Colloquium
Speaker: Patrick Polo (Visiting Professor IIT Bombay)
Host: Dipendra Prasad
Title: Representation theory of Algebraic groups in characteristic p
Time, day and date: 4:00:00 PM, Monday, March 3
Abstract: Representation theory of Algebraic groups in characteristic 
zero was completed by Cartan and Weyl a century back. But the 
representation theory of Algebraic groups, such as SL(n), in 
characteristic p, is still not understood in spite of efforts of many 
mathematicians. This Colloquium talk will be an introduction to the 
subject as well as to a course which the speaker will give here of about 
6 to 8 lectures, starting on Tuesday 4th March.
 

Commutative Algebra seminar
Speaker: P. M. S Sai Krishna (IIT Bombay)
Host: Tony Puthenpurakal
Title: Number of generators of a Cohen-Macaulay ideal- II
Time, day and date: 4:00:00 PM, Tuesday, March 3
Venue: Ramanujan Hall
Abstract: We look at some results related to the bound on the number of 
generators of an ideal of a Noetherian local ring. Under the assumption 
that the ideal is Cohen-Macaulay, we get a bound on the number of 
generators of an ideal in a Noetherian local ring (R) in terms of the 
embedding dimension of the ring, the dimension, and the multiplicity of 
the quotient ring. We extend this result to any Cohen-Macaulay ideal of 
a Noetherian ring.

Speaker: Martin Ulirsch.

Affiliation: Goethe University Frankfurt am Main.

Venue: Ramanujan Hall,

Time: 5 March, 4pm.

Title: What is the combinatorial shadow of a matrix?

Abstract: Tropicalization is a process that associates to an
algebro-geometric object a piecewise linear polyhedral shadow that
captures its essential combinatorial structure. In this talk, I will give
an overview of the numerous ways of how to extract tropical information
from a matrix. Our focus will be on naturally occurring logarithmically
concave sequences associated to a matrix (or more generally a matroid
respectively a bimatroid), an area of study, which in recent years has
tremendous progress due to the introduction of methods originating from
Hodge theory. A particular emphasis will be made to make this story
accessible; a background in algebraic geometry is not necessary to follow
this talk.

This talk draws from joint work with Felix Röhrle as well as with Jeff
Giansiracusa, Felipe Rincon, and Victoria Schleis.

CACAAG seminar
Speaker: Martin Ulirsch (Goethe University Frankfurt am Main)
Host: Madhusudan Manjunath
Title: Vector bundles in tropical geometry: An elementary approach
Time, day and date: 4:00:00 PM, , March 6
Venue: Room No 216, Department of Mathematics
Abstract: Tropical geometry studies a piecewise linear combinatorial 
shadow of degenerations and compactifications of algebraic varieties. A 
typical phenomenon is that many of the usual algebro-geometric objects 
have a tropical analogue that is intimately tied to its classical 
counterpart. An example is the theory of divisors and line bundles on 
algebraic curves, whose tropical counterparts have been crucial in 
numerous surprising applications to classical Brill--Noether theory and 
the birational geometry of moduli spaces.

One classical object that has resisted the effort of tropical geometers 
so far is the geometry of vector bundles beyond rank one. In this talk, 
I will outline an elementary approach to tropical vector bundles that 
builds on earlier work of Allermann. Although limited in scope, this 
theory leads to a satisfying tropical story for semistable vector 
bundles on elliptic curves and, more generally, semihomogeneous vector 
bundles on abelian varieties. The engines in the background that make 
these cases accessible to our methods are Atiyah's classification of 
vector bundles on elliptic curves, Fourier-Mukai transforms on abelian 
varieties, and the interactions with non-Archimedean uniformization.

This talk is based on joint work with Andreas Gross and Dmitry Zakharov 
as well as with Andreas Gross, Inder Kaur, and Annette Werner.