# The structure of instability in moduli theory

 Topology of Algebraic Varieties Topic: The structure of instability in moduli theory Speaker: Daniel Halpern-Leistner Affiliation: Member, School of Mathematics Date: Tuesday, October 21 Time/Room: 3:30pm - 4:30pm/S-101 Video Link: http://video.ias.edu/tav/2014/1021-DanielHalpernLeistner

In many examples of moduli stacks which come equipped with a notion of stable points, one tests stability by considering "iso-trivial one parameter degenerations" of a point in the stack. To such a degeneration one can often associate a real number which measures "how destabilizing" it is, and in these situations one can ask the question of whether there is a "maximal destabilizing" or "canonically destabilizing" degeneration of a given unstable point. I will discuss a framework for formulating and discussing this question which generalizes several commonly studied examples: geometric invariant theory, the moduli of bundles on a smooth curve, the moduli of Bridgeland-semistable complexes on a smooth projective variety, the moduli of $K$-semistable varieties. The key construction in this story may be of independent interest: it assigns to any point in an algebraic stack a topological space parameterizing all possible iso-trivial degenerations of that point. When the stack is $BG$ for a reductive $G$, this recovers the spherical building of $G$, and when the stack is $X/T$ for a toric variety $X$, this recovers the fan of $X$.