Operator decomposition methods are an attractive solution strategy
for computing complex phenomena involving multiple physical processes,
multiple scales or multiple domains. The general strategy is to
decompose the problem into components involving simpler physics
over a relatively limited range of scales, and then to seek the
solution of the entire system through an iterative procedure
involving solutions of the individual components.

We construct an operator decomposition finite element method for a
conjugate heat transfer problem consisting of a fluid and a
solid coupled through a common boundary. Accurate a posteriori
error estimates are then developed to account for both local
discretization errors and the transfer of error between fluid and
solid domains. These estimates can be used to guide adaptive mesh
refinement. We show that the order of convergence of the operator
decomposition finite element method is limited by the accuracy of
the transferred gradient information, and demonstrate how a simple
boundary flux recovery method can be used to regain the optimal
order of accuracy in an efficient manner.

This is joint work with Don Estep and Tim Wildey.

We will discuss our current research which describes and constructs polarizations of regular adjoint orbits for certain classical groups. This research generalizes recent work of Bertram Kostant and Nolan Wallach. Kostant and Wallach construct polarizations of regular adjoint orbits in the space of $n\times n$ complex matrices $M(n)$. They accomplish this by defining an $\frac{n(n-1)}{2}$ dimensional abelian complex Lie group $A$ that acts on $M(n)$ and stabilizes adjoint orbits. Note that the dimension of this group is exactly half the dimension of a regular adjoint orbit in $M(n)$. This fact allows $A$ orbits of dimension $\frac{n(n-1)}{2}$ contained in a given regular adjoint orbit to form the leaves of a polarization of an open submanifold of that orbit. We study the $A$ orbit structure on $M(n)$ and generalize the construction to complex orthogonal Lie algebras $\mathfrak{so}(n)$. In the case of $M(n)$, we obtain complete descriptions of $A$ orbits of dimension $\frac{n(n-1)}{2}$ and thus of leaves of polarizations of all regular adjoint orbits. For $\mathfrak{so}(n)$, we construct polarizations of certain regular semi-simple adjoint orbits.

There is a rich interplay between combinatorial and differential
geometry.
We will give first a geometric proof of a combinatorial result, and
then
a combinatorial analysis of a geometric moduli space. The first is
joint
work with Ivan Izmestiev, Rob Kusner, Guenter Rote, and Boris
Springborn;
the second with Karsten Grosse-Brauckmann, Nick Korevaar and Rob
Kusner.

In any triangulation of the torus, the average vertex valence is 6.
Can there be a triangulation where all vertices are regular (of valence 6) except for one of valence 5 and one of valence 7? The answer is no.
To prove this, we give the torus the metric where each triangle is
equilateral and then explicitly analyze its holonomy. Indeed,
techniques
from Riemann surfaces can characterize exactly which euclidean cone
metrics have full holonomy group no bigger than their restricted
holonomy group (at least when the latter is finite).

Next we consider the moduli space $M_k$ of Alexandrov-embedded surfaces
of constant mean curvature which have k ends and genus 0 and are contained in a slab. We showed earlier that $M_k$ is homeomorphic to an open manifold
$D_k$ of dimension $2k-3$, defined as the moduli space of spherical metrics
on an open disk with exactly k completion points. In fact, $D_k$ is the
ball $B^{2k-3}$; to show this we use the Voronoi diagram or Delaunay
triangulation of the k completion points to get a tree, labeled by
logarithms of cross-ratios. The combinatorics of the tree are tracked
by the associahedron, and the labels give us a complexification of the
cone over its dual. We note similarities to the spaces of labeled
trees
used in phylogenetic analysis.

In 1890 Stickelberger published his eponymous theorem in Math. Annalen giving an explicit annihilator for the `minus' (or imaginary) part of the class group of a cyclotomic field as a Galois module. However, Stickelberger's wonderful theorem raises more questions than it answers. And strangely, many obvious ones have only begun to receive serious attention - let alone answers - in the late 20th and early 21st centuries. For instance:

Is there a similar result for an arbitrary (abelian) extension of number fields?

Is the `Stickelberger ideal' the full annihilator of minus part the class group?

What about the `plus' (or real) part?

The first question leads to Brumer's Conjecture. The answer to the second question is certainly negative, for several different reasons which we shall try to disentangle. This leads to comparisons with the Fitting ideal of the class group and of its Pontrjagin dual, and so to very recent work by Greither, the speaker and others, which we shall survey.

If time allows we should like to report on some recent approaches to the third question.

Maxwell (1866) and Boltzmann (1872) developed a recipe to go from certain Newtonian laws of molecular dynamics to the Navier-Stokes system of gas dynamics. This recipe was controversial at the time. Mathematicians such as Hilbert, Klein, Poincare, and Zermelo were drawn into the debate. Hilbert featured it at the 1900 ICM in the articulation of his sixth problem, and made important contributions towards its resolution. The problem however remains largely open. Recent significant advances start with the DiPerna-Lions (1990) theory of global solutions to Boltzmann equations and lead to the Golse-Saint Raymond (2004) proof of the incompressible Navier-Stokes limit. This lecture will introduce the Boltzmann equation and survey some "new" connections to linear and weakly nonlinear gas dynamics that are the focus of recent research.