For strong solutions of the incompressible Navier-Stokes
equations in bounded domains with velocity specified at the boundary, we
proof unconditional stability and obtain error estimates of discretization
schemes that decouple the updates of pressure and velocity through
explicit time-stepping for pressure. The proofs are simple, based upon a
new, sharp estimate for the commutator of the Laplacian and Helmholtz
projection operators. This allows us to treat an unconstrained formulation
of the Navier-Stokes equations as a perturbed diffusion equation.

We study the subgroup B_0(G) of H^2(G,Q/Z) consisting of all elements which have trivial restrictions to every Abelian subgroup of G. The group B_0(G) serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties V/G where V is a faithful complex linear representation of the group G. We prove that B_0(G) is trivial for finite simple groups of Lie type A_{\ell}.

In this series of lectures, we shall give an introduction to
Dwork's unit root zeta functions, focusing on the Kloosteman
family example, its relation to the L-function of high
symmetric powers and p-adic information about zeros of
these L-functions.

I'll begin by describing our recent attempts to model the evolution of crypts in the colon, using methylation patterns as markers [see Yatabe et al.]. Mutations in colon crypts are thought to play an important role in pre-tumor progression, and therefore in understanding the time to cancer. One common complaint about such multistage and multihit models is that they require unrealistically high mutation rates to explain the observed incidence of cancer. I'll use our model, together with classical extreme value theory, to show that we can explain the SEER incidence data for colon cancer using typical mutation rates [Calabrese et al.]. A number of corroborative datasets and open problems will be discussed.