Abstract: In this work, we develop Frank-Wolfe-based first-order algorithms for composite convex optimization problems. More specifically, for problems whose objective involves a logarithmically-homogeneous, self-concordant function. Recently developed first-order methods for this class of problems assume that we can solve linear optimization (sub)problems over the given feasible set exactly. In our work, we relax this assumption and propose a first-order method (a variant of Frank-Wolfe method) for this class of problems. Our method relies on an approximate (rather than exact) solution to the linear optimization (sub)problem. We show that our ‘inexact’ first-order method enjoys a similar convergence guarantee to the ‘exact’ version proposed in the literature, while at the same time allowing more flexibility in solving the linear (sub)problem. In particular, our approach is applicable in both cases: when (sub)problem can be solved approximately up to (1) some prespecified ‘additive’ error, or (2) up to some prespecified ‘relative’ error. Furthermore, we also develop a first-order algorithm that can handle the situation where, with some nonzero probability, the (sub)problem might fail to generate an approximate solution within the prespecified error. The motivation for our ‘inexact’ method comes from large-scale semidefinite programming problems, where solving the (sub)problem exactly is not practical. Finally, we implement our method compare its performance against existing first-order methods, and show that the performance of our method (in terms of the number of iterations required for convergence) is similar to that of existing methods.
Abstract: We give a self-contained introduction to Macdonald polynomials starting from the basics of symmetric functions. Time permitting, we will also explain the relation to Hecke algebras.
Abstract: In the last few years, we have seen remarkable progress on the theory of random matrices with independent entries. For example, one might consider the successful resolution of the Dyson Mehta conjectures.
After the independent case being solved, recently some amount of interest has been shown in matrices with dependent entries. In this talk, we shall discuss our result on the spectral norm of Wigner matrices with dependent entries. When the entries of the matrix are correlated centered Gaussians, we show under some assumptions, the largest eigenvalue converges to the support of the limiting spectral distribution. Our result is in some sense optimal as we show counter examples (i.e. the largest eigenvalue goes beyond the support) under minor violations of the assumptions. We also have weak results in the non-centered case. Due course of the talk, I shall also discuss some ongoing projects with my collaborators. If time permits, I shall further talk about some of my earlier results in high dimensional hypothesis testing where random matrix theory plays an important role.
Abstract: Complex quadrics of prescribed rank and number of variables are an instance of complex spherical varieties. After introducing the latter varieties, I shall present a generalisation of the celebrated Sylvester's law of inertia for real loci of those varieties. This is a joint work with Dmitry Timashev.
Abstract: Understanding how neural networks learn features, or relevant patterns in data, for prediction is necessary for their reliable use in technological and scientific applications. In this work, we presented a unifying mathematical mechanism, known as average gradient outer product (AGOP), that characterized feature learning in neural networks. We provided empirical evidence that AGOP captured features learned by various neural network architectures, including transformer-based language models, convolutional networks, multilayer perceptrons, and recurrent neural networks. Moreover, we demonstrated that AGOP, which is backpropagation-free, enabled feature learning in machine learning models, such as kernel machines, that a priori could not identify task-specific features. Overall, we established a fundamental mechanism that captured feature learning in neural networks and enabled feature learning in general machine learning models.
Abstract: Algebraic Statistics is a relatively new field of research where tools from Algebraic Geometry, Combinatorics and Commutative Algebra are used to solve statistical problems. A key area of research in this field is the Gaussian graphical models, where the dependence structure between jointly normal random variables is determined by a graph. In this talk, I will explain the algebraic perspectives on Gaussian graphical models and present some of my key results on understanding the defining equations of these models. In the end, I will talk about the problem of structural identifiability and causal discovery and how algebraic techniques can be implemented to tackle them.
Abstract: Consider a well-shuffled deck of cards that contains distinct types of cards each with multiplicity . Consider the following card-guessing game: At each step, the player (guesser) guesses the topmost card in the deck. After each guess, the topmost card is shown to the player (guesser) and removed from the deck. The game continues till the deck is exhausted. This game is often referred to as a complete feedback game. The player's goal in the complete feedback game is to maximize the number of correct guesses. Diaconis and Graham showed that the greedy strategy maximizes the total number of correct guesses in expectation. In this talk, we give a leisurely survey of many interesting questions arising from this setup and present a central limit theorem for the total number of correct guesses under the greedy strategy. This is based on joint work with Ottolini.
Abstract: The speaker will begin with the motivation behind the motivic homotopy theory and its applications to cohomology theories of schemes and vector bundles on affine schemes. Later, he will present an ongoing research work with Neeraj Deshmukh on computing motivic invariants of stacky curves and using those invariants to compute Brauer group of moduli stack of curves of genus 1.
Abstract: The design and analysis of finite element methods for the incompressible Navier–Stokes equations that are robust and accurate for a wide range of Reynolds numbers remain a challenging problem. In this talk, we will first consider the implicit-explicit (IMEX) time discretizations using equal order interpolation for the incompressible Oseen equations at high Reynolds numbers and discuss its stability and convergence results.
Then, in the case of high Reynolds numbers or inviscid flow problems, we show how the implicit-explicit method can naturally be written as a split method based on Poisson pressure projection steps without splitting error. The analysis remains valid for the splitting scheme and satisfies close to optimal error estimates. Finally, the validation of the proposed stabilization scheme and verification of the derived estimates are presented with appropriate numerical experiments.
Abstract: The question of which functions acting entry-wise preserve positive semi-definiteness has a long history, beginning with the Schur product theorem [Crelle, 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions.
Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entry-wise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at "opposite ends'', and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entry-wise except on specified "diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entry-wise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic.
Abstract: Given a group G and two Gelfand subgroups H and K of G, associated to an irreducible representation \pi of G, there is a notion of H and K being correlated with respect to \pi in G (introduced by Benedict Gross in 1991). We discuss this theme and give some details in some specific examples.
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I plan to talk about the notion of Hamiltonian reduction from the point of view of physics (specifically classical mechanics) and mathematics (specifically symplectic geometry), focusing on a concrete example involving the Calogero-Moser space. Time permitting, I will motivate the notion of quantum Hamiltonian reduction and its applications to representation theory.
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Weyl's law is a fundamental result that governs the asymptotics of the eigenvalues of the Laplacian. Stated very simply, it states that the number of eigenvalues of the Laplacian less than or equal to t, is asymptotically equal to the Volume of the domain times t^{d/2}, where d is the dimension of the domain.
As with any asymptotic problem, it is a natural question to investigate the error term. After important work by Courant in 1920's, it was shown by Hormander in 1968 that the error term is of the order of O(t^{d/2 -1/2}) (for manifolds without boundary). It can also been shown that the error term is actually sharp for the round sphere.
In 2019, Iosevich and Wyman showed that the error term actually can be improved for a product of spheres. There proof involves a highly nontrivial estimate for lattice points inside an ellipsoid.
We have recently extended the idea of Iosevich and Wyman to projective spaces. This is joint work with Anupam Pal Choudhury and Sai Sriharsha.
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To accurately quantify landslide hazard in a region of Turkey, we develop new marked point process models within a Bayesian hierarchical framework for the joint prediction of landslide counts and sizes. We leverage mark distributions justified by extreme-value theory and specifically propose “subasymptotic`` distributions to flexibly model landslide sizes from low to high quantiles. The use of intrinsic conditional autoregressive priors, and a customized adaptive Markov chain Monte Carlo algorithm, allow for fast fully Bayesian inference. We show that subasymptotic mark distributions provide improved predictions of large landslide sizes, and use our model for risk assessment and hazard mapping. Furthermore, within the general modeling framework, a submodel known as the areal model is utilized when data are aggregated at a coarser slope unit resolution. We applied this framework to jointly model Wenchuan landslide counts and sizes data, highlighting the benefit of the joint modeling approach in the landslide literature for hazard and risk assessment.
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Let R(G) denote the category of smooth complex representation of G(F), where G is a connected reductive group defined over a non-archimedean local field F. Bernstein decomposition expresses R(G) as a product of indecomposable subcategories called Bernstein blocks. Each Bernstein block is equivalent to the module category of the "Hecke algebra" associated with that "type". I will go over the basic theory mentioned above. To each Bernstein block, the theory of Moy and Prasad associates a number called depth. I will describe a result, part of a work in progress jointly being done with Jeff Adler, Jessica Fintzen and Kazuma Ohara, which states that each Bernstein block is equivalent to a depth-zero Bernstein block of a certain subgroup of G, when the residue characteristic is not too small.
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With advancements in technology for measuring environmental variables, there has been a significant increase in the volume and accessibility of data available to hydrologists and climate scientists.
Statistics plays an important role in the analysis of these datasets for various applications such as risk assessment, forecasting and identification of drivers of extreme events. The objective of this talk is to introduce some of these applications and to discuss the challenges involved in applying statistical techniques in hydrology and climate sciences which arise from the nature of problems in the domain and the characteristics of available datasets. These issues will be discussed in the context of three problems: (1) spatiotemporal analysis of extreme events, (2) attribution of physical mechanisms of extreme events and (3) quantifying the contributions of natural and anthropogenic processes to changes in the water cycle.
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Large-scale assessment surveys typically collect data via tests about cognitive or socio-emotional skills from a heterogeneous sample. The methods or approaches for scoring these tests vary significantly. This leaves applied researchers uncertain about the assumptions and computational methods involved in scoring such tests. In this presentation, we will delve into three primary methods of scoring and using them in subsequent analyses: (1) test scores (provides point estimates of individual ability), (2) structural equation modeling (SEM), and (3) plausible values (PV). We will explore the biases inherent in each approach and present findings from a simulation study comparing the three methods under conditions typical of socio-emotional skill and personality assessments. Our results demonstrate that while different test scores may exhibit high correlation, the resulting bias in regression coefficients can vary significantly.
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In this talk, I will present some of my work on designing and analyzing algorithms for learning in large and structured environments, where the state and action spaces are huge or even infinite. I will focus on two main topics: (1) Bayesian optimization for hyperparameter tuning in large-scale machine learning models, and (2) Policy optimization for language models using human feedback. For the first topic, I will introduce the Gaussian process optimization framework and design multi-armed-bandit algorithms for hyperparameter optimization. I will show sublinear regret bounds for the proposed algorithms that depend on the information complexity of the objective function to be optimized. Along the way, I will present a self-normalized concentration inequality for vector-valued martingales of arbitrary, possibly infinite, dimension, and discuss some applications of this concentration bound. For the second topic, I will talk about the effects of noisy preference data that can negatively impact language model alignment. I will propose a robust loss function for language model policy optimization in the presence of random preference flips. I will show that the proposed language model policy is provably tolerant to noise and characterize its sub-optimality gap as a function of noise rate, dimension of the policy parameter, and sample size. I will also demonstrate the empirical performance of the proposed policy on various tasks, such as dialogue generation and sentiment analysis. I will conclude with some open problems and future directions of research in large scale machine learning.
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Handling higher-dimensional data poses a familiar challenge for most machine learning applications such as cancer genomics. We first discuss traditional regression models that induce shrinkage, such as LASSO, and their Bayesian variants. We then look at spike and slab priors that have become popular for high-dimensional Bayesian
modeling and how we use them to model shrinkage in deep neural networks, aiming at feature selection. Lastly, we discuss computational challenges, a scalable solution based on variational Bayes methods, and some work on the theoretical guarantees of the variational posterior (briefly).
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In this talk, we define Eisenstein cycles in the first homology groups of quotients of the hyperbolic three spaces as linear combinations of Cremona symbols (generalization of Manin symbols) for the imaginary quadratic field. They generate the Eisenstein part of the homology groups. We also discuss the Eisenstein part of the cohomology groups. As an application, we find asymptotic dimension formula (level aspect) for the cuspidal cohomology groups of congruence subgroups of certain form inside the full Bianchi groups. This is a joint work with Pranjal Vishwakarama.
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Research in machine learning and data science is increasingly entering the realm of staggeringly large Multiview data collections (concurrent measurements (views) collected on the same subjects from multiple sources).Fueled by an explosion in recent high-throughput and AI technologies, we are now ready to enter the world of personalized medicine and individualized solutions, where clinical or other non-therapeutic interventions can be custom-tailored to individuals to achieve better outcomes based on their Multiview profiles. Although analyses of such multimodal datasets have the potential to provide new insights into the underlying mechanistic processes that cannot be inferred with a single modality, the integration of very large, complex, multimodal data represents a considerable statistical and computational challenge. An understanding of the principles of data integration and visualization methods is thus necessary to determine which methods are best applied to a particular integration problem. In this talk, I will discuss open challenges in multimodal integration, including methodological issues that must be resolved to establish the resources needed to move beyond incremental advances toward translational intervention while keeping machine learning and data science at the forefront of the next generation of Multiview research.
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Fix an odd prime p. Let f be a p-ordinary newform of weight k and h be a normalized cuspidal p-ordinary Hecke eigenform of weight l< k. In this talk we will discuss the p-adic L function and the structure of the p∞-Greenberg Selmer group of the Rankin-Selberg convolution of f and h. In the special cases when the residual representation of h at p is reducible, we also discuss certain congruences between the associated characteristic ideal of the dual Selmer group and the p-Adi Rankin-Selberg L-function. This is a joint work with Somnath Jha and Ravitheja Vangala.
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The parity of Selmer ranks for elliptic curves defined over the rational numbers $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$ has been studied by Shekhar. The proof of Shekhar relies on proving a parity result for the $\lambda$-invariants of Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension$\mathbb{Q}_\infty$ of $\mathbb{Q}$. This has been further generalized for elliptic curves with super singular reduction at $p$ by Hatley and for modular forms by Hatley--Lei.
In this talk, we will present a parity result for the $\lambda$-invariants of Selmer groups over $\mathbb{Q}_\infty$ for the symmetric square representations associated to two modular forms, both ordinary at $p$ with congruent residual irreducible Galois representations.
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There has been a tremendous amount of work in the past few decades in arithmetic statistics and especially about counting number fields. In this talk, after giving a general view of the sort of results available in the literature, I will report on the results of an ongoing joint work with Qiyao (Vivian) Yu. We consider the sub-class of totally imaginary number fields and ask the question: how likely is such a number field to contain a CM subfield? A CM field is a totally imaginary quadratic extension of a totally real number field. We show that about 67% of quartic totally imaginary fields do not contain a CM subfield. I will also discuss the case of sextic fields, where the complexity of the problem in the general case becomes apparent.
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The classical Diophantine problem of determining which integers can be expressed as sum of two rational cubes has a long history; it includes works of Sylvester, Selmer, Stage, Leiman and the recent work of Alpöge-Bhargava-Shnidman-Burungale-Skinner. In this talk, we will use Selmer groups of elliptic curves and integral binary cubic forms to study some cases of the rational cube sum problem. This talk is based on joint works with D. Majumdar, P.Shingavekar and B. Sury.
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Neeman recently settled a conjecture by Antieau, Gepner and Heller on the existence of bounded t - structures on the derived category of perfect complexes. We prove a triangulated categorical generalisation of that theorem. In particular, we will show that the existence of a bounded t - structure implies that the singularity category, appropriately defined, vanishes. To achieve this, we also introduce the notion of finitistic dimension for a classically generated triangulated category. Finally, we also show that all t - structures on the completion under these hypotheses are equivalent. This proves that all bounded t - structures on the bounded derived category of a Noetherian finite dimensional scheme are equivalent, generalising a result by Neeman. This is joint work with Rudra dip Biswas, Hong Xing Chen, Chris J. Parker,and Junhua Zheng. (https://arxiv.org/abs/2401.00130)
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We study random walks on a d-dimensional torus by affine expanding maps. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. From this, we deduce uniform distribution of almost every orbits modulo 1 in certain self-similar sets in R^d. As this conclusion amounts to normality of numbers in the one-dimensional case, thus we obtain the version of Borel’s theorem on Normal numbers for a class of fractals in R, for instance, cantor typesets. The talk is based on a joint work with Yiftach Dayanand Barak Weiss.
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In this talk, we deal with the problem of efficient sampling from a stochastic differential equation, given the drift function and the diffusion matrix. The proposed approach leverages a recent model for probabilities (the positive semi-definite -- PSD model) from which it is possible to obtain independent and identically distributed (i.i.d.) samples at precision with a cost that is where is the dimension of the model, the dimension of the space. The proposed approach consists of: first, computing the PSD model that satisfies the Fokker-Planck equation (or its fractional variant) associated with the SDE, up to error, and then sampling from the resulting PSD model. Assuming some regularity of the Fokker-Planck solution (i.e. -times differentiability plus some geometric condition onits zeros) We obtain an algorithm that: (a) in the preparatory phase obtains a PSD model with L2 distance from the solution of the equation, with a model of dimension where is the fractional power to the Laplacian, and total computational complexity of and then (b) for Fokker-Planck equation, it can produce i.i.d. samples with error in Wasserstein-1 distance, with a cost that is per sample. Our results suggest that as the true solution gets smoother, we can circumvent the curse of dimensionality without requiring any sort of convexity.
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The study of the Selmer group for an Artin representation was initiated by Greenberg. In this talk, we will discuss the algebraic functional equation of the dual Selmer group coming from Artin's representation of the cyclotomic Zp-extension of Q. One usually studies the algebraic functional equation of the dual Selmer group because of their connection with the Iwasawa Main Conjecture.
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The dimer model, also referred to as dominotilings or perfect matching, are tilings of the Z^d lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture isa little daunting because most of these tools are limited to two dimensions. In this talk I will describe what techniques can be extended to higher dimensions and give a brief account of a large deviations principle for dimer tilings in three dimensions that we prove analogous to the results by Cohn, Kenyon and Propp (2000).
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In this talk, we present a well posed ness result of fluid-structure interaction model regarding the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We prove the existence of a weak solution of the fluid-structure system up to collision. We will also discuss the case of several rigid bodies.
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We shall state and prove the Rank-Nullity theorem and give some of its applications.
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One of the most dynamically evolving research areas, with both practical and theoretical interests, is that of sampling and reconstruction. In 1949, the celebrated Shannon sampling theorem was proved which turned out to be a milestone in this field of study and set the foundation for information theory. Over these years, the theory of sampling has been intensively studied.
In this talk, we discuss the problem of regular and irregular average sampling over certain classes of shift-invariant space of functions. We also consider the random average sampling problems for certain suitable subsets of shift-invariant subspaces of mixed Lebesgue spaces. Finally, based on this work, we will see some future directions.
Short Biography:
Dr. Ankush Kumar Garg is a visiting scientist at the Indian Statistical Institute Bangalore, specializing in the field of Harmonic Analysis. He earned his doctorate from the Indian Institute of Science Education and Research Thiruvananthapuram, where he was mentored by Dr. P. Devaraj. His dissertation, entitled ”A Study on Reconstruction from Local Average and Random Average Samples over Shift-Invariant Spaces,” was successfully defended in November2022. Dr. Garg’s research interests primarily revolve around frame theory, sampling and reconstruction theory, as well as invariant subspace characterization. His contributions to the field have been published in numerous peer-reviewed journals. In addition to his research accomplishments, Dr. Garg is also a passionate educator who enjoys sharing his knowledge with others. He has a strong commitment to teaching
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Initiated by Elias Stein in late1960's the Fourier restriction conjecture has played a central part in the development of modern harmonic analysis. Despite continuous progress over the last five decades, currently this remains out of reach in dimensions bigger than two. To get a better sense of restriction inequalities, we consider Fourier restriction estimates onto curves $\gamma : \R \to \Rd$. Even in this well-explored setting, there are many basic questions that remain open such as the question of existence of maximizers for such inequalities. This talk will be a gentle introduction to such questions and some recent progress on these. This is based on our recent works with Betsy Stovall (at University of Wisconsin Madison).
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This talk will be an introduction to the notion of derivative in several variables Calculus. We will introduce the notion of derivatives and will observe that many of the fundamental properties such as the chain rule from one variable theory continue to hold inseveral variables.
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The main purpose of this talk is to describe the invariant subspaces of a class of operators on Hilbert space.
This presentation is devoted to generalizing the famous Beurling’s Invariant sub space Theorem for the shift operator to the case of the tuple of operators, where the operators assumed are weaker than isometries, we will be referring to this weaker condition of operators as near-isometries. To begin with, we first derive a generalization of Slocinski’s well-known Wold type decomposition of a pair of doubly commuting isometries to the case of an n-tuple of doubly commuting operators near-isometries. Then, with the help of Wold decomposition for the n-tuple of doubly commuting near-isometries, we will represent in concrete fashion those Hilbert spaces that are vector subspaces of the Hardy spaces Hp(Dn) (1 ≤ p ≤ ∞) that remain invariant under the action of coordinate wise multiplication by an n-tuple(TB1,...,TBn ) of operators where for each 1 ≤ i ≤n, Bi is a finite Blaschke factor on the open unit disc. The critical point to be noted is that these TBi are assumed to be near-isometries.
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In this talk, we discuss non-trivial bounds for the Rankin Selberg L- function associated with a GL(3) form of levelP_1 and a
GL(2) form of level P_2 when P_1 and P_2 are large. This family of L-functions is of particular interest because of its connection with the quantum unique Ergodicity conjecture formulated by Rudnik and Sarnak. This is joint work with Saurabh Singh and Ritabrata Munshi.
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We will start by recalling the definition of a topological space and will see some examples. We will move on to discuss connected topological spaces and see various examples and properties of such spaces.
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Short biography:
Manisha Aggarwal completed her Ph.D. in Mathematics at Indian Institute of Technology Delhi in 2016 under the supervision of Prof. Subiman Kundu. Since then, she has been teaching at St. Stephen's College, University of Delhi. Her research interest is analysis on metric spaces. The passion for teaching and research has motivated her to write textbooks for undergraduates and postgraduates in collaboration with Prof. Kundu. The research monograph, titled Cofinally Complete Metric Spaces and related functions, written in collaboration with Prof. Kundu and Dr. Gupta is to be published by World Scientific in 2023.
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We are already familiar with the signicance of the study of convergence of sequences in real analysis. In the talk, we will discuss regarding pointwise and uniform convergence of a sequence of real-valued functions. The main focus will be Dini's theorem, Stone-Weierstrass theorem and their generalizations.
Short biography:
Manisha Aggarwal completed her Ph.D. in Mathematics at Indian Institute of Technology Delhi in 2016 under the supervision of Prof. Subiman Kundu. Since then, she has been teaching at St. Stephen's College, University of Delhi. Her research interest is analysis on metric spaces. The passion for teaching and research has motivated her to write textbooks for undergraduates and postgraduates in collaboration with Prof. Kundu. The research monograph, titled Cofinally Complete Metric Spaces and related functions, written in collaboration with Prof. Kundu and Dr. Gupta is to be published by World Scientific in 2023.
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.
Abstract:
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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
Abstract:
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 (?) 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.
Abstract:
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
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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.
Abstract:
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/.