Moduli of vector bundles on gerbes and the period index problem

Pre-seminar title: Structure and problems in Brauer groups
Pre-seminar abstract: There are many perspectives of the Brauer group, from the classical central simple algebras to gerbes. In this talk, I am going to review the associations Brauer group brings together, and state a few structure theorems and main open problems for Brauer groups.

Seminar title: Moduli of vector bundles on gerbes and the period index problem
Seminar abstract: The period index problem has been suggested by Colliot-Thélène in early 2000s, and some of its low dimensional cases has been solved by de Jong, Lieblich and more recently Perry-Hotchkiss. We adapt the point of view of moduli of twisted sheaves, and by combining classical results in the theory of moduli of vector bundles, we were able to prove the period index problem for the 2-torsion Brauer classes on genus 2 curves over C1 fields, and derive a bound for those over arbitrary fields, meanwhile shed light to other cases with their moduli structure.