# Abstracts

#### Prakash Belkale: Gauss-Manin representations of conformal block local systems

Conformal blocks give projective local systems on moduli spaces of curves with marked points. One can ask if they are “realizable in geometry”, i.e., as local subsystems of suitable Gauss-Manin local systems of cohomology of families of smooth projective varieties.

I will discuss (in genus 0) the proof of Gawedzki et al’s conjecture that Schechtman-Varchenko forms are square integrable (this was proved first for sl(2) by Ramadas). Together with the flatness results of Schechtman-Varchenko, and the work of Ramadas, one obtains the desired realization and a unitary metric on conformal blocks. I will also speculate on the nature of local systems which correspond to conformal blocks in higher genus.

#### Jean-Louis Colliot-Thélène: Homogeneous spaces over function fields of curves over a local field

Let K be the field of fractions of a complete discrete valuation ring. Let F be the field of functions of a curve over K. Given an F-variety which is a homogeneous space of a connected linear algebraic group over F, one would like to decide whether this variety has an F-rational point. Various local-global principles have been envisioned, as well as various obstructions to such a local-global principle. I shall survey work of several authors (Harbater, Hartmann, Krashen; Parimala, Suresh, the speaker; Harari, Szamuely).

#### Izzet Coskun: The Higher Rank Interpolation problem and Bridgeland stability

Given a sheaf $F$ on $\mathbb{P}^2$, the higher rank interpolation problem asks for the classification of slopes $\mu$ for which there exists a vector bundle $E$ such that $H^i(E \otimes F) = 0$ for every $i$. In this talk, I will explain the solution of the interpolation problem for ideal shaves $I_Z$ of monomial schemes and discuss implications for the stable base locus decomposition of the effective cone of the Hilbert scheme. The solution depends on finding a new resolution of $I_Z$ which is motivated by Bridgeland stability. This is joint work with Jack Huizenga.

#### Brendan Hassett: Moduli of quartic del Pezzo surfaces and classification of fibrations

We analyze the moduli stack of quartic del Pezzo surfaces with mild singularities with a view toward classifying fibrations over $\mathbb P^1$
of given height. (joint with Kresch and Tschinkel)

#### Margarida Melo: Comparing toroidal compactifications of the moduli space of abelian varieties

Any polyhehdral decompostion of the cone of positive definite quadratic forms gives rise to a toroidal compactification of the moduli space of abelian varieties. The aim of this talk (based on joint work with Filippo Viviani, arXiv:1106.3291) is to compare two well-known such polyhedral decompositions: the perfect cone decomposition and the 2nd Voronoi decomposition.
Using the theory of regular matroids, we determine which cones belong to both the decompositions, thus providing a positive answer to a conjecture of Alexeev and Brunyate. As an application, we compare the two associated toroidal compactifications, each of which has a special role in the geometry of the moduli space of abelian varieties.

#### R. Parimala: Period-index questions for function fields

Class field theory tells us that over a number field or a p-adic field, the period and the index coincide for central simple algebras. A theorem of de Jong asserts that the same is true for function fields of surfaces over algebraically closed fields. The question, whether indices of algebras over function fields over number fields or p-adic fields are bounded in terms of their period, has deep connections with nontrivial representability of zeros by quadratic forms. We explain some recent results concerning period/index bounds for function fields of curves over complete discrete valued fields. (Joint work with Suresh.)

#### Sam Payne: Nonarchimedean methods for multiplication maps

Multiplication maps on linear series are among the most basic structures in algebraic geometry, encoding, for instance, the product structure on the graded pieces of the homogeneous coordinate ring of a projective variety. In this talk, I will discuss joint work with Dave Jensen, developing tropical and nonarchimedean analytic methods for studying multiplication maps of linear series on algebraic curves in terms of piecewise linear functions on graphs, with a view toward applications in classical complex algebraic geometry.

#### Yuri Tschinkel: Universal spaces for birational invariants

Anabelian geometry techniques allow the construction of explicit universal spaces which capture birational properties of algebraic varieties. I will describe this theory and its applications (joint with F. Bogomolov).

#### Filippo Viviani: On the cone of Moriwaki divisors

We give a characterization of the cone of Moriwaki divisors on the moduli space of stable curves in terms of augmented and restricted base loci. Then we draw some interesting consequences on the Zariski decomposition of divisors, on the minimal model program of and on the log canonical models of the moduli space of stable curves. This is a joint work with S. Cacciola and A. Lopez.