**Title:** Moduli of Twisted Sheaves and Azumaya Algebras

**Speaker:** Max Lieblich

**Speaker Info:** MIT

**Brief Description:**

**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 $[D]=\alpha$?

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