The beta ensembles of random matrix theory are natural generalizations of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles, these classical cases corresponding to beta = 1, 2, and 4. We prove that the extremal eigenvalues for the general ensembles have limit laws described by the low lying spectrum of certain raandom Schroedinger operators, as conjectured by Edelman-Sutton. As a corollary, a second characterization of these laws is made the explosion probability of a simple one-dimensional diffusion. A complementary pictures is developed for beta versions of random sample-covariance matrices. (Based on work with J. Ramirez and B. Virag.)

We will discuss the existence and uniqueness of the foliations by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the parity condition at infinity. The concept of center of mass in general relativity will also be discussed. This work generalizes the earlier results of Huisken-Yau, R. Ye, and Metzger.

Many constructions in algebraic geometry require one to choose a point
outside a countable union of subvarieties. Over $\C$ this is always
possible. Over a countable field, a countable union of subvarieties
can cover all the closed points. Let $k$ be a finitely generated
field of characteristic zero and let $\kbar$ be an algebraic closure.
Let $A$ be a semiabelian variety defined over $k$, and let $\End(A)$
be the ring of endomorphisms of $A$ over $\kbar$. Let $X\subset A$ be
a subvariety of smaller dimension. We show that $\Union_{f\in
\End(A)} f(X(\kbar))$ does not equal $A(\kbar)$. Bogomolov and
Tschinkel show that the above is false for $k$ equal to an algebraic
closure of a finite field, and use the result to show that on any
Kummer surface over such $k$, the union of all rational curves covers
all of the closed points. We give further examples of such problems.