dispersion coefficients which describe the whole dispersion process in the system are determined. The mean concentration of solute is analyzed at all time. A comparative study of the solute dispersion is made among the Newtonian and non-Newtonian fluid models. Also, the comparison of solute dispersion between single- and two-phase models is made at all time for different radius of micro blood vessels.

 Abstract:

Quasilinear symmetric and symmetrizable hyperbolic system has a wide range of applications in engineering and physics including unsteady Euler and potential equations of gas dynamics, inviscid magnetohydrodynamic (MHD) equations, shallow water equations, non-Newtonian fluid dynamics, and Einstein field equations of general relativity. In the past, the Cauchy problem of smooth solutions for these systems has been studied by several mathematicians using semigroup approach and fixed point arguments. In a recent work of M. T. Mohan and S. S. Sritharan, the local solvability of symmetric hyperbolic system is established using two different methods, viz. local monotonicity method and a frequency truncation method. The local existence and uniqueness of solutions of symmetrizable hyperbolic system is also proved by them using a frequency truncation method. Later they established the local solvability of the stochastic quasilinear symmetric hyperbolic system perturbed by Levy noise using a stochastic generalization of the localized Minty-Browder technique. Under a smallness assumption on the initial data, a global solvability for the multiplicative noise case is also proved. The essence of this talk is to give an overview of these new local solvability methods and their applications.

 Abstract:

In this talk, we present the moduli problem of rank 2 torsion free Hitchin pairs of fixed Euler characteristic χ on a reducible nodal curve. We describe the moduli space of the Hitchin pairs. We define the analogue of the classical Hitchin map and describe the geometry of general Hitchin fibre. Time permits,talk on collaborated work with Balaji and Nagaraj on degeneration of moduli space of Hitchin pairs.

 Abstract:

The formalism of an ``abelian category'' is meant to axiomatize the operations of linear algebra. From there, the notion of ``derived category'' as the category of complexes ``upto quasi-isomorphisms'' is natural, motivated in part by topology. The formalism of t-structures allows one to construct new abelian categories which are quite useful in practice (giving rise to new cohomology theories like intersection cohomology, for example).
In this talk we want to discuss a notion of punctual (=``point-wise'') gluing of t-structures which we formulated in the context of algebraic geometry. The essence of the construction is classical and well known, but the new language leads to several applications in the motivic world.

 Abstract:

Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, product of elementary Abelian groups and dihedral groups.
Restricting them to finite skeleta constructs equivariant maps between spaces which are related to the topological Tverberg conjecture. This answers negatively a question of \"Ozaydin posed in relation to weaker versions of the same conjecture. Further, it also has consequences for Borsuk-Ulam properties of representations of cyclic and dihedral groups.
This is joint work with Samik Basu.

 Abstract:

 Abstract:

Quantile regression provides a more comprehensive relationship between a response and covariates of interest compared to mean regression. When the response is subject to censoring, estimating conditional mean requires strong distributional assumptions on the error whereas (most) conditional quantiles can be estimated distribution-free. Although conceptually appealing, quantile regression for censored data is challenging due to computational and theoretical difficulties arising from non-convexity of the objective function involved. We consider a working likelihood based on Powell's objective function and place appropriate priors on the regression parameters in a Bayesian framework. In spite of the non-convexity and misspecification issues, we show that the posterior distribution is strong selection consistent. We provide a “Skinny” Gibbs algorithm that can be used to sample the posterior distribution with complexity linear in the number of variables and provide empirical evidence demonstrating the fine performance of our approach.

 Abstract:

 Abstract:

Given a closed smooth Riemannian manifold M, the Laplace operator is known to possess a discrete spectrum of eigenvalues going to infinity. We are interested in the properties of the nodal sets and nodal domains of corresponding eigen functions in the high energy (semiclassical) limit. We focus on some recent results on the size of nodal domains and tubular neighbourhoods of nodal sets of such high energy eigenfunctions (joint work with Bogdan Georgiev).

 Abstract:

I will define affine Kac-Moody algebras, toroidal Lie algebras and full toroidal Lie algebras twisted by several finite order automorphisms and classify integrable representations of twisted full toroidal Lie algebras.

 Abstract:

In recent years, one major focus of modeling spatial data has been to connect two contrasting approaches, namely, the Markov random field approach and the geostatistical approach. While the geostatistical approach allows flexible modeling of the spatial processes and can accommodate continuum spatial variation, it faces formidable computational burden for large spatial data. On the other hand, spatial Markov random fields facilitate fast statistical computations but they lack in flexibly accommodating continuum spatial variations. In this talk, I will discuss novel statistical models and methods which allow us to accommodate continuum spatial variation as well as fast matrix-free statistical computations for large spatial data. I will discuss an h-likelihood method for REML estimation and I will show that the standard errors of these estimates attain their Rao-Cramer lower bound and thus are statistically efficient. I will discuss applications on ground-water Arsenic contamination and chlorophyll concentration in ocean. This is a joint work with Debashis Mondal at Oregon State University

 Abstract:

It is a well-known result from Hermann Weyl that if alpha is an irrational number in [0,1) then the number of visits of successive multiples of alpha modulo one in an interval contained in [0,1) is proportional to the size of the interval. In this talk we will revisit this problem, now looking at finer joint asymptotics of visits to several intervals with rational end points. We observe that the visit distribution can be modelled using random affine transformations; in the case when the irrational is quadratic we obtain a central limit theorem as well. Not much background in probability will be assumed. This is in joint work with Jon Aaronson and Michael Bromberg.

 Abstract:

The action of Gl_n(F_q) on the polynomial ring over n variables has been studied extensively by Dickson and the invariant ring can be explicitly described. However, the action of the same group on the ring when we go modulo Frobenius powers is not completely solved. I'll talk about some interesting aspects of this modified version of the problem. More specifically, I'll discuss a conjecture by Lewis, Reiner and Stanton regarding the Hilbert series corresponding to this action and try to prove some special cases of this conjecture.

 Abstract:

One way to understand representations of a group is to ‘restrict’ the representation to its various subgroups, especially to those subgroups which give multiplicity one or finite multiplicity. We shall discuss a few examples of restriction for the representations of p-adic groups. Our main examples will be the pairs $(GL_2(F), E^*) and (GL_2(E), GL_2(F))$, where $E/F$ is a quadratic extension of $p$-adic fields. These examples can be considered as low-rank cases of the well known Gross-Prasad conjectures, where one considers various ‘restrictions’ simultaneously. Further, we consider a similar ‘restriction problem’ when the groups under consideration are certain central extensions of $F$-point of a linear algebraic groups by a finite cyclic group. These are topological central extensions and called ‘covering groups’ or ‘metaplectic groups’. These covering groups are not $F$-point of any linear algebraic group. We restrict ourselves to only a two fold covers of these groups and their ‘genuine’ representations. Covering groups naturally arise in the study of modular form of half-integral weight. Some results that we will discuss are outcome of a joint work with D. Prasad.

 Abstract:

 Abstract:

It is a well-accepted practice in experimental situations to use auxiliary information to enhance the accuracy of the experiment i.e., to reduce the experimental error. In its simplest form of use of auxiliary information, data generated through an experiment are statistically modeled in terms of some assignable source(s) of variation, besides a chance cause of variation. The assignable causes comprise ‘treatment’ parameters and the ‘covariate’ parameter(s). This generates a family of ‘covariate models’ - serving as a ‘combination’ of ‘varietal design models’ and ‘regression models’. These are the well-known Analysis of Covariance
(ANCOVA) Models. Generally, for such models, emphasis is given on analysis of the data [in terms of inference on treatment effects contrasts] and not so much on the choice of the covariate(s) values. In this presentation, we consider the situation where there is some flexibility in the choice of the experimental units with specified values of the covariates. The notion of 'optimal' choice of values of the covariates for a given design set-up so as to minimize variance for parameter estimates has attracted attention of researchers in recent times. Hadamard matrices and Mixed orthogonal array have been conveniently used to construct optimum covariate designs with as many covariates as possible.

 Abstract:

 Abstract:

Recent increase in the use of 3-D magnetic resonance images (MRI) and analysis of functional magnetic resonance images (fMRI) in medical diagnostics makes imaging, especially 3-D imaging very important. Observed images often contain noise which should be removed in such a way that important image features, e.g., edges, edge structures, and other image details should be preserved, so that subsequent image analyses are reliable. Direct generalizations of existing 2-D image denoising techniques to 3-D images cannot preserve complicated edge structures well, because, the edge structures in a 3-D edge surface can be much more complicated than the edge structures in a 2-D edge curve. Moreover, the amount of smoothing should be determined locally, depending on local image features and local signal to noise ratio, which is much more challenging in 3-D images due to large number of voxels. In this talk, I will talk about a multi-resolution and locally adaptive 3-D image denoising procedure based on local clustering of the voxels. I will provide a few numerical studies which show that the denoising method can work well in many real world applications. Finally, I will talk about a few future research directions along with some introductory research problems for interested students. Most parts of my talk should be accessible to the audience of diverse academic background.

 Abstract:

We shall discuss a new method of computing (integral) homotopy groups of certain manifolds in terms of the homotopy groups of spheres. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we verify a conjecture of J. C.
Moore. This is joint work with Somnath Basu.

 Abstract:

The theory of pseudo-differential operators provides a flexible tool for treating certain problems in linear partial differential equations. The Gohberg lemma on unit circle estimates the distance (in norm) from a given zero-order operator to the set of the compact operators from below in terms of the symbol. In this talk, I will introduce a version of the Gohberg lemma on compact Lie groups using the global calculus of pseudo-differential operators. Applying this, I will obtain the bounds for the essential spectrum and a criterion for an operator to be compact. The conditions used will be given in terms of the matrix-valued symbols of operators

 Abstract:

Let $F$ be a $p$-adic field. The restriction of an irreducible admissible representation of $GL_{2}(F)$ to its maximal tori was studied by Tunnell and Saito; and they provide a very precise answer. In particular, one gets multiplicity one. This can be considered as the first case of the Gross-Prasad conjectures.

We will discuss a metaplectic variation of this question. More precisely, we will talk about the restriction of an irreducible admissible genuine representation of the two fold metaplectic cover $\widetilde{GL}_2(F)$ of $GL_2(F)$ to the inverse image in $\widetilde{GL}_2(F)$ of a maximal torus in $GL_2(F)$. We utilize a correspondence between irreducible admissible genuine supercuspidal representations of the metaplectic group widetilde{SL}_2(F)$ to irreducible admissible supercuspidal representations of linear group $SL_2(F)$. This is a joint work with D. Prasad.

 Abstract:

We will describe the problem of mod p reduction of p-adic Galois representations. For two dimensional crystalline representations of the local Galois group Gal(Q¯ p |Qp ), the reduction can be computed using the compatibility of p-adic and mod p Local Langlands Correspondences; this method was first introduced by Christophe Breuil in 2003. After giving a brief sketch of the history of the problem, we will discuss how the reductions behave for representations with slopes in the half- open interval [1, 2). In the relevant cases of reducible reduction, one may also ask if the reduction is peu or tr`es ramifi´ee. We will try to sketch an answer to this question, if time permits. (Joint works with Eknath Ghate, and also with Sandra Rozensztajn for slope 1.)

 Abstract:

 Abstract:

Let M be a compact manifold without boundary. Define a smooth real valued function of the space of Riemannian metrics of M by taking Lp-norm of Riemannian curvature for p >= 2. Compact irreducible locally symmetric spaces are critical metrics for this functional. I will prove that rank 1 symmetric spaces are local minima for this functional by studying stability of the functional at those metrics. I will also show examples of irreducible symmetric metrics which are not local minima for it.

 Abstract:

Unfolding operators have been introduced and used to study homogenization problems. Initially, it was introduced for problems with rapidly oscillating coefficients and porous domains. Later, this has been developed for domains with oscillating boundaries, typically with rectangular or pillar type boundaries which are classified as non-smooth.
In this talk, we will demonstrate the development of generalized unfolding operators, where the oscillations of the domain can be smooth and hence it has wider applications. We will also see the further adaptation of this new unfolding operators for circular domains with rapid oscillations with high amplitude of O(1). This has been applied to homogenization problems in circular domains as well.

 Abstract:

We describe three approaches to the classical p-completion(or localization) of a topological space: as spaces, through cosimplicial space resolutions, and through mapping algebras – and show how they are related through appropriate "universal" systems of higher cohomology operations. All terms involved will be explained in the talk.

 Abstract:

We deal with the following eigenvalue optimization problem: Given a bounded open disk $B$ in a plane, how to place an obstacle $P$ of fixed shape and size within $B$ so as to maximize or minimize the fundamental eigenvalue $lambda_1$ of the Dirichlet Laplacian on $B setmunus P$. This means that we want to extremize the function $rho \rightarrow lambda_1(B setminus rho(P))$, where $rho$ runs over the set of rigid motions such that $rho(P) subset B$. We answer this problem in the case where $P$ is invariant under the action of a dihedral group $D_{2n}$, and where the distance from the center of the obstacle $P$ to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of $P$ coincide with a diameter of $B$. The maximizing and the minimizing configurations are identified.

 Abstract:

We deal with the following eigenvalue optimization problem: Given a bounded open disk B in a plane, how to place an obstacle P of fixed shape and size within B so as to maximize or minimize the fundamental eigenvalue
λ1 of the Dirichlet Laplacian on B \ P . This means that we want to extremize the function ρ → λ1 (B \ ρ(P)), where ρ runs over the set of rigid motions such that ρ(P) ⊂ B. We answer this problem in the case where P is invariant under the action of a dihedral group D 2n, and where the distance from the center of the obstacle P to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of P coincide with a diameter of B. The maximizing and the minimizing configurations are identified.

 Abstract:

Matched case-control studies are popular designs used in epidemiology for assessing the effects of exposures on binary traits. Modern investigations increasingly enjoy the ability to examine a large number of exposures in a comprehensive manner. However, risk factors often tend to be related in a non-trivial way, undermining efforts to identify true important ones using standard analytic methods. Epidemiologists often use data reduction techniques by grouping the prognostic factors using a thematic approach, with themes deriving from biological considerations. However, it is important to account for potential misspecification of the themes to avoid false positive findings. To this end, we propose shrinkage type estimators based on Bayesian penalization methods. Extensive simulation is reported that compares the Bayesian and Frequentist estimates under various scenarios. The methodology is illustrated using data from a matched case-control study investigating the role of polychlorinated biphenyls in understanding the etiology of non-Hodgkin's lymphoma.

 Abstract:

Cowen and Douglas have shown that the curvature is a complete invariant for a certain class of operators. Several ramifications of this result will be discussed.

 Abstract:

It is well known that the center of U(glN) is _nitely generated as an algebra. Gelfand de_ned central elements (Gelfand invariants) Tk for every positive integer k. It is known that the _rst N generates the center. The decomposition of tensor product modules for a reductive Lie algebra is a classical problem. It is known that each Gelfand invariant acts as a scaler on an irreducible submodule of a tensor product module. In this talk, for each k we de_ne several operators which commute with glN action but does not act as a scalar. This means these operators take one highest weight vector to another highest weight vector. Thus it is a practical algorithm to produce more highest weight vectors once we known one of them. Further the sum of these operators is Tk. If time permits we will de_ne some of these operators in the generality of Kac-Moody Lie algebras.

 Abstract:

Recent increase in the use of 3-D magnetic resonance images
(MRI) and analysis of functional magnetic resonance images (fMRI) in medical diagnostics makes imaging, especially 3-D imaging very important. Observed images often contain noise which should be removed in such a way that important image features, e.g., edges, edge structures, and other image details should be preserved, so that subsequent image analyses are reliable. Direct generalizations of existing 2-D image denoising techniques to 3-D images cannot preserve complicated edge structures well, because, the edge structures in a 3-D edge surface can be much more complicated than the edge structures in a 2-D edge curve. Moreover, the amount of smoothing should be determined locally, depending on local image features and local signal to noise ratio, which is much more challenging in 3-D images due to large number of voxels. In this talk, I will talk about a multi-resolution and locally adaptive 3-D image denoising procedure based on local clustering of the voxels. I will provide a few numerical studies which show that the denoising method can work well in many real world applications.
Finally, I will talk about a few future research directions along with some introductory research problems for interested students. Most parts of my talk should be accessible to the audience of diverse academic background.

 Abstract:

The theory of pseudo-di_erential operators has provided a very powerful and exible tool for treating certain problems in linear partial differential equations. The importance of the Heisenberg group in general harmonic analysis and problems involving partial di_erential operators on manifolds is well established. In this talk, I will introduce the pseudo-di_erential operators with operator-valued symbols on the Heisenberg group. I will give the necessary and su_cient conditions on the symbols for which these operators are in the Hilbert-Schmidt class. I will identify these HilbertSchmidt operators with the Weyl transforms with symbols in L2(R2n+1 _ R2n+1). I will also provide a characterization of trace class pseudo-di_erential operators on the Heisenberg group. A trace formula for these trace class operators would be presented.

 Abstract:

A finite separable extension E of a field F is called primitive if there are no intermediate extensions. It is called a solvable extension if the group of automorphisms of its galoisian closure over F is a solvable group. We show that a solvable primitive extension E of F is uniquely determined (up to F-isomorphism) by its galoisian closure and characterise the extensions D of F which are the galoisian closures of some solvable primitive extension E of F.

 Abstract:

The genesis of “zero” as a number, that even a child so casually uses today, is a long and involved one. A great many persons concerned with the history of its evolution, today accepts that the number “zero”, in its true potential, as we use it in our present day mathematics, has its root, conceptually as well as etymologically, in the word ‘´S¯unya’ of the Indian antiquity. It was introduced in India by the Hindu mathematicians, which eventually became a numeral for mathematical expression for “nothing”, and via the Arabs, went to Europe, where it survived a prolonged battle with the Church (which once banned the use of ‘zero’ !) through centuries. However, the time frame of its origin in Indian antiquity is still hotly debated. Furthermore, some recent works even try to suggest that a trace of the concept, if not in total operational perspective, might have a Greek origin that traveled to India during the Greek invasion of the northern part of the country. However, from the works on Vedic prosody by Pi ˙ngala (Pi ˙ngalacchandah. s¯utra) [3rd Century BC] to the concept of “lopa” in the grammarian Panini, (As. t.¯adhy¯ay¯ı) (400-700 BC, by some modern estimates) it appears very likely that the thread of rich philosophical and socio academic ambiances of Indian antiquity was quite pregnant with the immensity of the concept of ‘´S¯unya’ - a dichotomy as well as a simultaneity between nothing and everything, the ‘zero’ of void and that of an all pervading ‘fathomless’ infinite.

A wide variety of number systems were used across various ancient civilizations, like the Inca, Egyptian, Mayan, Babylonian, Greek, Roman, Chinese, Arab, Indian etc. Some of them even had ‘some sort of zero’ in their system! Why then, only the Indian zero is generally accepted as the ancestor of our modern mathematical zero? Why is it only as late as in 1491, that one may find the first ever mention of ‘zero’ in a book from Europe? In this popular level lecture, meant for a general audience, based on the mindboggling natural history of ‘zero’, we would like to discuss, through numerous slides and pictures, the available references to the evolution and struggle of the concept of place-value based enumeration system along with a “zero” in it, in its broader social and philosophical contexts.

 Abstract:

Logistic regression is an important and widely used regression model for binary responses and is extensively used in many applied fields.
In the presence of misclassified binary responses using internal validation data, a pseudo-likelihood method is proposed for estimation of logistic regression parameters. Under minimal assumptions we establish rigorous asymptotic results for the pseudo-likelihood based estimators. A bootstrapped version of the pseudo likelihood based estimators is also proposed and its distributional consistency is proved enabling us to effectively use bootstrap method for statistical inference. The results of the simulation studies clearly indicate the superiority of pseudo-likelihood based estimators to the full likelihood based estimators, and the likelihood estimators based on misclassified binary responses only.
Also, inferences on the regression parameters using asymptotic distribution of pseudo-likelihood estimators and its bootstrap version are found to be similar.
Joint work with T. Bandyopadhyay (IIM-A) and Sumanta Adhya (WBSU).

 Abstract:

High dimension, low sample size data pose great challenges for the existing statistical methods. For instance, several popular methods for cluster analysis based on the Euclidean distance often fail to yield satisfactory results for high dimensional data. In this talk, we will discuss a new measure of dissimilarity, called MADD and see how it can be used to achieve perfect clustering in high dimensions. Another important problem in cluster analysis is to find the number of clusters in a data set. We will see that many existing methods for this problem can be modified using MADD to achieve superior performance. A new method for estimating the number of clusters will also be discussed. We will present some theoretical and numerical results in this connection.
This presentation is based on joint works with Anil K. Ghosh.

 Abstract:

Most of todays experimentally verifiable scientific research, not only requires us to resolve the physical features over several spatial and temporal scales but also demand suitable techniques to bridge the information over these scales. In this talk I will provide two examples in mathematical biology to describe these systems at two levels: the micro level and the macro (continuum) level. I will then detail suitable tools in homogenization theory to link these different scales.
The first problem arises in mathematical physiology: swelling-de-swelling mechanism of mucus, an ionic gel. Mucus is packaged inside cells at high concentration (volume fraction) and when released into the extracellular environment, it expands in volume by two orders of magnitude in a matter of seconds. This rapid expansion is due to the rapid exchange of calcium and sodium that changes the cross-linked structure of the mucus polymers, thereby causing it to swell. Modelling this problem involves a two-phase, polymer/solvent mixture theory (in the continuum level description), together with the chemistry of the polymer, its nearest neighbor interaction and its binding with the dissolved ionic species (in the micro-scale description). The problem is posed as a free-boundary problem, with the boundary conditions derived from a combination of variational principle and perturbation analysis. The equilibrium-states of the ionic gels are analyzed.
In the second example, we numerically study the adhesion-fragmentation dynamics of rigid, round particles clusters subject to a homogeneous shear flow. In the macro level we describe the dynamics of the number density of these cluster. The description in the micro-scale includes (a) binding/unbinding of the bonds attached on the particle surface, (b) bond torsion, (c) surface potential due to ionic medium, and (d) flow hydrodynamics due to shear flow.

 Abstract:

In this talk, I shall consider the high-dimensional moving average (MA) and autoregressive (AR) processes. My goal will be to explore the asymptotics for eigenvalues of the sample autocovariance matrices. This asymptotics will help in the estimation of unknown order of the high-dimensional MA and AR processes. Our results will also provide tests of different hypotheses on coefficient matrices. This talk will be based on joint works with Prof. Arup