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Quivers, curves, Kac polynomials and the number of stable Higgs bundles

Introductory Workshop: Geometric Representation Theory September 02, 2014 - September 05, 2014

September 03, 2014 (03:30 PM PDT - 04:30 PM PDT)
Speaker(s): olivier schiffmann (Université de Paris XI)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC



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.

22256?type=thumb Schiffmann Notes 969 KB application/pdf Download
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