Let E be an elliptic curve over an algebraically closed field k of
characteristic p>0. Then the physical p-torsion E[p](k) is either trivial,
and E is called supersingular, or E[p](k) is a group of order p. More
generally, if X/k is an abelian variety of dimension g, then X[p](k)
is isomorphic to (Z/p)^f for some number f, called the p-rank of X.
The p-rank induces a stratification of the moduli space of abelian
varieties; via the Torelli functor, it induces a stratification of the
moduli space of (hyperelliptic) curves.
I'll discuss recent results on the geometry of these strata, with
special attention to their structure at the boundary of the moduli
space. This information yields new applications about the prime-to-p
part of the class group of a quadratic function field with specified geometric
p-rank; the existence of absolutely simple hyperelliptic Jacobians of
every p-rank; and the stratification of the moduli space of curves by
Newton polygon.

Generalized ideal class groups can be described adelically in terms of a coset space for the group GL1, and this in turn leads to a notion of "class number" (as the size of a certain set, if finite) for an arbitrary affine algebraic group over a global field. Related to this is the notion of the "Tate-Shafarevich set" of an algebraic group, which is tied up with questions relating global and local information. Finiteness of class numbers and Tate-Shafarevich sets for affine algebraic groups was proved by Borel and his coworkers over number fields, andif one grants the finiteness of Tate-Shafarevich groups for abelian varieties then Mazur showed how to get such finiteness for all algebraic group varieties over number fields (which has applications to the local-to-global principle for projective varieties over number fields).

The above methods do not apply over global function fields. After reviewing some history, I will explain the content of a recent classification theorem of "pseudo-reductive groups" proved jointly with Gabber and G. Prasad that makes it possible to prove the analogous finiteness theorems in the function field case away from characteristic 2. If time permits I will say something about how this classification theorem is used to get such results.

We present numerical schemes for the incompressible Navier-Stokes
equation with open or traction boundaries. We use pressure Poisson
equation formulation and propose new boundary conditions for
pressure on the open or traction boundaries. For Stokes equation
with open boundary condition, we prove unconditional stability of
a first order semi-discrete scheme with explicit treatment of the
pressure. Using either boundary condition, the schemes for full
Navier-Stokes equations that treat both convection and pressure
terms explicitly work well with various spatial discretization
including spectral collocation and $C^0$ finite elements. Besides
standard stability and accuracy check, various numerical results
including backward facing step, flow past a cylinder and a
bifurcation tube (or h-shaped tube) are reported. In all the
numerics, we do not have to require the inf-sup compatibility
condition between finite element spaces for velocity and pressure.
Even though we treat pressure and convection terms explicitly, time
step size of $O(1)$ is allowed in benchmark computations with
either boundary condition when Reynolds number is of $O(1)$ and
when first order time stepping is used. Our results extend that of
H. Johnston and J.-G. Liu (J. Comp. Phys. 199 (1) 2004, 221--259)
which deals with Dirichlet boundary condition.