EVENT DETAILS AND ABSTRACT

Title: Moduli of Twisted Sheaves and Azumaya Algebras
Speaker: Max Lieblich
Speaker Info: MIT
Brief Description:
Special Note: special time
Abstract:

We construct and describe moduli spaces of Azumaya algebras on a smooth projective surface. These spaces are the algebro-geometric version of the spaces of principal $\operatorname{PGL}_n$-bundles and they also have strong connections to arithmetic. A geometric approach to the problem leads one to study moduli spaces of twisted sheaves.

We show that these spaces are very similar to the moduli spaces of semi-stable sheaves. On the arithmetic side, we use the geometry of these moduli spaces to answer a classical question about the Brauer group of a function field $K$ in two variables over a finite field, known as the ``period-index problem'': for which classes $\alpha$ in $\operatorname{Br}(K)$ of order $n$ does there exist a division algebra $D$ of rank $n^2$ with $[D]=\alpha$?