Abstracts of Talks

Resolution of singularities in zero and positive characteristic

Herwig Hauser

I will discuss main ideas how to prove resolution in zero characteristic, main obstructions why this does not work in positive characteristic and recent approaches to overcome these obstructions.

Resolution of Singularities in Algebraic Geometry

Heisuke Hironaka

I will present my way of proving resolution of singularities of an algebraic variety of any dimension over a field of any characteristic. There are some points of general interest, I hope, technically and conceptually more than just the end result. The resolution problem for all arithmetic varieties (meaning algebraic schemes of finite type over the ring of integers) is reduced to the question of how to extend the result from modulo pm to modulo p(m+1) after a resolution of singularities over Q. I want to discuss certain problems which arise in this approach.

Some Remarks on SLE and an Extended Sullivan Dictionary

Peter Jones

The Sullivan dictionary translates statements about Kleinian groups into statements about Julia Sets and vice versa. For example, a limit set on the Kleinian group side corresponds to a Julia Set, and the orbit of a point under a Kleinian group corresponds to the inverse images of a point by a rational map. We will discuss adding another category to the dictionary, namely SLE, Schramm Loewner Evolution. Here limit sets and Julia sets correspond to the SLE "trace". We will point out that with suitable modifications, the Sullivan dictionary can be enlarged to include SLE. As one example we will discuss the various analogues of the Ahlfors conjecture for Kleinian groups. Another subject we will discuss is the various versions of rigidity that appear in the dictionary. This lecture is aimed at a general mathematical audience and we will stress ideas, not technicalities appearing in proofs.

Lecture notes

Topology and Geometry of Ends of Hyperbolic 3-Manifolds

Yair Minsky

The classification of non-compact hyperbolic 3-manifolds with finitely-generated fundamental groups depends on an understanding of the topology and asymptotic geometry of their ends. A number of advances in recent years have made this classification possible, and more. I will discuss the background and features of this theory, and its applications to a fuller understanding of the ways in which these manifolds (compact and non-compact) cover and approximate each other.

Billiards and Moduli Spaces

Curtis T. McMullen

We will discuss ergodic theory over the moduli space of compact Riemann surfaces, and its connections with algebraic geometry, âTeichmüller theory and billiard tables with optimal dynamics.

Functoriality: Ubiquity and Progress

Dinakar Ramakrishnan

Questions in Automorphic forms and Number theory often get tied up with the magnificent, largely conjectural, edifice of functoriality, a simple instance being the desire to know if certain four-dimensional Galois representations occurring inside the cohomology of Siegel modular threefolds are symplectic. Of particular importance, besides base change, is the transfer of automorphic forms from orthogonal and symplectic groups to the general linear group, which sheds light on many problems. Crucial progress has been made of late in the work of Arthur via the twisted trace formula, extending the earlier results known for generic cusp forms, which had relied on the elegant converse theorem insight of Piatetski-Shapiro. Part of what makes Arthur's approach work is the incredible recent progress on the (different guises of) fundamental lemma due to Ngo, Waldspurger and others. This talk will try to introduce the basic global statements, a few ideas, and applications.

Quantum Unique Ergodicity and Number Theory

Kannan Soundararajan

A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic manifolds arising as a quotient of the upper half-plane by a discrete "arithmetic" subgroup of SL_2(R) (for example, SL_2(Z), and in this case the corresponding eigenfunctions are called Maass cusp forms). In this case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions become equi-distributed. I will discuss some recent progress which has led to a resolution of this conjecture, and also on a holomorphic analog for classical modular forms.

Endoscopy and Harmonic Analysis on Reductive Groups

Jean-Loup Waldspurger

Let G be a connected reductive group over a number field and let H be an endoscopic group of G. A conjecture of Langlands predicts that there exists a correspondence between automorphic representations of H(A) and automorphic representations of G(A), where A is the ring of adeles of the ground field. Langlands' idea of proof is to compare the Arthur-Selberg's trace formulas of H and G. It's necessary to solve many problems, in particular two problems of harmonic analysis over a local field: the transfer conjecture and the fundamental lemma. These two questions have remained open until the decisive result of Ngo Bao Chau, two years ago. In my ta lk, I'll try to explain what is the endoscopic transfer and what is the fundamental lemma. I will give several statements of that lemma, more or less sophisticated. I'll try to explain the situation at the present time. In fact, all the useful problems are resolved even if certain related questions of harmonic analysis remain open.

Lecture Notes