Ph. D. Defence seminar

Date and time: Monday, January 9, 2023 Time: 11 AM-12 PM 
Venue: Ramanujan Hall
Host: Sanjay Pusti
Google meet link: https://meet.google.com/mjb-ghwp-tgk

Speaker: Mr. Tapendu Rana

Title: Wiener Tauberian theorems on Lie groups and Pseudo-differential operators on symmetric spaces and homogeneous trees 
Abstract: In this seminar, first, we will discuss the L^p-boundedness property of the pseudo-differential operators associated with a symbol on the rank one Riemannian symmetric spaces of noncompact type, where the symbol satisfies Hörmander-type conditions near infinity. We will also investigate the same problem in the setting of homogeneous trees, which are considered to be the discrete version of the rank one noncompact symmetric spaces.
We will talk about the Wiener Tauberian theorem on Lie groups in the second part of our seminar. We will discuss a genuine analogue of Wiener Tauberian theorem for L^{p,1}(SL(2, R)) (1 ≤ p < 2). Finally, we will prove Wiener Tauberian theorem type results for various Banach algebras and Lorentz spaces of radial functions on real rank one semisimple Lie group G, which is noncompact, connected, and has a finite center. This is a natural generalization of the Wiener Tauberian theorem for the commutative Banach algebra of the radial integrable functions on G.
 

Lecture series on Lie groups 

Date and Time: Six Mondays at 4 pm
Tea: 3.50 pm
Venue: A1-A2, CDEEP, Mathematics Department
Host: Dipendra Prasad

Speaker: M. S. Raghunathan, CEBS, Mumbai

Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G, G]=G, π_1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is  compact. 

Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  

I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups, also with some basic results from algebraic topology.
 

Algebraic geometry seminar

Date and time: Tuesday, January 10, 2023, 2:15 pm

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker : Atharva Korde

Affiliation University of British Columbia, Vancouver, Canada

Title: Gromov Witten invariants and Donaldson Thomas invariants-II

Abstract: I will give an introduction to Gromov-Witten invariants (via algebraic geometry, rather than symplectic geometry) and Donaldson-Thomas invariants. Then I will explain the GW-DT correspondence, which relates these two sets of numbers for a Calabi-Yau threefold. If time permits, I will talk about quantum cohomology later, an idea of Kontsevich which is used to answer the question of counting the following GW-invariant – how many rational curves of degree d pass through 3d-1 general points in the plane? The first talk will be less technical and accessible to anyway with minimal knowledge of algebraic geometry. The second talk will be more technical.

Seminar on Algebraic Groups 

Date and time: Tuesday, January 10, 2023, 4.00 pm - 5.30 pm. 
Venue: Ramanujan Hall
Host: Shripad Garge

Speaker: Arghya Pramanik
Affiliation: IIT Bombay

Title: Some algebraic geometry- I

Abstract: This is the first lecture on this topic. We begin with basic notions of algebraic geometry keeping in mind that we want to learn (linear) algebraic groups.  We follow the first chapter of Tony Springer's book.
 

Seminar on Data Science

Date and time: Wednesday, January 11, 2023, 4 pm
Venue: Ramanujan Hall
Host: Ashish Das

Speaker: Rakhi Singh
Affiliation: The State University of New York at Binghamton, New York.

Title: Subdata selection: Introduction and Recent Works

Abstract: Data reduction or summarization methods for large datasets (full data) aim at making inferences by replacing the full data by the reduced or summarized data. Data storage and computational costs are among the primary motivations for this. In this presentation, data reduction will mean the selection of a subset (subdata) of the observations in the full data. While data reduction has been around for decades, its impact continues to grow with approximately 2.5 exabytes (2.5 x 10 18 bytes) of data collected per day. We will begin by discussing an information-based method for subdata selection under the assumption that a linear regression model is adequate . A strength of this method, which is inspired by ideas from optimal design of experiments, is that it is superior to competing methods in terms of statistical performance and computational cost when the model is correct. A weakness of the method, shared with other model-based methods, is that it can give poor results if the model is incorrect. We will therefore conclude with a discussion of a model-free method. The work discussed here is a joint work with John Stufken at George Mason University, USA.

Algebraic Geometry seminar

Date and Time: Thursday, 12 Jan 2023, 11.30-12.45

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Atharva Korde

Affiliation: University of British Columbia, Vancouver, Canada

Title: Gromov Witten invariants and Donaldson Thomas invariants-III

Abstract: I will give an introduction to Gromov-Witten invariants (via algebraic geometry, rather than symplectic geometry) and Donaldson-Thomas invariants. Then I will explain the GW-DT correspondence, which relates these two sets of numbers for a Calabi-Yau threefold. If time permits, I will talk about quantum cohomology later, an idea of Kontsevich which is used to answer the question of counting the following GW-invariant – how many rational curves of degree d pass through 3d-1 general points in the plane? The first talk will be less technical and accessible to anyway with minimal knowledge of algebraic geometry. The second talk will be more technical.

Seminar on Coding Theory

Date and time:: Thursday, 12th January 2023 at 2.30 pm 
Venue: Ramanujan Hall

Speaker: Mahir Bilen Can
Affiliation: Tulane University, New Orleans, USA
Host: Sudhir Ghorpade

Title: Dual Higher Grassman Codes

Abstract: The Grassmann variety of k-dimensional subspaces of an n-dimensional vector space over a finite field with q-elements can be thought of as the ``moduli space'' of all linear q-ary (n,k)-codes. At the same time, each Grassmann variety naturally provides an algebraic geometry code via its Plucker embedding. The structure of Grassmann codes has been parsed by many researchers, most notably by Ghorpade. In this talk, we will discuss a fruitful generalization of the Grassmann codes by using the embeddings of Grassmannians into higher dimensional projective spaces. This new family of ``higher Grassmann codes'' has interesting connections with representation theory of SL_n over finite fields.
 

Commutative algebra seminar

Date and time: Thursday, 12 January 2023, 4 pm
Venue: Ramanujan Hall
Host: Manoj Keshari

Speaker: Soumi Tikedar, Diamond Harbour Women's University

Title: On a question of Moshe Roitman and its applications 
Abstract: Let A be a ring of dimension d and P be a projective A[T]-module of rank n. We say that p ∈ P is a unimodular element if there exists a homomorphism f in P*  such that f(p) = 1. When n > d, then Plumstead proved that P has a unimodular element. But this is not the case for n=d and n< d.

 In this talk, we will discuss the following results:

Theorem:  Let A be a ring of dimension d containing an infinite field k,  P be a projective A[T]-module of rank n such that  2n is not less than d + 3 and singular locus of Spec(A) is a closed set V(J) with ht J  is atleast d − n + 2. If P_f has a unimodular element for some monic polynomial f(T). Then P has a unimodular element.

Next, we will discuss some applications of Roitman's question to define the Euler class group, which serves as an obstruction group to detect the existence of unimodular elements in the Projective module with certain conditions. In this talk, we associate a stably free module to the Euler class group and show that the vanishing of this is the precise obstruction having P unimodular element.

Mathematics Colloquium

Date and time: Friday, 13 January, 2023, 4 pm
Venue: Ramanujan Hall
Host: Mayukh Mukherjee

Speaker: Prashant G. Mehta
Affiliation:  University of Illinois at Urbana-Champaign, USA

Title:  A variational formulation of nonlinear filtering
Abstract. There is a certain magic involved in recasting the equations in Physics in variational
terms. The most classical of these ‘magics’ is the Lagrange’s formulation of the Newtonian mechanics. An accessible take on all this and more appears in the February 19, 2019 issue of The New Yorker magazine. My talk is concerned with a variational (optimal control-type) formulation of the problem of nonlinear filtering/estimation. Such formulations are broadly referred to as duality between optimal estimation and optimal control. The first duality principle appears in the original (1961) paper of Kalman-Bucy, where the problem of minimum variance estimation is shown to be dual to a linear quadratic optimal control problem.

In my talk, I will describe a newly discovered generalization of the Kalman-Bucy duality to nonlinear filtering. As an application, I will discuss some comparisons between the stochastic stability of a Markov process and the (filter) stability of a conditioned process. Either of these is shown to arise from assuming a respective Poincaré inequality (PI). This is joint work with Jin Won Kim. The talk is based on the following papers:

https://arxiv.org/abs/2208.06586 and https://arxiv.org/abs/2208.06587
 

Virtual Commutative algebra seminar

Date and Time: Friday, 13 January 2023, 5:30 pm 
Gmeet link: https://meet.google.com/ekz-uhiv-grs
Host: J. K. Verma

Speaker: Lisa Seccia 
Affiliation: University of Genoa, Genoa, Italy

Title: Weakly-closed graphs and F-purity of binomial edge ideals
Abstract: Herzog et al. characterized closed graphs as the graphs whose binomial edge ideals have a quadratic Groebner basis. In this talk, we focus on a generalization of closed graphs, namely weakly-closed graphs (or co-comparability graphs). Building on known results about Knutson ideals of generic matrices, we characterize weakly-closed graphs as the only graphs whose binomial edge ideals are Knutson ideals (associated with a certain polynomial f). In doing so, we re-prove Matsuda's theorem about the F-purity of binomial edge ideals of weakly-closed graphs in prime characteristic and we extend it to generalized binomial edge ideals. 
Lastly, we will discuss some open conjectures on the F-purity of binomial edge ideals and on the relation between Knutson ideals and compatible ideals.