Topology and Related Topics Seminar

Date

Friday, 5 April 2024, 10.15 am

Venue

Ramanujan Hall

Host

Rekha Santhanam

speaker

Sudarshan Gurjar 

Affiliation

IIT Bombay

Title

Vector bundles and Characteristic Classes

Abstract

This is the third talk in the series of three talks. We will give an introduction to the characteristic classes of a vector bundle. Characteristic classes are invariants of a vector bundle taking values in the singular cohomology of the base and satisfying the obvious functoriality property concerning pullback. They are the measure of the non-triviality of the vector bundle.
The background assumed will depend on the audience present.

Algebraic geometry seminar

Date

Friday, 5 April, 11.30 am

Venue

Ramanujan Hall

Host

Sudarshan Gurjar

speaker

Nitin Nitsure

Affiliation

Bhaskaracharya Pratishthana, Pune

Title

 Introduction to the Tohoku (1957) paper of Grothendieck-part 1

Abstract

 In 1957, Alexander Grothendieck published a long paper in
two parts in the Tohoku Mathematical Journal, Japan,
whose title can be translated as `On some points of homological
algebra'.  This paper revolutionized the subject of homological
algebra and it sheds unifying light on topics in diverse areas such as group theory, Lie theory, number theory, algebraic topology, etc. It had profound applications to Algebraic Geometry, which were made apparent by Grothendieck and his school in the next few decades. The ideas of abelian categories and additive functors between them, and their derived functors, come from this paper. The famous `Grothendieck spectral sequence' first appeared here. I will give a series of two talks, introducing this paper's ideas and reporting some subsequent developments.
The first talk will assume no prior knowledge except of the basics of groups, rings, and modules at an undergraduate level. In it,
I will explain the historical context, and introduce the paper's themes in simple terms. The second talk will assume some
acquaintance with Algebraic Geometry.

Analysis seminar

Date

Friday, April 5, 4 pm - 5 pm

Venue

Ramanujan Hall

Host

Chandan Biswas

speaker

 Prachi Mahajan

Affiliation

IIT Bombay

Title

 The Squeezing function & the Fridman invariant

Abstract

The squeezing function and its dual, the Fridman invariant, are biholomorphic invariants, both of which capture the coarse metric geometry of the given domain. I will describe some results on the squeezing function and Fridman invariant such as their boundary behavior, their utility in classifying the unit ball under various hypotheses, and estimates near the boundary of the given domain. In the second part, I will compare this pair of invariants by showing that they are both equally capable of determining the boundary geometry of a bounded domain if their boundary behavior is apriori known.