Daniel Halpern-Leistner

Columbia University

March 16, 2017

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.

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.