L4

I will describe 5 definitions of Euler characteristic for a space with perfect obstruction theory (i.e. a well-behaved moduli space), and their inter-relations. This is joint work with Yunfeng Jiang. Then I will describe work of Yuuji Tanaka on how to this can be used to give two possible definitions of Vafa-Witten invariants of projective surfaces in the stable=semistable case.

Abstract: n-homological algebra was initiated by Iyama
via his notion of n-cluster tilting subcategories.
It was turned into an abstract theory by the definition
of n-abelian categories (Jasso) and (n+2)-angulated categories
(Geiss-Keller-Oppermann).
The talk explains some elementary aspects of these notions.
We also consider the special case of an n-representation finite algebra.
Such an algebra gives rise to an n-abelian
category which can be "derived" to an (n+2)-angulated category.
This case is particularly nice because it is
analogous to the classic relationship between
the module category and the derived category of a
hereditary algebra of finite representation type.
 

Recent results showed that the low energy expansion of closed superstring amplitudes can be expressed in terms of

single-valued multiple elliptic polylogarithms. I will explain how these functions may be defined as iterated integrals on the torus and

sketch how they arise from Feynman integrals.

The Ginzburg-Landau functional serves as a model for the formation of vortices in many physical contexts. The natural gradient flow, the parabolic Ginzburg-Landau equation, converges in the limit of small vortex size and finite number of vortices to a system of ODEs. Passing to the limit of many vortices in this ODE, one can derive a mean field PDE, similar to the passage from point vortex systems to the 2D Euler equations. In the talk, I will present quantitative estimates that allow us to directly connect the parabolic GL equation to the limiting mean field PDE. In contrast to recent work by Serfaty, our work is restricted to a fairly low number of vortices, but can handle vortex sheet initial data in bounded domains. This is joint work with Daniel Spirn (University of Minnesota).

Graded affine Hecke algebras were introduced by Lusztig for studying the representation theory of p-adic groups. In particular, some problems about extensions of representations of p-adic groups can be transferred to problems in the graded Hecke algebra setting. The study of extensions gives insight to the structure of various reducible modules. In this talk, I shall discuss some methods of computing Ext-groups for graded Hecke algebras.
The talk is based on arXiv:1410.1495, arXiv:1510.05410 and forthcoming work.