Title: Importance of the General Equivalence Theorem on Optimal Designs
Speaker:  Soumadeb Pain
Abstract: The General Equivalence Theorem (GET) plays a pivotal role in the theory and practice of optimal experimental design, providing a rigorous framework for identifying designs that maximize statistical efficiency. By establishing conditions under which different optimality criteria, such as D-optimality and G-optimality, are equivalent, GET enables a unified approach to evaluating and constructing designs that minimize imprecision in parameter estimation. This equivalence is crucial because it allows researchers to select designs that not only satisfy one criterion but are also optimal under alternative criteria, leading to more robust and versatile experimental designs. In this seminar, I will try to explain how the GET provides a foundational framework for constructing and verifying optimal experimental designs that are efficient, robust, and adaptable across various statistical criteria.
Date: 13 November 2024
Time:  03:00 PM - 04:00 PM
Venue: FB567

Title: Exact MCMC for Intractable Proposals
Speaker:  Dwija Kakkad (4th year, BS - Math. & Sc. Comp.)
Abstract: Accept-reject based Markov chain Monte Carlo (MCMC) methods are the workhorse algorithm for Bayesian inference. These algorithms, like Metropolis-Hastings, require the choice of a proposal distribution which is typically informed by the desired target distribution. Surprisingly, proposal distributions with unknown normalizing constants are not uncommon, even though for such a choice of a proposal, the Metropolis-Hastings acceptance ratio cannot be evaluated exactly. Across the literature, authors resort to approximation methods that yield inexact MCMC or develop specialized algorithms to combat this problem. We show how Bernoulli factory MCMC algorithms, originally proposed for doubly intractable target distributions, can quite naturally be adapted to this situation. We present three diverse and relevant examples demonstrating the usefulness of the Bernoulli factory approach to this problem.
Date: 06 November 2024
Time:  03:00 PM - 04:00 PM
Venue: FB567

Title: Approximation method of Matern Gaussian field through SPDE approach
Speaker:  Sayan Bhowmik
Abstract: Handling large spatial data is quite challenging if there are large number of spatial locations due to inversion of dense covariance matrix. Assuming Gaussian field across spatial locations may not be a suitable choice as the computational time is of order O(n^3) for n spatial locations. Discretizing the continuous domain by defining fine mesh across the space quite helpful by using an explicit link between Gaussian field (GF) and Gaussian Markov random field (GMRF). Gaussian field with having Matern covariance function, can be represented as a solution of a specific stochastic partial differential equation (SPDE). A possible analytical method is integrated nested Laplace approximation (INLA). I will try to explain different kind of SPDE equations which are used in different situations.
Date: 30 October 2024
Time:  03:00 PM - 04:00 PM
Venue: FB567

Title: Classification of symplectic toric manifolds
Speaker:  Yogendra Singh
Abstract: Symplectic toric manifolds are smooth compact connected 2n-manifolds equipped with a symplectic structure and an action of a torus T^n with associated moment maps. The moment polytope, a convex shape in R^n, encodes the geometry of these manifolds.

In this talk, we will discuss the classification of symplectic toric manifolds in terms of specific polytopes, known as Delzant polytopes. More precisely, there exists a bijective correspondence between symplectic toric manifolds and Delzant polytopes. This correspondence, called Delzant's correspondence theorem, plays a fundamental role in understanding the geometry and topology of these manifolds.
Date: 23 October 2024
Time:  03:00 PM - 04:00 PM
Venue: FB567

Title: Von Neumann-Wold decomposition
Speaker:  Amritha K S
Abstract: We shall discuss a decomposition theorem for isometric operators on a Hilbert space by von Neumann and Wold. The theorem states that any isometric operator can be written as a direct sum of a unitary operator and copies of shift operator. We will see a proof of this theorem.
Date: 16 October 2024
Time:  03:00 PM - 04:00 PM
Venue: FB557

Title: Abelian and Tauberian Theory
Speaker:  Stuti Das
Abstract: Since early days of mathematics, summability methods have been used to assign a reasonable sum to an infinite series, whether it is convergent or not. In its simplest form, Tauberian theory deals with the problem of finding conditions under which a summable series is actually convergent. One of the first results in this direction, which applies to Abel summability was given by Alfred Tauber in 1897. However, Tauberian theory began in earnest only around 1910 with the work of Hardy and Littlewood.
In the this talk, I will briefly go through the notions of Abelian and Tauberian results, in particular focusing on the celebrated Hardy-Littlewood Tauberian theorem and elaborate the proof(Karamata’s version). In the remaining time, I will introduce integral versions of Tauberian theorems and if time permits, we will see one application of Karamta Tauberian theorem to prove the prime number theorem.
Date: 01 October 2024
Time:  06:00 PM - 07:00 PM
Venue: FB567

Title: The Müntz-Szász theorem
Speaker:  Prakhar Chaubey
Abstract: Click Here
Date: 24 September 2024
Time:  06:00 PM - 07:00 PM
Venue: FB567

Title: Denjoy- Wolff Theorem
Speaker:  Kanha Behera
Abstract: The Denjoy- Wolff theorem is a beautiful result in complex analysis which you may not find in most analysis books. The beauty of this theorem lies in its simplicity and usefulness. Apart from its use in complex function theory, the Denjoy- Wolff theorem also has applications in operator theory and the study of dynamic systems.
To explore this theorem, we need to understand the concept of function iteration. In this context, iteration refers to the repeated composition of a function. Specifically, for an analytic self-mapping of the open unit disc, the theorem addresses the behavior of its iterates. We shall discuss the statement and the proof of the theorem highlighting its applications.
Date: 10 September 2024
Time:  06:00 PM - 07:00 PM
Venue: FB567

Title: Random Walks on Homogeneous Spaces
Speaker: Tarun Goyal
Abstract: Click Here
Date: 02 September 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Navier Stokes Equation
Speaker: Dr. Prabir Barman
Abstract: Click Here
Date: 27 August 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Feynman-Kac formula
Speaker: Mangala Prasad
Abstract: Itô's formula is a fundamental result in stochastic calculus, which provides a way to compute the differential of function of a stochastic process. It works as a bridge between stochastic processes and partial differential equations.
Starting with basics of stochastic calculus, I shall explain Brownian motion, stochastic integral and Itô's formula. In this talk, we shall see an application of Itô's formula to get stochastic representation for the solution of the Cauchy problem for the backward heat equation with potential and Langrangian functions. This reprentaion is known as Feynman-Kac formula.
Date: 20 August 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Nevanlinna-Pick interpolation
Speaker: Santu Bera
Abstract: Click Here
Date: 13 August 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Nevanlinna-Pick interpolation
Speaker: Santu Bera
Abstract: Click Here
Date: 05 August 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: A Glimpse of Borel–Harish Chandra Theorem
Speaker: Sabyasachi Dhar
Abstract: Click Here
Date: 03 June 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Probability of Correct Selection: Insights into Ranking and Selection Procedures
Speaker: Yogesh Katariya
Abstract: In many practical situations, it is of interest to choose the best (or worst) of k (≥ 2) populations among several populations, where the quality of populations is assessed in terms of an unknown parameter associated with it. In the literature, such procedures are classified as “Ranking and Selection Procedures.” The goal is to develop effective and optimal selection/decision rules, ensuring a high probability of correctly identifying the best population or a nonempty subset of populations that include the best population.
In this talk, we will discuss some practical real-life examples of Ranking and Selection problems and emphasize the calculation of the probability of correctly selecting the best population by using standard selection rules. Key insights and general results for calculating the probability of correct selection for a given selection rule will be discussed to facilitate understanding and application in diverse contexts. Additionally, we will discuss critical general findings and insights essential for tackling such challenges across various domains.
Date: 20 May 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Homogeneous Dynamics and Its Application to Number Theory
Speaker: Sourav Das
Abstract: Recently, it has been observed that the ergodic theory of group actions on homogeneous spaces plays a crucial role in solving remarkable number-theoretic problems. Some notable instances include Margulis's proof of the Oppenheim conjecture, Furstenberg's proof of the Szemerédi theorem, Einsiedler, Katok, and Lindenstrauss's work on Littlewood's conjecture, and Kleinbock and Margulis's work on the Baker-Sprindžuk conjecture.
Starting with the basics of Diophantine approximation and Homogeneous dynamics, I will explain how the Diophantine properties of vectors in Euclidean space can be studied by examining the orbit behavior of diagonal flows on the space of all unimodular Euclidean lattices. In particular, I will delve into the details of the Baker-Sprindžuk conjecture, and time permitting, demonstrate how this problem of Diophantine approximation can be solved through Homogeneous dynamics. The content of this talk will be kept elementary to ensure accessibility to a broader audience.
Date: 15 April 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Frequentist validation of the Bayesian problems: A brief note on posterior contraction rate
Speaker: Arghya Mukherjee
Abstract: Click Here
Date: 08 April 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Solutions that explode
Speaker: N N Dattatreya
Abstract: The word 'explode' isn't a gimmick, we do have solutions that explode, so to say, to infinity, these solutions are called explosive solutions or large solutions. We will look at such solutions for ∆u=f(u) in one dimension; more precisely these are solutions to equations with singular boundary data. We will state the existence theorem in any bounded set in a Euclidean space and non-existence results in a whole Euclidean space. Finally, we will construct maximal and minimal large solutions in any bounded domain. Perhaps we will also discuss one of the most important tools to study such solutions, the comparison principle. We rely mostly on intuition and geometry rather than technicalities.
Date: 01 April 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Fully Homomorphic Encryption: Cryptography's Holy Grail
Speaker: Indranil Thakur(Ph.D. student, CSE IITK).
Abstract: Fully Homomorphic Encryption (FHE) has long been hailed as the "holy grail" of cryptography, promising to revolutionize data security and privacy. With FHE, arbitrary computations can be performed directly on encrypted data without decrypting it. It is very beneficial in the context of privacy-preserving outsourced storage and computation. This talk will explore the fascinating journey of FHE, from its theoretical inception to recent breakthroughs in practical implementations. We delve into the mathematical foundations of FHE, discussing the challenges and advancements that have propelled its development.
Date: 20 March 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Topological Complexity
Speaker: Dr. Gopal Chandra Dutta
Abstract: Click Here
Date: 11 March 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Exploring the Johnson-Lindenstrauss Lemma with Random Projection
Speaker: Annesha Deb
Abstract: Click Here
Date: 26 February 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Shannon Sampling Theorem
Speaker: Dr. Ankus Kumar Garg
Abstract: Click Here
Date: 20 February 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: On Pimsner-Popa probability constant.
Speaker: Mr. Guruprasad
Abstract: Click Here
Date: 12 February 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Proof of Riemann mapping theorem using potential theory
Speaker: Mr. Nishith Mandal
Abstract: Click Here
Date: 06 February 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Frostman's Theorem
Speaker: Mr. Chandan Sur
Abstract: Click Here
Date: 29 January 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Next word prediction using Markov Chains.
Speaker: Mr. Ojasvi Rajput
Abstract: In our increasingly digitized world, the ability to predict the next word in a sequence has become a fundamental aspect of natural language processing and human-computer interaction. This talk delves into the realm of next word prediction using Markov Chains, a probabilistic model that captures the essence of sequential dependencies within language.
The talk commences with an exploration of the underlying principles of Markov Chains, elucidating how these mathematical models encapsulate the idea that the probability of a future event depends on the current state. We delve into the application of Markov Chains to language, showcasing their versatility in capturing patterns and dependencies in text data.
Through engaging examples and demonstrations, we will be exploring the methodology behind implementing a Markov Chain-based next word prediction system.

Date: 22 January 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: Von Neumann's inequality for contractions
Speaker: Paramita Pramanick
Abstract: Click Here
Date: 15 January 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567

Title: When two commuting isometries are doubly commuting
Speaker: Shubham Jain
Abstract: Click Here
Date: 08 January 2024
Time: 06:00 PM - 07:00 PM
Venue: FB567