We will begin by discussing the following theorem proven in joint work with Jack Hall and David Rydh:  every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is etale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. After briefly discussing extensions of this result to arbitrary fields and base schemes, we will focus on applications.  First, we will show how this result allows us to extend classical theorems concerning algebraic groups.  Second, we will apply this theorem to construct projective moduli spaces of objects (such as semistable vector bundles over a smooth projective curve) which may have infinite automorphism groups.

Brill-Noether theory on curves is the classical study of linear series on curves: essentially, maps of curves to projective space. On a smooth compact curve X of genus g, the Brill-Noether variety G^r_d(X) parametrizes linear series on X of rank r and degree d. I will discuss joint work with Alberto Lopez Martin, Nathan Pflueger, and Montserrat Teixidor i Bigas, in which we use combinatorics related to Buch’s set-valued tableaux, along with Osserman’s machinery of degenerations to Eisenbud-Harris schemes of limit linear series, to study the geometry of G^r_d(X).

Developments in high energy physics, specifically the theory of mirror symmetry, have led to deep conjectures regarding the geometry of a special class of complex manifolds called 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, leading to the first substantial progress for compact Calabi-Yau manifolds of dimension greater than three. The key technique is the new theory of "Theta-stratifications" and "Theta-stability" -- a generalization of geometric invariant theory -- which allows one to bring ideas from equivariant Morse theory into the setting of algebraic geometry.

I will present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan’s theory of motivic volumes of semi-algebraic sets.  As time permits, I will discuss applications of this method, which include the solution to a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson-Thomas theory. This is based on joint work with Johannes Nicaise.

The problem of understanding semistable degenerations of K3 surfaces has been greatly studied and is completely understood. The aim of this talk is to present joint work in progress with J. Kollár, R. Laza, and C. Voisin giving partial generalizations to higher dimensional hyperkähler (HK) manifolds. I will also present some applications, including a generalization of theorem of Huybrechts to possibly singular symplectic varieties and shortcuts to showing that certain HK manifolds are of a given deformation type.

Mixtures of Gaussians are ubiquitous in data science. We give an introduction to the geometry of these statistical models, with focus on the projective varieties represented by their moments. Recent work with Carlos Amendola and Kristian Ranestad characterizes circumstances under which these moment varieties have the expected dimensions.