Winter School

in Mathematical Physics - 2014

Abstracts

Ezra Getzler, "Differential forms on stacks"

[Slides]

A typical example of a stack is a Lie groupoid. I will give a survey of applications of differential forms on stacks, using the language of simplicial manifolds. Some of the topics to be covered: Chern-Weil theory for stacks; quasi-symplectic Lie groupoids; the Polyakov-Wiegmann formula and its generalizations; shifted symplectic forms on higher stacks.

Iain Gordon, "Schubert Calculus and Bethe Algebras"

Works over the last decade or so of Mukhin, Tarasov and Varchenko have established a direct relationship between Schubert calculus for Grassmannians and tensor products of representations of the general linear group. The relationship goes through a study of the spectrum of a family of commuting operators - Gaudin operators - and this is approached using the Bethe Ansatz for this system. These lectures will discuss this work, and, if time allows, some recent developments related to it.

Marco Gualtieri, "The Stokes Groupoids"

Ordinary differential equations with singularities exhibit solutions with interesting singular and asymptotic behaviour, leading to the so-called Stokes phenomenon as well as a significantly more complicated Riemann-Hilbert correspondence which involves generalized monodromy data in the form of Stokes matrices. I will describe a new approach to the study of such differential equations which emphasizes the natural domain of definition for their solutions, which is a simple class of Lie groupoids which we call the Stokes groupoids. In this minicourse no previous knowledge of the Stokes phenomenon will be assumed.

Jan Manschot, "Quivers and BPS bound states"

In these lectures I will discuss moduli spaces of semi-stable quiver representations and their topological invariants. I will explain the relation of quivers with the physics of Bogomolny'i-Prasad-Sommerfield (BPS) bound states, leading to a new formula for the computation of quiver invariants, called the "Coulomb branch formula". I will also elaborate on various aspects of this formula, for example Abelianization and the action of (generalized) quiver mutations.