We will report on the construction of energy minimizing coarse spaces built by patching solutions to appropriate saddle point problems. We first set an abstract framework for such constructions, and then we give an example of constructing coarse space and stable interpolation operator for the two level Schwarz method. We apply the theoretical results in the design of coarse spaces for discretizations of PDE with large varying coefficients. The stability and approximation bounds of the constructed interpolant are in a weighted norm and are independent of the variations in the coefficients. Such spaces can be used in two level overlapping Schwarz algorithms for elliptic PDEs with large coefficient jumps generally not resolved by a standard coarse grid. This is a joint work with Robert Scheichl (University of Bath, UK) and Panayot S. Vassilevski (Lawrence Livermore National Lab).

We review the asymptotic behavior of a class of Toeplitz (as well as related
Hankel and
Toeplitz + Hankel) determinants which arise in integrable models and other
contexts.
We discuss Szego, Fisher-Hartwig asymptotics, and a transition between them. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling
limits, to Fredholm
determinants which appear in the theory of group representations, in
random matrices, random permutations and partitions. The connection to
Toeplitz determinants
helps to evaluate the asymptotics of related Fredholm determinants in
situations of interest, and we
mention some of the corresponding results.

I will present the proofs of some recent results of Viale
and Weiss. Weiss introduced the notion of a slender function in his
dissertation: roughly, a function $M \mapsto F(M) \subset M$ (where
$M$ models a fragment of set theory) is slender iff for every
countable $Z \in M$, $Z \cap F(M) \in M$; i.e. $M$ can see countable
fragments of $F(M)$. Viale and Weiss proved that under the Proper
Forcing Axiom, for every regular $\theta \ge \omega_2$, there are
stationarily many $M \in P_{\omega_2}(H_{(2^\theta)^+})$ which
``catch'' $F(M \cap H_\theta)$ whenever $F$ is slender (i.e. whenever
$F$ is slender then there is some $X_F \in M$ such that $F(M \cap
H_\theta) = M \cap X_F$). The stationarity of this collection implies
many of the known consequences of PFA; e.g. failure of weak square at
every regular $\theta \ge \omega_2$; and separating internally
approachable sets from sets of uniform uncountable cofinality.

Linear ordering of objects is important in many applications.
For example, destined to live with the RAM model of computing for a foreseeable future, optimal linear ordering of elements to improve cache coherency and performance of out of core algorithms becomes crucial. While ordering the elements, the access pattern has to be taken into
account, which in turn is application dependent. Assuming, between
pairs of elements, we have the probability estimates of the second
element being accessed after the first, we propose a solution to the
problem of linear ordering of elements using 1-factor graph
partitioning algorithm.

Primarily, we will motivate the need for linear ordering using its
application to various problems in computer graphics including cache-coherent triangle ordering (also called stripification), simplification,
compression, efficient back-face culling, quadrilateral mesh
stripification, and tetrahedral mesh stripification. In simplicial
complex realization of manifold spaces, the algorithm can be extended
to generate space-filling curves. The graph abstraction of the
problem makes the solution seamlessly extendable to elements in
higher dimensions including higher dimensional databases and nodes of
the hierarchical partitioning of the objects like quadtrees and
octrees in computer graphics.