Abstracts of Talks

Stationary measure on finite volume homogeneous spaces (I)

Jean-François Quint

Consider a Lie group G, a finite volume quotient X of G, a subgroup H of G, and a probability measure m on H whose support is compact and generates H. We study the dynamics of H on X when the Zariski closure of the adjoint group of H is semisimple. We describe the H-orbit closures in X and various equidistribution results on X. To this end, we classify the m-stationary probability measures on X.

Stationary measure on finite volume homogeneous spaces (II)

Yves Benoist

Consider a group H of matrices with integer coefficients acting on a torus X, and a probability measure m on H whose support is finite and generates H. We study the dynamics of H on X when the Zariski closure of H is semisimple. We develop new tools in order to classify the m-stationary probability measures on X: the positive m-unstability of the diagonal, the horocyclic flow, and the exponential drift. To this end, we check various equidistribution results for the associated random walk.

The SL(2,R) action on moduli space of Riemann surfaces

Alex Eskin

I will discuss some applications of the ideas introduced by Y. Benoist and J.F. Quint in their breakthrough work on stationary measures on homogeneous spaces to the seemingly unrelated problem of counting periodic billiard trajectories in polygons all of whose angles are rational multiples of p. The connection is via the study of the ergodic theory of the SL(2,R) action on the moduli space of compact Riemann surfaces. This is joint work with M. Mirzakhani.

The relevance of logic to transcendental number theory: a motivated account

Alex Wilkie

My talk will describe the role that a certain branch of mathematical logic, known as o-minimality, plays in the work of Jonathan Pila. I begin by stating a fundamental result of Pila and Bombieri giving a bound (in terms of height) on the number of rational values that a sufficiently differentiable function (in several real variables) can take at rational arguments. However, in order to apply this result to interesting (usually analytic) subsets of euclidean space one needs to represent the subset as a finite union of sets, each being the range of a sufficiently differentiable function with bounded derivatives. To do this one proceeds by induction on dimension, but the problem is to control the logical complexity of the definitions of the sets that one encounters along the way. The key point is that one never has to leave the collection of sets logically definable in an o-minimal structure. My aim will be to explain this remark.

Diophantine geometry via o-minimality

Jonathan Pila

Starting with a 1989 joint paper with Enrico Bombieri, I studied rational points on algebraic and (certain) non-algebraic sets, culminating in a general theorem, joint with Alex Wilkie, concerning the distribution of rational points in a set in real space that is "definable in an o-minimal structure over the real numbers". This is a model-theoretic notion of geometric "tameness". Bounded semi-analytic sets are examples of such sets. Employing a strategy proposed by Umberto Zannier, we used this result to give a new proof of the Manin-Mumford conjecture (Raynaud"s theorem) about torsion points on subvarieties of abelian varieties. Further applications of this general strategy to diophantine problems in the Andre-Oort-Manin-Mumford circle of problems and the broader "unlikely intersection" conjectures of Zilber and Pink have been made by Masser-Zannier, by me, and by others. I will describe these problems, indicate the main elements involved in carrying out this strategy, and sketch an unconditional proof of the Andre-Oort conjecture for products of modular curves. I will discuss further applications and questions.

Recent progress in mathematical general relativity

Mihalis Dafermos

General relativity has witnessed a resurgence of activity in the last few years from the point of view of the analysis of hyperbolic partial differential equations. I will review recent progress on long-standing open problems in the field, including the formation, uniqueness and stability of black holes, and the structure of singularities in gravitational collapse.

Thin integer matrix groups and the affine sieve

Peter Sarnak

Infinite index subgroups of groups like SL(n,Z), which are Zariski dense in SL(n) arise in geometric diophantine problems(e.g. Apollonian packings)and in problems connected with Monodromy groups. One of the key features in diophantine applications is a super form of strong approximation connected with congruence graphs associated to these groups. We will review the recent developments by a number of authors and discuss the applications to the general affine sieve.

The average rank of elliptic curves

Manjul Bhargava

A rational elliptic curve may be viewed as the set of solutions to an equation of the form y2=x3+Ax+B, where A and B are rational numbers. It is known that the rational points on this curve possess a natural abelian group structure, and the Mordell-Weil theorem states that this group is always finitely generated. The rank of a rational elliptic curve measures how many rational points are needed to generate all the rational points on the curve. There is a standard conjecture -- originating in work of Goldfeld and Katz-Sarnak -- that states that the average rank of all elliptic curves should be 1/2; however, it has not previously been known that the average rank is even finite! In this lecture, we describe recent work that shows that the average rank is finite (in fact, we show that the average rank is less than one). This is joint work with Arul Shankar.