Abstract:

The idea of using stability conditions to construct moduli spaces of sheaves has been around for a few decades - one of the earliest examples was Mumford's slope stability for coherent sheaves on curves.  As the theory of moduli spaces progressed over the years, equivalences of derived categories were discovered and used to study moduli spaces.  New types of stability conditions, such as Bridgeland stability and polynomial stability, were also developed in connection with homological mirror symmetry and enumerative geometry.

In this talk, I will discuss some recent results on autoequivalences of derived categories.  I will explain how they can be used to recover prior results in algebraic geometry and representation theory, as well as prove new results on stability and moduli in algebraic geometry.