# Abstracts

**Jarod Alper (University of Washington) Evolution of Luna’s etale slice theorem**

**Abstract:** Luna’s etale slice theorem is a beautiful result in equivariant geometry with important applications to moduli theory. Luna’s result has inspired recent investigations into the local structure of algebraic stacks. The goal of this talk is to explain various stacky generalizations of Luna’s slice theorem and their applications both to equivariant geometry and moduli theory. This talk is based on joint work with J. Hall, D. Halpern-Leistner, J. Heinloth and D. Rydh.

**Kenneth Ascher (Princeton University) Wall crossings for K-moduli spaces**

**Abstract: **K-stability has become a central tool in the study of compact moduli of Fano varieties. In this talk I will discuss K-stability compactifications of the moduli space of log Fano pairs (P2, aC), where C is a plane curve of degree at least 4 and a is a rational number. We establish a wall-crossing framework to study the behavior of these moduli spaces as the weight a varies. We show that when a is small, the K-moduli compactification is isomorphic to the GIT moduli space, and that the first wall crossing is a weighted blowup of Kirwan type. We describe all wall-crossings for degree 4, 5 and 6 and relate the final K-moduli spaces to Hacking's moduli space and some compact moduli of K3 surfaces. This is joint work with K. DeVleming and Y. Liu.

**Benjamin Bakker** **(University of Georgia)** **o-minimal GAGA and applications to Hodge theory**

**Abstract: **For a complex projective variety, Serre's classical GAGA theorem asserts that the analytification functor from algebraic coherent sheaves to analytic coherent sheaves is an equivalence of categories. For non-proper varieties, however, this theorem easily fails. In joint work with Y. Brunebarbe and J. Tsimerman, we show that a GAGA theorem holds even in the non-proper case if one restricts to analytic structures that are "tame" in a sense made precise by the notion of o-minimality. This result has particularly important applications to Hodge theory, and we will explain how it can be used to prove a conjecture of Griffiths on the quasiprojectivity of the images of period maps. We will also discuss some applications to moduli theory.

**Ivan Cheltsov (University of Edinburgh) K-stability of asymptotically log del Pezzo surfaces**

**Abstract:** In 2013, I and Yanir Rubinstein introduced special class of log Fano varieties which are known now as asymptotically log Fano varieties. We classified all of them in dimension 2 (asymptotically log del Pezzo surfaces) and studied the existence of Kahler-Einstein edge metrics on them (K-stability). Recently we almost completed this study together with Kewei Zhang. In my talk, I will explain what we did and what we were not able to do (but Kento Fujita did during this summer).

**Izzet Coskun (University of Illinois at Chicago) Brill-Noether type theorems for moduli spaces of sheaves on surfaces**

**Abstract:** In this talk, I will survey recent work on computing the cohomology of the general sheaf on a moduli space of sheaves on surfaces. I will concentrate on rational and K3 surfaces.

In joint work with Jack Huizenga, we solve the problem completely for Hirzebruch surfaces. As consequences, we determine the Chern characters of moduli spaces where the general sheaf is globally generated and obtain a Gaeta-type resolution for the general sheaf. These in turn yield a classification of Chern characters of stable sheaves for any polarization on a Hirzebruch surface. In joint work with Howard Nuer and Kota Yoshioka, we study the problem for K3 surfaces. Our approach crucially uses Bridgeland stability. If time permits, I will discuss applications to construction and classification of Ulrich bundles on surfaces.

**Sarah Koch (University of Michigan) Irreducibility in complex dynamics**

**Abstract: ** A major goal in complex dynamics is to understand dynamical moduli spaces; that is, conformal conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to $\mathbb C$. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. In this talk, we will begin with an overview of complex dynamics, focusing on the moduli space of quadratic rational maps, which is isomorphic to $\mathbb C^2$. This moduli space contains special algebraic curves, called Milnor curves. In general, it is unknown if Milnor curves are irreducible (over $\mathbb C$). We will find an infinite collection of Milnor curves that are irreducible. This is joint work with X. Buff and A. Epstein.

**Alina Marian (Northeastern University) On the Chow ring of holomorphic symplectic manifolds**

**Abstract:** I will propose a series of basic conjectural identities in the Chow rings of holomorphic symplectic manifolds of K3 type, and will discuss evidence for them. The emerging structure naturally generalizes in higher dimensions a set of key properties of cycles on a K3 surface. The talk is based on joint work with Ignacio Barros, Laure Flapan, and Rob Silversmith.

**Anton Zorich (Institut de Mathématiques de Jussieu) Bridges between flat and hyperbolic enumerative geometry**

**Abstract:** I will give a formula for the Masur-Veech volume of the moduli space of quadratic differentials in terms of psi-classes (in the spirit of Mirzakhani's formula for the Weil-Petersson volume of the moduli space of hyperbolic surfaces). I will also show that Mirzakhani's frequencies of simple closed hyperbolic geodesics of different combinatorial types coincide with the frequencies of the corresponding square-tiled surfaces. I will conclude with (mostly conjectural) description of geometry of a "random" square-tiled surface of large genus and of a "random" multicurve on a topological surface of large genus. The talk is based on the joint work with V. Delecroix, E. Goujard and P. Zograf. It is aimed to a broad audience, so I will try to include all necessary background.