A family of affine Weyl group representations

In this paper we explicitly determine the virtual representations of the finite Weyl subgroups of the affine Weyl group on the cohomology of the space of affine flags containing a family of elementsn _t in an affine Lie algebra. We also compute the Euler characteristic of the space of partial flags containingn _t and give a connection with hyperplane arrangements.This paper forms part of my Ph.D. thesis at M.I.T. I was supported by an NSF Graduate Fellowship and NSF grant DMS 9304580.     

Density of stationary points in a high dimensional random energy landscape and the onset of glassy behavior

The density of stationary points and minima of a N ≫ 1 dimensional Gaussian energy landscape has been calculated. It is used to show that the point of zero-temperature replica symmetry breaking in the equilibrium statistical mechanics of a particle placed in such a landscape in a spherical box of size L = R √N corresponds to the onset of exponential in N growth of the cumulative number of stationary points, but not necessarily the minima. For finite temperatures, a simple variational upper bound on the true free energy of the R = ∞ version of the problem has been constructed and it has been shown that this approximation can recover the position of the whole de-Almeida-Thouless line. The text was submitted by the authors in English.