# Abstracts

Lucia Caporaso:

Title: N\'eron models, compactified jacobians and combinatorial divisor theory

Abstract: The N\'eron model of a family of jacobians has a finite group

associated to it which plays a significant role in several situations, and

has also some remarkable incarnations in combinatorics.

In recent years this group has been used to classify compactified

jacobians, and to set up a divisor theory for graphs and tropical curves.

The lecture will give an overview, illustrating recent progress both in

algebraic geometry and combinatorics.

Christopher Hacon:

Title: Boundedness of varieties of log general type and the ACC for log

canonical thresholds

Abstract: In this talk I will discuss recent joint work with J. McKernan

and C. Xu on the boundedness of varieties of log general type and I will

explain how these results imply Shokurov's conjecture known as the ACC

for log canonical thresholds.

Wei Ho:

Title: Explicit Moduli Spaces for Decorated Curves and Arithmetic Applications

Abstract: We discuss parametrizations of geometric data, such as curves with

specified line bundles or vector bundles, by orbits of representations of

algebraic groups. Such explicit realizations of these moduli spaces lend

themselves to computations, (uni)rationality results, and arithmetic

applications. We will focus on the uniform constructions of many cases

where the curves that arise have genus one, using ideas involving the four

so-called Severi varieties (and generalizations) and some special Cremona

transformations. Finally, we explain how understanding the invariant

theory of these spaces has led to some recent applications on bounding the

ranks of elliptic curves over Q (in certain natural families). Much of

this is joint work with Manjul Bhargava.

Mikhail Kapranov:

Title: From Arakelov bundles to Arakelov sheaves

Abstract: The general approach of Arakelov geometry provides

a replacement for "compactification" (integer structure) at infinity

of an arithmetic scheme or a vector bundle on such a scheme.

This is done in terms of Hermitian metrics.

The talk, based on joint work with E. Vasserot, will address the

natural question of what is the Arakelov analog of a coherent sheaf,

including sheaves with singularities at the infinity.

Sam Payne:

Title: Nonarchimedean geometry, tropicalization, and metrics on curves

Abstract: I will discuss the relationship between the nonarchimedean

analytification of an algebraic variety and the tropicalizations of

its various embeddings in toric varieties, with attention to the

metrics on both sides in the special case of curves. This is joint

work with Matt Baker and Joe Rabinoff.

Ravi Vakil:

Title: Eight points on the projective line, and in projective three-space

Abstract: The GIT quotient of a small number of points on the projective line

has long been known to have beautiful geometry. For example, the case

of six points is intimately connected to the outer automorphism of

S_6. We describe a richer story in the case of 8 points. As a

start of the story: the space of 8 points in P^1 lies in P^{13}, and

is the singular locus of the unique S_8-skew cubic. The projective

dual of this cubic is a skew quintic, whose singular locus is

canonically the Gale-quotient of the space of 8 points in P^3. This

work was begun as the key initial step in understanding the equations

of all GIT quotients of any number of points in P^1, and I will

discuss these ideas at length as an extended introduction. This is

joint work with Ben Howard, John Millson, and Andrew Snowden.

Zhiwei Yun:

Title: Symmetry and duality for Hitchin fibrations

Abstract: Hitchin fibration is a natural geometric object associated

to an algebraic curve and a reductive group G. It plays important

roles in gauge theory, integrable systems and (somewhat surprisingly)

in the Langlands program. In this talk, I will describe a natural

symmetry on the cohomology of the fibers of the (parabolic) Hitchin

fibration, which allows us to give a geometric construction of

representations of the double affine Hecke algebra. This symmetry also

behaves in an interesting way when we compare the Hitchin fibrations

for G and its Langlands dual, which can be viewed as a topological

shadow of the expected mirror symmetry. An example when G=SL(2) will

be presented in details.