MIT 2011‎ > ‎

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.