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. |

MIT 2011 >