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.