**Ana-Maria Castravet:** *Birational geometry of moduli spaces of stable rational curves*

I will report on joint work with Jenia Tevelev on the birational geometry of the Grothendieck-Knudsen moduli space of stable rational curves with n markings. We prove that for n large, this space is not a Mori Dream Space, thus answering a question of Hu and Keel.

**Christopher Hacon:** *On the boundedness of the functor of KSBA stable varieties*

Let X be a canonically polarized smooth n-dimensional projective variety over $\mathbb C$ (so that $\omega_X$ is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of X in projective space. It then follows easily that if we fix certain invariants of X, then X belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized n-dimensional projective varieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.

This is joint work with C. Xu and J. McKernan

**Jun-Muk Hwang:** *Cartan-Fubini type extension of holomorphic maps preserving webs of rational curves*

Let $X_1$ and $X_2$ with $\dim X_1 = \dim X_2$ be two projective manifolds of Picard number 1 in projective space. Assume that both $X_1$ and $X_2$ are covered by lines. Let $\varphi: U_1 \to U_2$ be a biholomorphic map between two connected Euclidean open subsets $U_1 \subset X_1$ and $U_2 \subset X_2$. Suppose that both $\varphi$ and $\varphi^{-1}$ send pieces of lines to pieces of lines. We show that $\varphi$ can be extended to a biregular morphism $\Phi: X_1 \to X_2$. This was proved by Hwang-Mok in 2001 when the indices of $X_1$ and $X_2$ are bigger than 2 and the new result is when the indices are 2. In this case, the covering family of lines form webs of rational curves. We exploit the monodromy of the webs of lines to extend the holomorphic map.

**Robert Lazarsfeld:** *Syzygies and gonality of algebraic curves*

Mark Green and I conjectured in the mid-1980s that one could read off the gonality of an algebraic curve C from its syzygies in the embedding defined by any one sufficiently positive line bundle. Ein and I recently observed that a small variant of the ideas used by Voisin in her work on canonical curves leads to a surprisingly quick proof of this statement. I will discuss the conjecture and its proof.

**Diane Maclagan:** *Tropical schemes*

Tropicalization replaces a variety by a polyhedral complex that is a “combinatorial shadow” of the original variety. This allows algebraic geometric problems to be attacked using combinatorial and polyhedral techniques. While this idea has proved surprisingly effective over the last decade, it has so far been restricted to the study of varieties and algebraic cycles. I will discuss joint work with Felipe Rincon, building on work of Jeff and Noah Giansiracusa, to understand tropicalizing schemes, and more generally the concept of a tropical scheme.

**Davesh Maulik:** *Refined Donaldson-Thomas Theory*

Given an algebraic threefold, Donaldson-Thomas theory (in the classic sense) studies enumerative invariants of moduli spaces of sheaves on X. When the threefold is Calabi-Yau, these invariants admit different refinements – K-theoretic, Hodge-theoretic, and others less well-defined. In this talk, I’ll survey some of these ideas and explain situations when these refinements can be related explicitly.

**Mircea Mustaţă:** *The Decomposition Theorem for toric maps*

The connections between the topology of toric varieties and combinatorics have been much studied. In this talk I will discuss the Decomposition Theorem in the context of toric maps and a combinatorial invariant that comes out of these considerations. This is based on joint work with Marc de Cataldo and Luca Migliorini.

**Karl Schwede:** *Inversion of adjunction for rational and Du Bois pairs*

We prove a new inversion of adjunction statement for rational and Du Bois singularities. Roughly speaking, this says that if we have a family over a smooth base with Du Bois special fiber and generic fiber with rational singularities, then the total space also has rational singularities.

Furthermore, we even generalize this result to the context of rational and Du Bois pairs as defined by Kollár and Kovács. Imprecisely, a pair (X,D) is Du Bois if the failure of X to be Du Bois is equal to the failure of D to be Du Bois. In order to accomplish our inversion of adjunction result we need to generalize, for pairs, many recent results on Du Bois singularities. I will describe some of these ideas. This is joint work with Sandor Kovács.

**Chenyang Xu:** *Nonexistence of asymptotical GIT compactification*

(Joint with Xiaowei Wang) By comparing different stability notions and the related invariants, we show that there exists families of canonically polarized manifolds, e.g., hypersurfaces in $\mathbb P^3$, which don’t have asymptotical Chow semistable limits. This implies that unlike Giesker and Mumford’s result in the curve case, in higher dimension, the method of using asymptotic chow stability to construct moduli space of canonically polarized manifolds doesn’t yield a natural compactification.