Workshop

These talks will give an introductory survey of the following

topics: an introduction to the Langlands conjectures in their classical form as motivated from problems in number theory and automorphic representation theory, uses of the trace formula and problems such as the fundamental lemma that it motivates, and a survey of the proof of the fundamental lemma and its transfer back to characteristic zero

In the early 80's Kac proved that the number of indecomposable representations

of a given quiver (and a given dimension) over a finite field is a polynomial in the size of the finite field.

Hua later gave an explicit formula for these polynomials and subsequent representation-theoretic or

geometric interpretations for these polynomials were given by Crawley-Boevey, Van den Bergh, Hausel 

and others, leading to a beautiful and still mysterious picture.

The aim of this mini-course is to explain a 'global' analog of some of these results, in which the category

of representations of a quiver gets replaced by the category of coherent sheaves on a smooth projective curve.

As an application, we will give a formula for the number of stable Higgs bundles over such a curve defined 

over a finite field.

One of the most intriguing features of symplectic resolutions is that they tend to come in dual pairs.  I will give an introduction to this duality, with an emphasis on providing a link between two famous geometric constructions of simple representations of algebraic groups:  one via quiver varieties, and the other via the geometric Satake equivalence

In the early 80's Kac proved that the number of indecomposable representations

of a given quiver (and a given dimension) over a finite field is a polynomial in the size of the finite field.

Hua later gave an explicit formula for these polynomials and subsequent representation-theoretic or

geometric interpretations for these polynomials were given by Crawley-Boevey, Van den Bergh, Hausel 

and others, leading to a beautiful and still mysterious picture.

The aim of this mini-course is to explain a 'global' analog of some of these results, in which the category

of representations of a quiver gets replaced by the category of coherent sheaves on a smooth projective curve.

As an application, we will give a formula for the number of stable Higgs bundles over such a curve defined 

over a finite field.