Algebraic Geometry

A theorem of Donaldson and Sun asserts that the metric tangent cone of a smoothable Kähler–Einstein Fano variety underlies some algebraic structure, and they conjecture that the metric tangent cone only depends on the algebraic structure of the singularity. Later Li and Xu extend this…

Vertex operator algebras (VOAs) are generalizations of commutative associative algebras and of Lie algebras. As I will illustrate, there are a number of interesting examples of VOAs that come from moduli spaces, and striking instances where the VOA formalism has been used to solve problems about…

Crepant resolutions have inspired connections between birational geometry, derived categories, representation theory, and motivic integration. In this talk, we prove that every variety with log-terminal singularities admits a crepant resolution by a smooth stack. We additionally prove a motivic…

I will present some joint work with Hannah Markwig and Dhruv Ranganathan, in which we interpret double Hurwitz numbers as intersection numbers of the double ramification cycle with a logarithmic boundary class on the moduli space of curves. This approach removes the "need" for a branch morphism…

Born as part of algebraic topology, Massey products have now made a surprising appearance in Galois cohomology. The Massey Vanishing Conjecture of Minac and Tan predicts that all Massey products in the Galois cohomology of a field vanish as soon as they are defined. This conjecture is motivated…

Classical Brill-Noether theory studies the cohomology jumping loci for line bundles on curves. On surfaces, even the generic cohomology of a sheaf in a moduli space may be hard to determine. In this talk, I will explain how to compute the cohomology of a general stable sheaf on a K3 surface…

A landmark result of Birkar, Prokhorov, and Shramov shows that automorphism groups of Fano (or more generally rationally connected) varieties over C of a fixed dimension are uniformly Jordan. This means in particular that there is some upper bound on the size of symmetric groups acting…

In this talk, I will discuss the behavior of positivity, hyperbolicity, and Kodaira dimension under smooth morphisms of complex quasi-projective manifolds. This includes a vast generalization of a classical result: a fibration from a projective surface of non-negative Kodaira dimension to a…

I will introduce and discuss a remarkable class of algebraic varieties, called braid varieties. These include all open Richardson and positroid varieties, and are closely related to augmentation varieties for Legendrian links. The topology of braid varieties is related to various link invariants…

Wild ramification is known to be a major obstacle to solving various questions in positive characteristic, including resolution of singularities, compactifying Hurwitz spaces, etc., and the very terminology suggests that we are dealing with something not so controllable. Nevertheless,…