Abstract. Modular representation theory studies actions of finite groups (Lie algebras, algebraic groups, finite group schemes) on vector spaces over a field of positive characteristic. The simplest example is an action of the cyclic group Z/p on a vector space. Such an action is described by a single matrix which, in turn, is classified by its Jordan canonical form.

I shall describe an approach to the study of modular representations via their restrictions to certain elementary subalgebras which are analogs of one-parameter subgroups. As an application, we can recover the algebraic variety associated to the cohomology ring of a finite group scheme $G$ by purely representation-theoretic means, generalizing Quillen's stratification theorem" for group cohomology. As another application, we construct new numerical invariants of representations. These invariants are expressed in terms of Jordan forms.

Most of our results apply to any finite group scheme, but they are non-trivial even in the case of the finite group Z/p x Z/p, which is a baby example that will be used for illustrative purposes throughout the talk.

In this talk we shall discuss rigidity aspects of infinite discrete groups, which arise naturally in Geometry (as fundamental groups of manifolds), in Algebraic groups (as lattices) and, more generally, as symmetries of various mathematical objects.

Starting from classical by now rigidity results of Mostow, Margulis, Zimmer, we shall turn to the recently active area of Measurable Group Theory, which is closely related to Ergodic Theory, von Neumann algebras, and has applications to such fields as Descriptive Set Theory.

Consider a particle in a finite dimensional Euclidean space performing a Brownian motion with an instantaneous drift vector at every time point determined by the order in which the coordinates of its location can be arranged as a decreasing sequence. These processes appear naturally in a variety of areas from queueing theory, statistical physics, and economic modeling. One is generally interested in the spacings between the ordered coordinates under such a motion.

For finite n, the invariant distribution of the vector of spacings can be completely described and is a function of the drift. We show, as n grows to infinity, a curious phenomenon occurs. We look at a transformation of the original process by exponentiating the location coordinates and dividing them by their total sum. Irrespective of the drifts, under the invariant distribution, only one of three things can happen to the transformed values: either they all go to zero, or the maximum grows to one while the rest go to zero, or they stabilize and converge in law to some member of a two parameter family of random point processes. This family known as the Poisson-Dirichlet's appears in genetics and renewal theory and is well studied. The proof borrows ideas from Talagrand's analysis of Derrida's Random Energy Model of spin glasses. We also consider another alternative starting with a countable collection of Brownian motions. This countable model is related to the Harris model of elastic collisions and the discrete Ruzmaikina-Aizenmann model for competing particles.

This is based on separate joint works with Sourav Chatterjee and Jim Pitman.

A smooth plane projective cubic curve (also known as an elliptic curve or a curve of genus 1) carries a natural structure of a commutative group: the addition is defined geometrically by the "chord and tangent method". An attempt "to add" points on a curve of arbitrary positive genus g leads to the notion of the jacobian of the curve. This jacobian is a g-dimensional commutative algebraic group that is a projective algebraic variety; in particular, it cannot be realized as a matrix group. Geometric properties of jacobians play a crucial role in the study of arithmetic and geometric properties of curves involved. One of the most important geometric invariants of a jacobian is its endomorphism ring.

We discuss how to compute explicitly endomorphism rings of jacobians for certain interesting classes of curves that may be viewed as natural (and useful) generalizations of elliptic curves.

The principle of projective determinacy, being independent from the standard axiom system of set theory, produces a fairly complete picture of the theory of "definable" sets of reals. It is an amazing fact that projective determinacy is implied by many apparently entirely unrelated statements. One has to go through inner model theory in order to prove such implications.