In this talk we first review the critical threshold phenomena
for Euler-Poisson equations, which arise in the semiclassical
approximation of Schrodinger-Poisson equations and plasma
dynamics. We then present a phased space-based level set
method for the computation of multi-valued velocity and
electric fields of one-dimensional Euler-Poisson equations.
This method uses an implicit Eulerian formulation in an
extended space, which incorporates both velocity and electric
fields into the configuration space. Multi-valued velocity
and electric fields are captured through common zeros of two
level set functions, which solve a linear homogeneous
transport equation in the field space. The superposition
principle for multi-valued solutions is established.

Recently, using Gelfand-Zeitlin and the space of Hessenberg matrices, Wallach and I found natural Darboux coordinates (as a classical mechanical solution of the Gelfand-Zeitlin question) on $\frak g$ for the case where $\frak g$ is the space of all matrices. Now, at least locally, I do the same for any reductive $\frak g$ using a beautiful result of A. A. Tarasov on Fomenko-Miscenko theory and old results of mine on a generalization of the Hessenberg matrices.

To prove the Erd\H os-Ko-Rado theorem about extremal collections of
subsets of $1,\ldots,n$, they invented the {\em shifting} technique,
which preserves the number of subsets in a collection but simplifies
(in some senses) the collection. After a finite number of shifts,
one's collection becomes invariant under shifting, and easily studied.

Given a finite set of $n$ vectors in a $k$-dimensional vector space,
the collection of subsets that form bases of the vector space
satisfies some combinatorial properties. Abstracting them, Whitney
defined {\em matroids}. The matroids that are invariant under shifting
have been classified, and correspond to partitions inside a
$k \times (n-k)$ rectangle. The shift of a matroid usually is not a matroid.

I'll present a new version of the Littlewood-Richardson rule,
that starts with a certain matroid, and alternately shifts it
(breaking matroidness) and decomposes as a union of maximal submatroids.
The leaves of the tree so constructed are labeled with fully
shifted matroids, hence partitions. To actually carry out
such a calculation in practice requires some new algorithms.

Unlike all other known Littlewood-Richardson rules, this matroid
shifting rule has an easy generalization to multiplication of Schubert
(not just Schur) polynomials, where it is still a conjecture.

This work is joint with Ravi Vakil.

Let $N$ denote a manifold equipped with a finite Borel measure
$\gamma$. A vector field $Z$ on $N$ is said to
be admissible with respect
to $\gamma$
if $Z$ admits an integration by parts formula.
The measure $\gamma$ is
said to be quasi-invariant under $Z$
if the class of null sets of $\gamma$
is preserved by the flow generated
by $Z$. In this talk we study the law $\gamma$ of an elliptic
diffusion process with values in a closed compact manifold.
We construct a
class of admissible vector fields for $\gamma$, show that $\gamma$
is quasi-invariant under these vector fields,
and give a formula for the
associated family of Radon-Nikodym
derivatives $d\gamma_s\over d\gamma$.

In this talk, we use biological locomotion on small scales as an
inspiration (and an excuse) to solve a number of modeling
problems in small-scale fluid mechanics. We first solve for the
swimming kinematics of elastic swimmers, devices which exploit
flow-induced deformation of elastic filaments for
propulsion. More generally, we then show how soft surfaces can
be exploited for propulsion without inertia. Finally, we
describe how the viscoelastic nature of the surrounding fluid
can affect the kinematics and energetics of simple swimmers.

I will try to discuss Abstract Algebra from its emergence to
the present day and its place among other mathematical areas.

Abstract: Brumer's conjecture states that Stickelberger elements
combining values of L-functions at s=0 for an abelian extension of
number fields E/F should annihilate the ideal class group of E when it
is considered as module over the appropriate group ring.
In some cases, an ideal obtained from these
Stickelberger elements has been shown to equal
a Fitting ideal connected with the ideal class group.
We consider the analog of this at s=-1, in which the class group
is replaced by the tame kernel, which we will define.
For a field extension of degree 2, we show that there is an exact equality
between the Fitting ideal of the tame kernel and the most natural
higher Stickelberger ideal; the 2-part of this equality is conditional on
the Birch-Tate conjecture.

A number of recent results, including special discretization
schemes, adaptive methods and multilevel iterative methods for
the resulting algebraic systems, will be presented in this talk
for various partial differential equations (PDEs). With a
careful and combined use of qualitative properties of PDEs, the
underlying functional spaces and their discretizations, many
different kinds of equations will be treated with similar
techniques. After an introduction to some practically efficient
methods such as the algebraic multigrid method for the Poisson
equations, it will be shown how more complicated systems such as
linear elasticity equations, electro- magnetic equations, porous
media, Stokes equations and more general Newtonian/non-Newtonian
models can be reduced to the solution of a sequence of Poisson
equation and its simple variants. The efficiency of these
algorithms will be illustrated by theoretical analysis, numerical
examples and engineering applications.