The subject of the talk is the spectrum of a two-dimensional
Schrodinger operator with constant magnetic field and a compactly supported electric potential. The eigenvalues of such an operator form clusters around the Landau levels.
The eigenvalues in these clusters accumulate towards the Landau levels super-exponentially fast. It appears that these eigenvalues can be related to a certain sequence of orthogonal polynomials in the complex domain. This allows one to accurately describe the rate of accumulation of eigenvalues towards the Landau levels. This description involves the logarithmic capacity of the support of the electric potential. The talk is based on a joint work with Nikolai FIlonov from St.Petersburg.

A Einstein metric is stable if the second variation of the total scalar curvature functional is nonpositive in the direction of changes in conformal structures. Using spin^c structure we prove that a compact Einstein metric with nonpositive scalar curvature admits a nonzero parallel spin$^c$ spinor is stable. In particular, all metrics with nonzero parallel spinor (these are Ricci flat with special holonomy such as Calabi-Yau and $G_2$) and Kahler-Einstein metrics with nonpositive scalar curvature are stable. In fact we show that metrics with nonzero parallel spinor are local maxima for the Yamabe invariant and any metric of positive scalar curvature cannot lie too close to them. Similar results also hold for Kahler-Einstein metrics with nonpositive scalar curvature. This is a joint work with Xianzhe Dai and Xiaodong Wang.

The Euler-Poincare formula gives a relation between the local
properties of an l-adic sheaf (like ramification) and its global
properties (like the Euler characteristic). In this talk we will see how
to apply it to compute the rank of some pure exponential sums.