Abstract:

Given a polarized variation of Hodge structures, the Hodge locus is a countable union of proper algebraic subvarieties where extra Hodge classes appear. In this talk, I will explain a general equidistribution theorem for these Hodge loci and explain several applications: equidistribution of higher codimension Noether-Lefschetz

loci, equidistribution of Hecke translates of a curve in the moduli space of abelian varieties and equidistribution of some families of CM points in Shimura varieties. The results of this talk are joint work with Nicolas Tholozan.

Abstract:

Theoretical computer science has given rise to new, exciting questions in algebraic geometry and representation theory. In this talk I will focus on the problem of matrix multiplication. It is generally conjectured by computer scientists that as n grows very large, it becomes almost as easy to multiply nxn matrices as it is to add them! After giving a brief history of the problem I will focus on algebraic geometry and representation theory relevant for the problem and conclude by discussing recent work with A. Conner, A. Harper, and H. Huang.

Abstract:

In this talk, we determine when there is a Brill--Noether curve of given degree and given genus that passes through a given number of general points in any projective space.

Abstract:

I will explain a notion of rank for the relations amongst the equations of a projective variety, generalizing the classical notion of rank of a quadric. I will then turn explain a result telling us that, for a general canonical curve, all syzygies are linear combinations of syzygies of minimal rank. This is a sweeping generalization of old results of Andreotti-Mayer, Harris-Arbarello and Green, which tell us that canonical curves are defined by quadrics of rank four. As a special case, this perspective gives us a new, and simpler, proof of Green's conjecture for general curves.

Abstract:

I'll report on joint work with Temkin and Wlodarczyk, with Quek, and with Schober, using stacks to give resolution algorithms. These include Deligne--Mumford stacks (with weighted blowups) and Artin stacks (with multi-weighted blowups). That said, having looked at what's out there, I thought I'll teach you how to blow up.

Abstract:

I will discuss a derived version of microlocalization of sheaves à la Kashiwara-Schapira, and applications to topics such as singular support of étale sheaves and categorified Donaldson-Thomas theory.

Abstract:

Consider a smooth family of hypersurfaces of degree d in P^{n+1}. When is every smooth projective limit of this family also a hypersurface? While it is easy to construct example of limits that are not hypersurfaces when the degree d is composite, Mori conjectured that, if d is prime and n>2, every smooth projective limit is indeed a hypersurface. However, there are counterexamples when n=1 or 2; for example, one can take a family of degree 5 plane curves and degenerate to a smooth hyperelliptic (non-planar) curve. In this talk, we will propose a re-formulation of Mori's conjecture that explains the failure in low dimensions, provide results in dimension one, and discuss a general approach to the problem using moduli spaces of pairs. This is joint work with David Stapleton.

Abstract:

How often does a general cubic surface appear in a 4-dimensional linear system? As a slice of a general cubic 3-fold? I will describe how we can solve problems of this kind using equivariant geometry. This is joint work with Anand Patel and Dennis Tseng.

Abstract:

The universal centralizer of a complex semisimple adjoint group G is the family of regular centralizers in G, parametrized by the regular conjugacy classes. It has a natural symplectic structure which is inherited from the cotangent bundle of G. I will construct a smooth, log-symplectic relative compactification of this family using the wonderful compactification of G. Its compactified centralizer fibers are isomorphic to Hessenberg varieties, and its symplectic leaves are indexed by root system combinatorics. I will also explain how to produce a multiplicative analogue of this construction, by moving from the Poisson to the quasi-Poisson setting.

Abstract:

The extrinsic geometry of the canonical model of a nonhyperelliptic curve captures many aspects of the intrinsic geometry of the curve. In this talk I will discuss joint work with Izzet Coskun and Eric Larson in which we show that the normal bundle of a general canonical curve of genus at least 7 is always semistable. This makes substantial progress towards a conjecture of Aprodu--Farkas--Ortega, and answers it completely in a third of all cases.

Abstract:

For a prime number p, the crystalline cohomology of an F_p-scheme can be regarded as an analogue of the singular cohomology with Z_p coefficients of a topological space. On the topological side, there are other "generalized" cohomology theories, e.g. K-theory and cobordism, and these are related to natural operations on singular cohomology. In this talk, I will discuss analogues of these generalized cohomology theories and cohomology operations in the crystalline setting.

Abstract:

Supersingular twistor spaces are certain families of K3 surfaces over A^1 associated to a supersingular K3 surface. We will describe a geometric construction that produces families of K3 surfaces over P^1 which compactify supersingular twistor spaces. The key input is a construction relating Brauer classes of order p on a scheme of characteristic p to certain sheaves of twisted differential operators. We will give some results on the geometry of compactified supersingular twistor spaces, and some applications.

Abstract:

Benoist and Wittenberg recently introduced a new rationality obstruction that refines the classical the Clemens--Griffiths intermediate Jacobian obstruction to rationality, and exhibited its strength by showing that this new obstruction characterizes rationality for intersections of two quadrics. We show that this phenomenon does not extend to all geometrically rational threefolds. We construct examples of conic bundle threefolds over P^2 that have no refined intermediate Jacobian obstruction to rationality, yet fail to be rational. This is joint work with S. Frei, L. Ji, S. Sankar, and I. Vogt.

Abstract:

In this talk, we will present some results and expectations in understanding or classifying birational maps in 3fold MMP. Particularly, we will explain ideas toward the classification of divisorial contractions to curves.

We will also demonstrate some geometric applications to the study of threefolds of general type.