Lecture series on Hodge Theory

Wednesday, 1 November, 11.30-1.00

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

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: V. Srinivas, IIT Bombay

Title: Kahler condition and its consequences
Abstract: I will continue the sketch of the proof of the Hodge Theorem. After that, I will give some introduction to variation of Hodge structures, in a couple of lectures.

Lecture series on Hodge Theory

Thursday, 2 November, 11.30-1.00

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

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: V. Srinivas, IIT Bombay

Title: Kahler condition and its consequences

Abstract: I will continue the sketch of the proof of the Hodge Theorem. After that, I will give some introduction to variations of Hodge structures, in a couple of lectures.

Topology and Related Topics Seminar

Thursday, 2  November  2023, 2:30 -3:30 pm

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

Venue: Room 215

Host: Rekha Santhanam

Speaker:  Aparajita Karmakar

Affiliation: ISI Kolkatta 

Title:  Regularity in Ring Spectra 

Abstract: In this talk, our goal will be to translate the notion of regularity in rings to regularity of ring spectra. Although the structural definition and the definition in terms of Ext can be extended for ring spectra, another equivalent definition of regularity involving finitely generated modules is not readily adaptable for spectra. 

Analysis and PDE seminar

Thursday, 2 Nov. 5.15 pm

=========================
Venue: Ramanujan Hall
Host: Chandan Biswas

Speaker: Chandan Biswas, IIT Bombay

Title: A basic introduction to Fourier restriction estimates
Abstract: This is the fourth talk of the series. We will talk about Fourier restriction to the moment curve.

Lecture series on algebraic stacks

Monday, 06 Nov 2023, 11:30 am

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

Host: Sudarshan Gurjar

Venue: Ramanujan Hall

Speaker: Nitin Nitsure

Title: Schematic loci of algebraic spaces.

Abstract: Michael Artin showed that any algebraic space has a unique largest open subscheme which is dense in it. In this talk, we will give a proof. The ingredients of the proof are of even more general interest: the chief one is Grothendieck's form of Zariski's main theorem. The other inputs are the Grothendieck-Gabriel quotient theorem and a typical use of generic points of schemes. The lecture will focus on conceptual explanations, relegating the detailed proof to notes in the form of a handout for the participants.

Geometric analysis seminar

Monday 6 Nov, 2023, 5:30 - 6:30 pm

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

Host: Saikat Mazumdar

Venue: Ramanujan Hall, Zoom meet, Link TBA

Speaker: Ayush Khaitan

Affiliation: Rutgers University

Title: Conformal Geometry and Ricci flow 

Abstract: We study a surprising duality between conformal geometry and Ricci flow. Using a classical construction called the Fefferman-Graham ambient space, we construct an infinite family of fully nonlinear analogs of Perelman's F and W functionals and study their monotonicity under several natural conditions.

Topology and Related Topics Seminar 

Tuesday, 7  November 2023, 2:30 -3:45 pm

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

Venue: Ramanujan hall 

Host: Rekha Santhanam

Speaker: Sayed Sadiqul Islam

Affiliation: IIT Bombay

Title: Kozul Complexes

Abstract: We'll begin by discussing the Koszul complex and regular rings. Then, we'll look into some fundamental properties of these ideas. Afterward, we'll explain the necessary and sufficient conditions that make a Noetherian local ring regular, without diving into complex proofs.

Analysis of PDE Seminar

Tuesday, 07 November 2023, 4:00 pm

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

Venue: Room 113, Department of Mathematics

Host: Neela Nataraj

Speaker: Ricardo Ruiz Baier

Affiliation: Monash University

Title: New mixed finite element formulations for the coupled Stokes /Poisson-Nernst-Planck equations

Abstract: I will discuss a Banach spaces-based framework and new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson--Nernst--Planck equations (a nonlinear model describing the dynamics of electrically charged incompressible fluids). The pseudostress tensor, the electric field (rescaled gradient of the potential) and total ionic fluxes are used as new mixed unknowns. The resulting fully mixed variational formulation consists of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddle-point problem with linear source terms, which is in turn
additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babu\v{s}ka--Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach--Ne\v{c}as--Babu\v{s}ka theorem. An analogous approach (but using now both the Brouwer and Banach theorems and stability conditions on arbitrary FE subspaces) is employed at the discrete level. A priori error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., Raviart--Thomas elements of order $k$ along with piecewise polynomials of degree $\le k$. Finally,  several numerical experiments confirm the theoretical error bounds and illustrate the
balance-preserving properties and applicability of the proposed family of
methods.

Lecture series on Hodge Theory

8 Nov., Wed.  at 11.30 am
=========================

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: V. Srinivas, IIT Bombay

Title: Some elements of variation of Hodge structures

Abstract: I will touch on some aspects of variations of Hodge structure, finishing up my series of lectures for this semester.

Combinatorics and TCS Seminar

Wednesday, 8th Nov. 2.30 pm

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

Host: Niranjan Balachandran

Venue:  Ramanujan Hall

Speaker: Venkitesh (University of Haifa, Israel)

Title: List Decoding Randomly Punctured Polynomial Ideal Codes
Abstract: List decoding is a paradigm in algorithmic coding theory, where for a received word (possibly corrupted to some extent), the objective is to efficiently obtain a small list of 'approximately correct' codewords. A classic problem in this context is to obtain constructions of 'polynomial-based' codes that can be optimally list decoded (up to the information-theoretic limit), with optimal field size and output list size.  Such codes will be said to 'achieve list decoding capacity'. While no such 'perfectly optimal' explicit constructions are known as yet, it has been observed in previous works that a promising trick of 'folding' codewords enables achieving capacity (non-optimally in other parameters), prominently at the cost of a blow-up in the folding size as the 'gap to capacity' approaches zero.  Some recent previous works showed a breakthrough where 'unfolded' polynomial codes (called Reed-Solomon codes) achieve capacity under a random choice of evaluation points.  We observe a similar result in the folded setting, with the folding size, held constant (independent of the gap to capacity), and for a much larger class of 'polynomial ideal codes'.  This talk is based on an ongoing joint work with Noga Ron-Zewi and Mary
Wootters.

CACAAG Seminar

Wednesday, 8 Nov. 5:30 pm

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

Venue: Ramanujan Hall.

Host: Madhusudan Manjunath.

Speaker: Trygve Johnsen. 

Affiliation: The Arctic University of Norway.

Title: Geometry of matroids, with a view toward application in coding theory.

Abstract: We will try to explain some of the material in "Matroid Theory for Algebraic Geometers" by Erik Katz, and "Simplicial Generation of Chow rings of Matroids" by Bachman, Eur & Simpson. Here one associates geometric objects like toric varieties with matroids and describes the fans that give rise to them.  One also describes Chow rings of matroids, and how the (Bergman fan of a ) matroid itself can be viewed as an element of the Chow ring, or Minkowski weight,  for a (usually) "larger " uniform matroid.  If time permits, we will mention briefly how Chow rings can be defined also for q-matroids, an object arising from rank metric codes, analogous to how usual matroids are a tool to describe codes with the Hamming distance.

Lecture series on Hodge Theory

9 Nov., Thu.  at 11.30 am
=========================

Venue: Ramanujan Hall

Host: Sudarshan Gurjar

Speaker: V. Srinivas, IIT Bombay

Title: Some elements of variation of Hodge structures

Abstract: I will touch on some aspects of variations of Hodge structure, finishing up my series of lectures for this semester.

Topology and Related Topics Seminar 

Thursday, 9  November 2023, 2:30-3.45  pm

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

Venue: 215

Host: Rekha Santhanam

Speaker: Navnath Daundkar

Affiliation: IIT Bombay

Title: Cohomology of moment angle complexes - III

Abstract: In this talk, we will present the proof of Buchstaber-Panov's theorem, which describes the cohomology ring of a moment angle complex associated with a simplicial complex K as the tor algebra of K.

Analysis of PDE Seminar

Thursday, 09 November 2023, 02:15 pm

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

Venue: Room 216, Department of Mathematics

Host: Harsha Hutridurga

Speaker: Rishabh Gvalani

Affiliation: Max Planck Institute, Leipzig.

Title: Exponential mixing by random velocity fields

Abstract: We establish exponentially fast mixing for passive scalars driven by two well-known examples of random divergence-free vector fields. The first one is the alternating shear flow model proposed by Pierrehumbert, in which case we set up a dynamics-based framework to construct such space-time smooth universal exponential mixers. The second example is the statistically stationary, homogeneous, isotropic Kraichnan model of fluid turbulence. In this case, the proof follows by a new explicit identity for the evolution of negative Sobolev norms of the scalar. This is based on joint works with Alex Blumenthal (Georgia Tech)
and Michele Coti Zelati (ICL), and Michele Coti Zelati and Theodore Drivas (Stony Brook).

Commutative algebra Seminar

Thursday, 9 Nov. 4-5 pm

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

Venue:  Ramanujan Hall

Host: Tony  Puthenpurakal

Speaker: R. V. Gurjar, IIT Bombay

Title:  Brieskorn-Pham Singularities.

Abstract: Let  B=k[X_1,...,X_{n+1}]/(X_1^{a_1}+...+X_{n+1}^{n+1}) where k is the field of complex numbers and X the corresponding affine variety. These have been studied from many angles: (1) Brieskorn-Pham, Milnor, (2) from the topological viewpoint, giving rise to exotic spheres. (3) Storch for calculating the divisor class group of B (4) Flenner, Keichi Watanabe for characterizing rational singularities among them. (4) Recently Michael Chitayat wrote a beautiful thesis at Univ. of Ottawa characterizing 3-dimensional B-P singularities that admit a non-trivial locally nilpotent derivation (which is equivalent to having a G_a action on X). He has solved a conjecture about this completely. (5) I asked Michael which of the 3-dimensional B-P singularities define rational varieties in the sense of function field. Using ideas in his thesis he answered this completely.

Analysis and PDE seminar

Thursday, 9th Nov. 5:15 pm - 6:15 pm

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

Host: Chandan Biswas

Venue: Ramanujan Hall

Speaker: Chandan Biswas, IIT Bombay

Title: A basic introduction to Fourier restriction estimates

Abstract: This is the fifth talk of the series. We will finish our discussion on Fourier restriction to the moment curve.

Analysis and PDE seminar

Monday 13th Nov, 2023, 4 pm - 5 pm

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

Venue: Ramanujan Hall

Host: Chandan Biswas

Speaker: Senthil Raani

Affiliation: IISER Berhampur

Title: Distance Set Problems

Abstract: The distance set ∆(E) of a set E in Euclidean space consists of all non-negative numbers that represent distances between pairs of points in E. How does the structure of E impact that of ∆(E)? Questions of this nature play a fundamental role in geometric measure theory. We will begin with a brief history of results and conjectures on ∆(E). Apart from measure theoretic techniques, the asymptotic of the Fourier transform of measures supported on E plays a vital role in this study. Our main goal in this talk is to discuss a few properties of ∆(E) when E is sparse but has a large Hausdorff dimension. This is based on recent joint work with Prof. Malabika Pramanik.

Mathematics Colloquium

Wednesday, 15 November, 4-5 pm

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

Venue: Ramanujan Hall

Hosts: Sanjiv Sabnis and Radhe Srivastava

Speaker: Prof. Sujit Ghosh

Affiliation: North Carolina State University

Title: Multivariate Dependence Beyond Correlation: Nonparametric Copulas

Abstract: In the field of climate, finance, insurance, system reliability, etc., it is often of interest to measure the dependence among variables by modeling a multivariate distribution using a copula. The copula models with parametric assumptions are easy to estimate but can be highly biased when such assumptions are false, while the empirical copulas are non-smooth and often not genuine copula, making the inference about dependence challenging in practice. As a compromise, the empirical Bernstein copula provides a smooth estimator, but the estimation of tuning parameters remains elusive. In this paper, by using the so-called empirical checkerboard copula, we build a hierarchical empirical Bayes model that enables the estimation of a smooth copula function for arbitrary dimensions. The proposed estimator based on the multivariate Bernstein polynomials is itself a genuine copula, and the selection of its dimension-varying degrees is data-dependent. We also show that the proposed copula estimator provides a more accurate estimate of several multivariate dependence measures, which can be obtained in closed form. We investigate the asymptotic and finite-sample performance of the proposed estimator and compare it with some nonparametric estimators through simulation studies. An application to portfolio risk management is presented, along with a quantification of estimation uncertainty.

The presentation is based on a recently published paper with Dr. Lu Lu:
https://doi.org/10.3390/math11204383

Analysis and PDE seminar

Thursday 16th Nov, 2023, 3.30-4.30 pm

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

Venue: Ramanujan Hall

Host: Chandan Biswas

Speaker : Chandan Biswas, IIT Bombay

Title: A basic introduction to Fourier restriction estimates

Abstract: This is the sixth talk of the series. We will finish our discussion on Fourier restriction to the moment curve.

Commutative algebra Seminar

Thursday  16 Nov 4-5 pm

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

Venue:  Room 215

Host: Tony  Puthenpurakal

Speaker: R. V. Gurjar, IIT Bombay

Title:  Brieskorn-Pham Singularities.

Abstract: Let  B=k[X_1,...,X_{n+1}]/(X_1^{a_1}+...+X_{n+1}^{n+1}) where k is the field of complex numbers and X the corresponding affine variety. These have been studied from many angles: (1) Brieskorn-Pham, Milnor, (2) from the topological viewpoint, giving rise to exotic spheres. (3) Storch for calculating the divisor class group of B (4) Flenner, Keichi Watanabe for characterizing rational singularities among them. (4) Recently Michael Chitayat wrote a beautiful thesis at Univ. of Ottawa characterizing 3-dimensional B-P singularities that admit a non-trivial locally nilpotent derivation (which is equivalent to having a G_a action on X). He has solved a conjecture about this completely. (5) I asked Michael which of the 3-dimensional B-P singularities define rational varieties in the sense of function field. Using ideas in his thesis he answered this completely.

Lecture series on algebraic stacks
Monday 20 November, 11.30 am

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

Venue: Room No 215, Department of Mathematics
Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (Retd)

Title: The classifying stack BG for an algebraic group

Abstract: To any Lie group, there is classically associated a  topological space BG with the requisite universal property in the homotopy category of paracompact topological spaces. For example, for G = GL(n) the space BG is the infinite Grassmannian. However, when we go to the algebraic category (say schemes or algebraic spaces and their morphisms), such a space BG does not exist. This is a paradigmatic example where algebraic stacks rescue the situation. In this lecture, we will explain the construction of an algebraic stack BG which has the requisite universal property of classifying principal G-bundles, where G is an algebraic group. The algebraic cohomology of this stack gives the algebraic cohomological version of the characteristic classes of principal G-bundles.

Analysis and PDE seminar

Monday 20th Nov. 4 pm - 5 pm

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

Venue: Ramanujan Hall

Host: Sanjay Pusti

Speaker: Ankit Bhojak
Affiliation: IISER Bhopal

Title: Sharp endpoint L^p-estimates for bilinear spherical maximal functions
Abstract: Attached.

Probability and Statistics Seminar

Thursday, 23  November  2023, 10:00 am

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

Venue: Ramanujan hall

Host: Ayan Bhattacharya

Speaker: Souvik Dhara

Affiliation: Purdue University 

Title: Community Detection with Censoring

Abstract: Recovering latent communities is a key unsupervised learning task in network data with applications spanning across a multitude of disciplines. For example, identifying communities in web pages can lead to faster searches, classifying regions of the human brain in communities can be used to predict the onset of psychosis, and identifying communities of assets can help investors manage risk by investing in different communities of assets. However, the scale of these massive networks has become so large that it is often impossible to work with the entire network data. In this talk, I will talk about some theoretical progress for community detection in a probabilistic setup especially when we have missing data about the network. Based on joint works with Julia Gaudio, Elchanan Mossel, and Colin Sandon.

Analysis and PDE seminar

Thursday 23rd Nov. 3 pm - 4 pm

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

Venue: Ramanujan Hall

Host: Chandan Biswas

Speaker: Chandan Biswas
Affiliation: IIT Bombay

Title:  A basic introduction to Fourier restriction estimate.
Abstract: We will finish our discussion on restriction onto the moment curve.

Lecture series on algebraic stacks
Monday 24 November, 11.30 am
=======================

Venue: Ramanujan Hall
Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (Retd)

Title: An Introduction to Gerbes

Abstract: Gerbes (for good reason) have become very fashionable objects in algebra, algebraic geometry, differential topology, and physics. Algebraists come across them for example in their study of Brauer groups (see Milne `Etale Cohomology' Chapter 4). Differential geometers study connections on these and their relation to characteristic classes. These feature in the works, for example, of Hitchin, Brylinski, Breen, etc. I will not be able to say anything about the physics applications. Gerbes are very important (in many ways) in Algebraic Geometry. This lecture will give an introduction to the subject. The basic technology of gerbes involves Stacks, and that is why this talk is in the ongoing series on Algebraic Stacks.

Combinartorics and TCS seminar

Friday, 24 November 11.30 am

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

Venue: Ramanujan Hall

Host: Niranjan Balachandran

Speaker: Shagnik Das

Affiliation: National University of Taiwan

Title: Covering grids with multiplicity

Abstract: It is a fairly simple exercise to determine the minimum number of
hyperplanes needed to cover all the points of a finite grid $S_1 \times S_2 \times \dots \times S_d \in \mathbb{F}^d$. However, the situation becomes much more complex if you add the condition that one of the points of the grid must be avoided, leading to a classic problem in combinatorial geometry. This was resolved by Alon and Füredi in a classic result that popularised the use of algebraic methods in combinatorics.  The recent work of Clifton and Huang, in which they considered the question a variant of the problem where the nonzero points of a hypercube should be covered multiple times while avoiding the origin, brought renewed interest to this problem. In addition to Alon-Füredi-style algebraic arguments, they used linear programming to asymptotically resolve the problem in certain ranges. Despite all the attention this problem has received, there remain many open problems, even in the case of two-dimensional grids over $\mathbb{R}$. In this talk, after surveying the background and introducing the techniques used, we shall present some recent results that resolve the problem for almost all two-dimensional grids.  The new results are joint work with Anurag Bishnoi, Simona Boyadzhiyska, and Yvonne den Bakker, and separately with Valjakas Djaljapayan, Yen-Chi Roger Lin, and Wei-Hsuan Yu.

Pre-synopsis seminar

Tuesday, November 28, 10:00 am

=====================
Venue:  https://meet.google.com/mrn-wpbn-hnh

Host: Prachi Mahajan

Speaker: Rahul Kumar
Title:  The Bergman-Friedman invariant on some classes of pseudo-convex domains

Combinatorics and TCS seminar

Tuesday, 28th Nov 2023, 11 am

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

Venue: Ramanujan Hall

Host: S. Sivaramakrishnan

Speaker: Haritha Cheriyath, TIFR-CAM
Title: ENUMERATION OF SUBWORDS ON SEQUENCES AND ITS APPLICATIONS
 

Abstract: We start with an enumeration problem studied by Guibas and Odlyzko in 1979 and its applications in seemingly unrelated scenarios that include game theory, pattern matching algorithm, graph theory, and symbolic dynamics. One of the main objects of our study is a subshift of finite type, which is used as a tool to model a large class of dynamical systems. It consists of a collection of all one-sided sequences over a finite symbol set which contains none of a given finite collection of words. We discuss its correspondence with an edge labeled multigraph and hence with its associated adjacency matrix. We see how some (topological as well as measure-theoretic) properties of a subshift of finite type are studied using this correspondence and solve a generalized version of the enumeration problem.

Algebraic Groups seminar

Tuesday, 28 November 2023, 4 pm

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

Venue: Ramanujan Hall

Host: Shripad Garge

Speaker: Chayan Karmakar

Affiliation: IIT Bombay

Title: Differentials & Smooth Points - IV
Abstract: We study the notion of Lie algebra of a linear algebraic group

Pre-Synopsis Seminar
Wednesday, 29 November 2023, 10:00 am
===========================
Venue: Conference Room, Math Office
Host: Koushik Saha

Speaker: Kiran Kumar A.S.
Title: On Eigenvalues of Linial-Meshulam Complexes and Generalized Patterned Random Matrices