We present some new results concerning the Dirichlet problem for the prescribed mean curvature equation over a bounded domain in R^n. In the case when the mean curvature is zero this can be posed variationally as the problem of finding a least area representative among functions of bounded variation with prescribed boundary values. We show that there is always a minimizer which is represented by a compact C^{1,alpha} manifold with boundary, with boundary given by the prescribed Dirichlet data, provided this data is C^{1,alpha} and it is of class C^{1,1} if the prescribed data is C^3.

One of the fundamental problems in the classification of complex surfaces is to find a new family of simply connected surfaces with p_g = 0 and K^2 > 0. In this
talk, I will sketch how to construct a new family of simply connected symplectic 4- manifolds using a rational blow-down surgery and how to show that such 4-manifolds
admit a complex structure using a Q-Gorenstein smoothing theory. In particular, I will show explicitly how to construct a simply connected minimal surface of general
type with p_g = 0 and K^2 = 3.

If time allows, I will also sketch how to construct a simply
connected, minimal, symplectic 4-manifold with b_+2 = 1 (equivalently, p_g = 0) and K^2 = 4 using a rational blow-down surgery.

We discuss how BMM affects the large cardinal
structure of V as well as the size of \theta^{L(R)}. BMM proves
that V is closed under sharps (and more), and BMM plus the
existence of a precipitous ideal on \omega_1 proves that
\delta^1_2 = \aleph_2. Part of this is joint work with my
student Ben Claverie.

The talk deals with the spectral analysis of Jacobi matrices superimposed
with random perturbations that decay in a certain sense.

We shall focus our attention on two problems: The first is the analysis of
spectral stability. We show that the absolutely continuous spectrum
associated with bounded generalized eigenfunctions, for Jacobi matrices with
a mild growth restriction on the off-diagonal terms, is stable under random
Hilbert-Schmidt perturbations. We also give some results for singular
spectral types. This is joint work with Yoram Last.

The second problem is the spectral analysis of Jacobi matrices arising in
the study of Gaussian \beta ensembles of Random Matrix Theory. These
matrices may be viewed as simple Jacobi matrices (with growing off-diagonal
terms) with a random perturbation that decays in a certain sense. With the
help of the appropriately modified methods, we analyze the behavior of the
generalized eigenfunctions and the Hausdorff dimension of the spectral
measure. Some of this work is joint with Peter Forrester and Uzy Smilansky.