Instability in algebraic geometry                                                                                                      
-- Daniel Halpern-Leistner, October 18, 2013                                                                                                                                             
<p>
 
Many algebraic geometers have encountered some notion
of semistability, either in studying a moduli problem or in the
more concrete context of geometric invariant theory. Often
along with a notion of a semistable point, one can stratify the
set of unstable points by some numerical quantity encoding the
degree of instability of each point. I will discuss some of the
very special properties of the resulting stratifications --
both topological and homological -- and how they can be
used to deduce information about the semistable locus itself. I
will also discuss a new framework for studying such
stratifications that generalizes the constructions in geometric
invariant theory and well-known modular examples.