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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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Abstract:

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.

Abstract:

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

Abstract:

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.

Abstract:

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.

Abstract:

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.

Abstract:

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