Speaker: Aryaman Jal.

Affiliation:  KTH, Stockholm.

Time: 5-6pm, Thursday, 6 July.

Venue:  Room 216, Mathematics Department.

Host : Madhusudan Manjunath

Title: Polyhedral geometry of bisectors and bisection fans.


Abstract: Every symmetric convex body induces a norm on its affine hull.
The object of our study is the bisector of two points with respect to this
norm. A topological description of bisectors is known in the 2 and
3-dimensional cases and recent work of Criado, Joswig and Santos (2021)
expanded this to a deeper study of the geometric, combinatorial and
topological properties of the bisector. A key object introduced was the
bisection fan of a polytope which they were able to explicitly describe in
the case of the tropical norm. We discuss the bisector as a polyhedral
complex, introduce the notion of bisection cones and describe the
bisection fan corresponding to other polyhedral norms. This is joint work
with Katharina Jochemko.

Commutative algebra seminar

M.Sc. project presentation

Monday, 17 July, 11 am

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

Venue:  Ramanujan Hall
Host: J. K. Verma
Speaker: Pradyumna Datta, IIT Bombay

Title: Syzygies of toric ideals  and integer programming
Abstract: We shall present algorithms to find integer solutions of linear Diophantine equations using computations of syzygies of toric ideals using Grobner bases of polynomial ideals.

Mathematics Colloquium:

Date and time: Monday, July 17, 4:00 p.m. 

Venue: Ramanujan Hall, Department of Mathematics

Host: Ravi Raghunathan

Speaker:  Subhajit Jana 

Affiliation:  Queen Mary's College

Title: Analytic Newvectors and moments of central L-values

Abstract: 

A reciprocity formula usually relates certain moments of two different families of L-functions that apparently have no connections between them. The first such formula was due to Motohashi who related a fourth moment of Riemann zeta values on the central line with a cubic moment of certain automorphic central L-values for GL(2). 

In the first talk, we describe some instances of reciprocity formulas both in low and high-rank groups and give certain applications to subconvexity and non-vanishing of central L-values. This talk will be partly based on a few joint works with Nunes and Blomer--Nelson.

Number Theory Seminar:

Date and time: Tuesday, July 18, 4:00 p.m

Venue: Ramanujan Hall, Department of Mathematics

Host: Ravi Raghunathan

Speaker:  Subhajit Jana 

Affiliation:  Queen Mary's College

Title: Analytic Newvectors and moments of central L-values

Abstract: In the second talk, we describe the concept of analytic newvectors and how they can be used for asymptotic evaluation of moments of central L-values in the conductor aspect. This talk will be partly based on joint work with Nelson.

Algebraic Groups seminar

Tuesday, 18 July 2023, 4.30 pm

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

Venue: Room 114

Host: Shripad Garge

Speaker: Dibyendu Biswas

Affiliation: IIT Bombay

Title: Diagonalizable groups and tori

Abstract: We resume our seminar with a quick review of the first two chapters of Springer's book.

Time permitting, the third chapter will be discussed.

Mathematics Colloquium

Wednesday, July 19, 4 pm

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

Venue: Ramanujan Hall

Host: Ravi Raghunathan

Speaker: Shreyasi Datta, Univ. of Michigan

Title: S-arithmetic Diophantine Approximation

Abstract: Diophantine approximation deals with quantitative and qualitative aspects of approximating numbers by rationals. A significant breakthrough by Kleinbock and Margulis in 1998 was to study Diophantine approximations for manifolds using homogeneous dynamics. After giving an overview of recent developments in this subject, I will talk about Diophantine approximation in the S-arithmetic set-up, where S is a finite set of valuations of rationals.

Dynamics, geometry and Number Theory Seminar
Thursday, July 20, 4 pm
=============================================
Venue: Ramanujan Hall
Host: Ravi Raghunathan
Speaker: Shreyasi Datta

Title: Quantitative Simultaneous Approximation (Seminar talk)

Abstract: One of the most fascinating problems in number theory is to study rationals that are close to a specific curve, more generally, close to a manifold. Showing that given a nice
decaying function, the points on any 'curved' manifold approximated at this function's rate are 'pnegligible' was a long-standing problem.  In recent ground-breaking work (arXiv:2105.13872), Beresnevich and Yang solved this problem.

In this talk, we will explain an effective version of their result. This is based on the work https://arxiv.org/abs/2209.14196.
 

CACAAG Seminar

Date: Monday, 24 July, 5 pm

----------------------------------

Host: Madhusudan Manjunath

Venue: Ramanujan Hall

Speaker: Aryaman Jal

Affiliation: KTH, Stockhom

Title: Polyhedral geometry of bisectors and bisection fans

 Abstract: Every symmetric convex body induces a norm on its affine hull.
The object of our study is the bisector of two points with respect to this
norm. A topological description of bisectors is known in the 2 and
3-dimensional cases and recent work of Criado, Joswig and Santos (2021)
expanded this to a deeper study of the geometric, combinatorial and
topological properties of the bisector. A key object introduced was the
bisection fan of a polytope which they were able to explicitly describe in
the case of the tropical norm. We discuss the bisector as a polyhedral
complex, introduce the notion of bisection cones and describe the
bisection fan corresponding to other polyhedral norms. This is joint work

Algebraic Groups seminar

Tuesday, 25 July 2023, 4 pm

--------------------------------

Venue: Room 114

Host: Shripad Garge

Speaker: Dibyendu Biswas

Affiliation: IIT Bombay

Title: Commutative unipotent groups - I

Abstract: We study additive functions with the aim of understanding commutative unipotent groups.

Ph.D. defense seminar

Date:  Wednesday, 26th July, 3 pm

---------------------------------------
Host: Debanjana Mitra

Venue: Ramanujan Hall

Speaker: Wasim Akram

Title: Feedback stabilization of certain evolution equations and its numerical study

Mathematics Colloquium

Wednesday, 26 July, 4 pm

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

Host: J. K. Verma

Venue: Ramanujan Hall

Speaker: Vaibhav Pandey

Affiliation: Purdue University, West Lafayette, IN, USA

Title: Classical invariant theory

Abstract: Invariant theory has played a key role in the development of modern commutative algebra. The classical invariant theory has a rich history going back at least to the seminal contributions of David Hilbert and Emmy Noether. The beauty of the subject lies in its strong insistence on computations and on the far-reaching consequences of these computations in modern algebra.

In this talk, we will focus on the classical invariant rings---they are the fixed subrings of the natural actions of linear algebraic groups on polynomial rings. Exploring key properties of these rings over rational numbers was a major research program in the late eighteenth century. We will talk about these properties and prove the corresponding ring theoretic properties of these invariant rings over fields of positive characteristic. This is joint work with Mel Hochster, Jack Jeffries, and Anurag Singh.

Commutative algebra seminar

Speaker: Vaibhav Pandey
Affiliation: Purdue University
Venue: Ramanujan Hall
Date: Thursday, 27 July 2023
Title: On the natural null cones of the classical invariant rings.

Abstract: The generic determinantal ring can be seen as a subring of a polynomial ring in a natural manner. When the field is infinite, this subring is, in fact, the invariant ring of the natural action of the general linear group on the ambient polynomial ring. Over a field of characteristic zero, the general linear group is linearly reductive, so the invariant ring splits from the polynomial ring. This immediately implies a wealth of nice properties for the generic determinantal rings including the Cohen-Macaulay property and rational singularities.

When the field has positive characteristic, the general linear group is typically not linearly reductive. It is natural to ask if the above embedding continues to split over an infinite field of positive characteristic. We show that strikingly, this embedding does NOT split in ANY positive characteristic if the general linear group is NOT linearly reductive.

Broadly speaking, this non-splitting is due to the Cohen-Macaulay property of the irreducible components of the null cone of this representation. We analyze the irreducible components of the null cones of the general linear group and also the null cone of the symplectic group in greater depth. We show that they are all F-regular in positive characteristic and calculate their divisor class groups independent of characteristic. This talk is based on two separate projects: One with Mel Hochster, Jack Jeffries, and Anurag Singh and another with Uli Walther and Yevgeniya Tarasova. 
 

Ph.D. Defence Seminar 

Friday, 28 July 2023, 2.30 pm

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

Venue: Ramanujan Hall

Host: Rekha Santhanam

Speaker: Soumyadip Thandar 

Affiliation: IIT Bombay

Title: Classifying Equivariant Rational Homotopy Type up to Isomorphic Cohomology Diagrams

Seminar on Optimal transport

Monday, 31 July, 3.30-4.45 pm

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

Host: Harsha Hutridurga

Venue: Ramanujan Hall

Speaker: Saumyajit Das

Affiliation: IIT Bombay

Title: Introduction to optimal transport

Abstract: The material covered in the first lecture will be recalled and then the duality formulation of the Kantorovich problem will be introduced.