Equivariant geometry and Calabi-Yau manifolds

Abstract: Mirror symmetry has led to deep conjectures regarding the geometry of Calabi-Yau manifolds. One of the most intriguing of these conjectures states that various geometric invariants, some classical and some more homological in nature, agree for any two Calabi-Yau manifolds which are birationally equivalent to one another. I will discuss how new methods in equivariant geometry have shed light on this conjecture over the past few years, ultimately leading to a proof of the conjecture for compact Calabi-Yau manifolds which are birationally equivalent to a moduli space of sheaves on a K3 surface. This represents the first substantial progress on the conjecture in dimension > 3 in several years. The key technique is the new theory of ``Theta-stratifications," which allows one to bring ideas from equivariant Morse theory into the setting of algebraic geometry.