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.