Georgia Institute of Technology College of Sciences

We will present probability inequalities for the sums of independent, selfadjoint random matrices. The focus is made on noncommutative generalizations of the classical bounds of Azuma, Bernstein, Cherno ff, Hoeffding, among others. These inequalities imply concentration results for the empirical covariance matrices. No preliminary knowledge of probability theory will be assumed. (The talk is based on a paper by J. Tropp).

In 1956 Mark Kac published his paper about the Foundation of Kinetic Theory in which he gave a mathematical, probabilistic description of a system of N particles colliding randomly. An interesting result that was found, though not causing any surprise, was the convergence to the stable equilibrium state. The question of the rate of the L2 convergence interested Kac and he conjectured that the spectral gap governing the convergence is uniformly bounded form below as N goes to infinity. While this was proved to be true, and even computed exactly, many situations show that the time scale of the convergence for very natural cases is proportional to N, while we would hope for an exponential decay. A different approach was considered, dealing with a more natural quantity, the entropy. In recent paper some advancement were made about evaluating the rate of change, and in 2003 Villani conjectured that the corresponding 'spectral gap', called the entropy production, is of order of 1/N. In our lecture we'll review the above topics and briefly discuss recently found results showing that the conjecture is essentially true.

n his 1994 ICM lecture, Borcherds famously introduced an entirely new conceptin the theory of modular forms. He established that modular forms with very specialdivisors can be explicitly constructed as infinite products. Motivated by problemsin geometry, number theorists recognized a need for an extension of this theory toinclude a richer class of automorphic form. In joint work with Bruinier, the speakerhas generalized Borcherds's construction to include modular forms whose divisors arethe twisted Heegner divisors introduced in the 1980s by Gross and Zagier in theircelebrated work on the Birch and Swinnerton-Dyer Conjecture. This generalization,which depends on the new theory of harmonic Maass forms, has many applications.The speaker will illustrate the utility of these products by resolving open problemson the following topics: 1) Parity of the partition function 2) Birch and Swinnerton-Dyer Conjecture and ranks of elliptic curves.

The Research Horizons seminar this week will be a panel discussion on the job market for mathematicians. Professor Doug Ulmer and Luca Dieci will give a presentation with general information on the academic job market and the experience of our recent students, in and out of academia. The panel will then take questions from the audience.