Number Theory Seminar

Date and time

Tuesday, 3 January 2023, 4:00 pm

Venue

Ramanujan Hall

Speaker

Saurabh Kumar Singh

Affiliation

IIT Kanpur

Title:

The Circle Method and the Subconvexity Problem

Abstract:

In this talk, we shall discuss about the early days of the Circle Method and the Subconvexity Problem. We will examine how the Circle Method developed into one of the most powerful techniques, and how Subconvexity became one of the central problems in Analytic Number Theory.

Mathematics Colloquium

Date

Wednesday, 4 January 2023, 4 pm

Venue

Ramanujan Hall

Speaker

Mahir Bilen Can

Affiliation:

Tulane University, New Orleans, USA 

Title:

Linear Algebraic Monoids From a Bird's Eye Viewpoint

Abstract:

The theory of algebraic groups, which began as an offshoot of Lie theory, has become one of modern mathematics' crowning achievements. It has connections and applications to almost all areas of mathematics. Surprisingly, the more general theory of algebraic monoids has a shorter history, dating only to the early 1980s. In this talk, after making a broad introduction to the theory of algebraic groups, we will review the theory of linear algebraic monoids. The purpose of our discussion is to explain why further investigation and focus should be given to this mathematical gold mine.

Algebraic Geometry  Seminar

Date:

Thursday, 5 January, 2023, 2.15 pm

Venue:

Ramanujan Hall

Speaker:

Atharva Korde

Affiliation:

Univ. of British Columbia,Vancouver, BC, Canada

Title:

An introduction to Gromov-Witten and Donaldson-Thomas theory

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.

Mathematics Colloquium

Date and time: 6 January 2023
Venue: Ramanujan Hall

Speaker: Amod Agashe, Florida State University, USA

Title: The second part of the Birch and Swinnerton-Dyer conjecture for elliptic curves

Abstract: We will define elliptic curves and state the Birch and Swinnerton-Dyer conjecture. The first part of the conjecture is a Clay millennium prize problem. We will focus on the second part, and mention some of our results relevant to it.

Virtual Commutative Algebra  Seminar

For more information and links to videos and lectures notes previous seminars held during 2020-2022 visit the website of VCAS:

 https://sites.google.com/view/virtual-comm-algebra-seminar

Date

Friday, 6 January 2023, 5:30 pm

Venue

Gmeet link: meet.google.com/utk-dhcw-oeu

Speaker

Vu Quang Thanh

Affiliation

 Hanoi University of Science and Technology, Hanoi, Vietnam

Title

Regularity of powers and symbolic powers of squarefree monomial ideals

Abstract

I will discuss the problem of comparing/computing the regularity of symbolic powers and regular powers of certain classes of square-free monomial ideals focusing on edge ideals of graphs.

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.
 

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 Groups Seminar
Date and time: Tuesday, 17 Jan. 2023, 4-5.30 pm

Venue: Room 215
Host: Shripad Garge

Speaker: Arghya Pramanik
Affiliation: Dept of Mathematics, IIT Bombay

Title: Some algebraic geometry – II
Abstract:  This is the second lecture on this topic. We are following the first chapter of Tony A. Springer's book. In this lecture, we will discuss locally ringed spaces and then give the definition of an affine algebraic variety.

Seminar on stochastic processes
Date and Time: Tuesday, 17 January, 2023, 4 pm

Venue: Ramanujan Hall

Host: Ayan Bhattacharya

Speaker: Siva Athreya, ICTS, Bengaluru

Title: Graphon-valued Stochastic Processes

Abstract: We will present our attempts thus far to develop a theory of 
graphon-valued stochastic processes. We will present a brief review of 
the theory of Graphons and dynamics constructed on the space of 
graphons. We shall construct and analyze a natural class of such 
processes arising from population genetics. In conclusion, we shall 
present the challenges in our ongoing work on constructing dynamics 
where the edges and vertices interact with each other. This is joint 
work with Frank den Hollander and Adrian Roellin.

Seminar on number theory

Date and time: Wednesday, 18 January 2023 at 11:30 AM

Venue: Ramanujan Hall

Host: Dipendra Prasad

Speaker:  Rahul Dalal

Affiliation: Johns Hopkins University

Title: Explicit Trace Formulas and Statistics of Families of "Nice" Automorphic Representations

Abstract: The lecture will discuss a bit of the famous Arthur-Selberg trace formula with a view to applying it to families of automorphic representations. 

Mathematics Colloquium
Date and Time: Wednesday, 18 January, 2023, 4 pm
Host: Sudhir Ghorpade

Speaker: Sudesh Kaur Khanduja
Affiliation: Panjab University, Chandigarh

Title: When is Z[θ] the ring of integers?
Abstract: Let K = Q(θ) be an algebraic number field with θ an algebraic integer having minimal polynomial f(x) over Q. Let AK denote the ring of algebraic integers of K. In this talk, we shall discuss some necessary and sufficient conditions to be satisfied by f(x) so that AK = Z[θ]. In particular when f(x) is an irreducible trinomial x^n+ax^m +b ∈ Z[x], then we shall describe a set of necessary and sufficient conditions in terms of prime powers dividing a, b, m and n, for any prime p to divide the group index [A_K : Z[θ]]. Using the well known Dedekind Criterion, we shall also discuss a generalisation of this result for a simple ring extension R[η] of a valuation ring R to be integrally closed when η is a root of an irreducible trinomial x^n+ax^m +b belonging to R[x]. The latter result yields interesting number theoretic applications. This is partly based on joint works with A. Jakhar, B. Jhorar, Sumandeep Kaur, M. Kumar, and N. Sangwan.
 

Commutative Algebra Seminar
Date and time: Thursday 19 January, 2023, 4 pm

Venue: Ramanujan Hall
Host: Tony Puthenpurakal

Speaker: Tony Puthenpurakal
Affiliation: Mathematics Department, IIT Bombay

Title: An analogue of Rees Theorem for filtrations
Abstract: Let A be an analytically unramified Cohen-Macaulay local ring. Let {I_n} be a filtration of m-primary ideals. Let I be an m-primary ideal contained in I_1. It is easily seen that the multiplicity of {I_n} is at least multiplicity of I. We show that if equality holds then the Rees algebra of the filtration is a finite module over Rees algebra of I.
 

Discrete Mathematics Seminar

Date and time: Monday, 23rd Jan 2023 at 2.30 pm

Venue: Ramanujan Hall

Host:Sudhir R Ghorpade

Speaker:  Rakhi Pratihar

Affiliation: IIIT Delhi

Title: Matroids, Euler characteristics, Möbius functions, and q-analogs

Abstract: For a co-loopless matroid M of rank r, the reduced Euler characteristic of the corresponding matroid complex S_M is determined by a Mobius function via the relation χ(S_M) = (−1)^{r−1} |μ_{L_M} (\hat{0}, \hat{1})|, where L_M is the lattice of cycles of M. The relation can be seen as a link between the poset of independent sets of M, and the geometric lattice of flats of the dual matroid M^*, which has a very interesting application to coding theory. It has been shown that the generalized Hamming weights of a linear code can be determined by the Betti numbers of the Stanley-Reisner ring of an associated matroid. In this talk, I will present a q-analogue of this relation where one consider the Euler characteristic of the order complex associated to a q-matroid. I will also briefly discuss its potential application to the theory of rank metric codes.

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.
 

Mathematics Colloquium
Tuesday, 24th Jan 2023 at 2.30 pm
Venue: Ramanujan Hall
Host: Sudhir R Ghorpade
Speaker: Mrinmoy Datta
Affiliation: IIT Hyderabad
Title: Codes from Schubert varieties and their parameters
Abstract: The codes from Schubert varieties, a natural generalization of the codes from Grassmann
varieties, have drawn the attention of several mathematicians in the recent past. In this talk, we will
review the known results regarding some basic parameters of these codes (such as dimension and
minimum distance) and present recent developments on a conjecture related to the characterization
of minimum distance codewords of these codes by Ghorpade and Singh. We will also present recent
results on the generalized Hamming weights of the codes from Schubert varieties. This is a joint work
with Sudhir Ghorpade and Avijit Panja.

Algebraic Groups Seminar

Date and time: Tuesday, 24 January 2023, 4 pm

Venue: Room 215

Host:Shripad M. Garge

Speaker:  Arghya Pramanik

Affiliation: IIT Bombay

Title: Some algebraic geometry. III

Abstract: We continue to follow the first chapter of Tonny A. Springer's book. We discuss projective varieties in this lecture.

Combinatrorics seminar
Date: 25th Jan 2023
Time: 14:30-15:30 hrs.
Venue: Room-113, Maths Dept.

Speaker: Dr. Shivani Goel, IISc Bangalore
Host: S. Krishnan

Title: Resistance matrices of balanced directed graphs
Abstract: Attached as a pdf file

Mathematics Colloquium

Date and time: Wednesday 25th January, 2023, 4:00 pm

Venue: Ramanujan Hall

Host: Sanjeev Sabnis

Speaker:  Prof. Sujit Ghosh

Affiliation: Department of Statistics, North Carolina State University, Raleigh

Title: A Gambler's Journey through Monte Carlo

Abstract: In the era of 'data science' we often require solving large dimensional optimization and integration problems for parameter estimation and/or for predictive analytics. Even in moderately large dimensions, the gradient-based deterministic numerical optimization and deterministic grid-based numerical integration methods suffer from the so-called 'curse of dimensionality.' In such cases stochastic or so-called Monte Carlo methods can be shown to work reasonably well even when the dimensions become much larger. This presentation will provide a brief tour of a Gambler's journey starting with basic notions of Monte Carlo methods with simple geometry based examples to more contemporary data science problems on the use of Bayesian methods for variable selection problems.

Virtual commutative algebra semnar
27 January 2023, 5:30 pm
For getting the Google meet link
Please contact the organisers and register by filling a form at the site
https://sites.google.com/view/virtual-comm-algebra-seminar/home
Host: J. K. Verma
Speaker: Alessio Caminata
Affiliation: University of Genoa, Italy
Title: Determinantal varieties from point configurations on hypersurfaces
Abstract: Point configurations appear naturally in different contexts, ranging from the study
of the geometry of data sets to questions in commutative algebra and algebraic geometry
concerning determinantal varieties and invariant theory. In this talk, we bring these
perspectives together: we consider the scheme X_{r,d,n} parametrizing n ordered points in r-
dimensional projective space that lie on a common hypersurface of degree d. We show that
this scheme has a determinantal structure and, if r>1, we prove that it is irreducible, Cohen-
Macaulay, and normal. Moreover, we give an algebraic and geometric description of the
singular locus of X_{r,d,n} in terms of Castelnuovo-Mumford regularity and d-normality.

This yields a complete characterization of the singular locus of X_{2,d,n} and X_{3,2,n}.
This is joint work with Han-Bom Moon and Luca Schaffler.

Algebraic Geometry Seminar

Monday 30th Jan, 11:30 am

Venue: Ramanujan hall

Host: Sudarshan Gurjar

Speaker: Arusha

Affiliation: TIFR, Mumbai

Title: Poincare and Picard bundles for moduli spaces of vector bundles over nodal curves

Abstract: Poincare and Picard bundles and their different variants have been a topic of interest ever since the quest for moduli spaces of vector bundles was initiated, owing to their universality. Though a great deal is known about these objects in the case of smooth curves, the study on singular curves has been relatively slow. Interestingly, the results for irreducible nodal curves are very similar to those for smooth curves; however, the proofs are different and difficult. It was known since late 1960s that there does not exist a Poincar´e bundle (a universal family) for the moduli problem of vector bundles on smooth curves if the rank and degree are not coprime. The primary aim of the talk is to discuss the non-existence of a Poincare bundle in the non-coprime case for nodal curves. There has also been ample interest to understand the stability of Poincar´e and projective Poincare bundles as well as Picard and projective Picard bundles. The secondary aim is to discuss the stability of projective Poincar´e and Picard bundles, again when the degree and rank are not relatively prime to each other in the context of nodal curves. On the way to achieve these goals, we compute the codimension of a few closed subsets of the moduli spaces. They are of independent interest and have other applications; we discuss a few of them. This is a joint work with Prof. Usha Bhosle and Dr. Sanjay Singh.

==================================

PDE seminar

Monday 30 January, 02:15 PM to 03:45 PM

==================================

Host: Harsha Hutridurga
Venue: Ramanujan Hall, Department of Mathematics

Speaker: Bishnu Prasad Lamichhane
Affiliation: University of Newcastle, UK


Title: A finite element method for a biharmonic equation using biorthogonal systems.
Abstract: In this talk we will discuss applications of biorthogonal systems in a finite element method for the biharmonic equation with clamped and simply supported boundary conditions. We also discuss the construction of biorthogonal systems and their approximation properties.

=========================

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  and Commutative Algebra seminar

Tuesday, 31 January, 11:30 am

==============================================

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR Mumbai (retd)

Title: Multi-variable calculus via commutative algebra

Abstract:  The nicest functions of all are the linear ones. But most of the important functions that we meet are non-linear. The derivative of a function gives a linear approximation to a nonlinear function. The Jacobian matrix is the version of this in several variables. Theorems such as the implicit function theorem of multivariable calculus say that under suitable hypothesis, a nonlinear function behaves in a small enough neighbourhood of a point quite like the linear function defined by its Jacobian matrix at that point. But what happens when we move from real numbers and Euclidean spaces (or smooth manifolds) to arbitrary fields, commutative rings, varieties and schemes? For example, what happens if functions are replaced by integers in a number field? What can differential calculus tell us about these? The amazing answer, discovered by Zariski, Grothendieck, Michael Artin and the modern algebraic geometers. is that we can essentially replicate the results of multivariable differential calculus via commutative algebra, in fact, it is possible to do much more, and then apply it to algebraic geometry. This led to a close study in the 1960s of the local behaviour of functions (or morphisms) on varieties and schemes, which occupies more than half the space in Grothendieck's famous `EGA'. This lecture is a semi-popular account of some of these ideas, and will not require any prior knowledge beyond standard undergraduate material. It is a special `opening lecture' of a more technical semester-long course on local structure of morphisms in algebraic geometry, which is in turn is the part II of the multi-semester course on Algebraic Stacks and Moduli which began in the last semester.

Algebraic Groups Seminar

Date and time: Tuesday, 31 January 2023, 4 pm

Venue: Room 215

Host:Shripad M. Garge

Speaker:  Deepkumar Makadiya

Affiliation: IIT Bombay

Title: Linear algebraic groups, basic themes. I

Abstract: We now get into the second chapter of T. A. Springer's book to start studying linear algebraic groups.