Abstract: The inhomogeneous multispecies PushTASEP is an interacting particle system with multiple species of particles on a finite ring where the hopping rates are site-dependent. (The homogeneous variant on Z is also known as the Hammersley–Aldous–Diaconis process.) In its simplest variant with a single species, a particle at a given site will hop to the first available site clockwise. We show that the partition function of this process is intimately related to the classical Macdonald polynomial. We also show that large families of observables satisfy the property of interchangeability, namely they have the same distribution even in finite time when the hopping rates are permuted. For some reason, Schur polynomials seem to appear as expectations in the stationary distribution of important observables. This is a joint work with James Martin and Lauren Williams and based on the preprints arXiv:2310.09740 and arXiv:2403.10485.
Abstract: Abstract Self-adjoint unitaries on Hilbert spaces are known as symmetries. These symmetries have a very simple structure as they have their spectrum contained in {-1, 1}. In 1958 Halmos and Kakutani showed that every unitary on an infinite dimensional complex Hilbert space is a product of four symmetries. We show that in type II von Neumann algebras every unitary is a product of six symmetries. In this setting at least four symmetries are needed but we don’t know whether four are enough. This talk is based on a joint work with Soumyashant Nayak and P. Shankar.
Abstract: There is a well observed philosophy in Mathematics that studying subobjects of a given object (e.g., a group, a field, an algebra, etc) is a very effective methodology employed to obtain a good understanding of the ambient object. Over last 4 or 5 decades, this philosophy has been employed very effectively in understanding certain families of normed algebras, called operator algebras. And among the tools available for this study, various notions of distance between subalgebras have had very significant impact on the theory.
In the same context, in a recent article with Keshab Chandra Bakshi, we introduced the notion of angle between appropriate intermediate subalgebras of an inclusion of some 'fascinating' algebras. In this talk, we shall try to acquaint the audience with this notion and, if time permits, we shall also present some calculations and applications achieved so far. An attempt will be made to keep a good portion of the talk accessible to the students.
Abstract: The plan of the talk is to give a gentle introduction to some aspects of the relative Langlands programme. The interest here is in understanding representations contributing to harmonic analysis on symmetric spaces (or more generally spherical varieties). For a symmetric space G/H, such representations of G are said to be H-distinguished. There are local (“p-adic”) and global (“adelic”) notions of distinction and questions of interest may be local or global or local-global. Analogous questions are often studied also over finite groups of Lie type. In this talk, we touch upon some of these topics in a few specific contexts.
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Abstract: I will introduce two notions of non-positively curved spaces - hyperbolic spaces and CAT(0) spaces and look at properties of groups acting 'nicely' on such spaces. I will also introduce two groups - the mapping class group of a surface and the group of automorphisms of a finitely generated free group. These groups unfortunately are neither hyperbolic nor CAT(0). However, we will see some spaces on which these groups act 'nicely enough' and explore the geometry of these spaces.
Abstract: We shall explore properties of moments and cumulants, their interrelation (via Mobius function), and their relation to the concept of probabilistic independence. Then we shall connect convergence in law (distribution) to convergence of moments or cumulants. This will lead us to moment or cumulant based proofs of the central limit theorem and the convergence of the binomial to Poisson.
The lecture is targeted towards a broad audience consisting of undergraduate students and graduate students from math-stats. Department and across departments as well.
Abstract: I am happy that IIT Kanpur is celebrating the birth centenary of Harish-Chandra. In this talk I will outline a biography of Harish-Chandra and say something about his mathematics in as non-technical away as possible. I am among the small number of living Indian mathematicians who had the privilege of knowing. I will take this opportunity to talk about some of my interactions with him.
Abstract: In this presentation, I will first define what is Statistical Science and then discuss about Statistical Science in action through various examples. Then move to the topic on Statistical Science and Machine Learning (ML) and formally define Statistical Learning (SL) which leads to Artificial Intelligence (AI). Through examples, I will go over two important concepts, margin of errors and Uncertainty Quantification (UQ). We will talk about notions of clustering, classification and Predictive Analytics as examples of AI. Finally, example will be provided from Randomized ClinicalTrials (RCT), Social Network, Disease Mapping for Spatial and Temporal Data indifferent application domains.
Abstract: Gram matrices are ubiquitous in the literature,from theoretical to applied settings. This talk will showcase(non-chronologically) some of these appearances: they are covariance/correlation = positive semidefinite matrices, they are useful in understanding GPS trilateration, and they appeared as Cayley-Menger matrices in understanding metric embeddings into Euclidean space [Schoenberg, Ann. of Math. 1935]. We will also see the entry wise transforms that send the class of Gram matrices into itself [Schoenberg, Duke Math. J. 1942], and will end with then-dimensional version of the well-known (and 2000-year old) Heron's formula.
Abstract: The basic ideas of Calculus started with Archimedes and reached a highly developed form in the 17th Century with Newton and Leibniz often being credited as its inventors. What was not so well-known until only a few decades ago is that between the 14th and 17th Century there was an unbroken lineage of profound mathematicians working in Kerala who had independently discovered many of the great themes of Calculus. This talk is an introduction to the lives and works of some of the prominent members of the Kerala School of Mathematics. Most of the talk will be accessible to a general audience. Only towards the end of the talk some elementary mathematics will be assumed to explain a few of their contributions.