The study of matrix structures makes it feasible to quickly solve some large discretized PDEs and integral equations. In particular, direct factorizations of some 2D and 3D elliptic problems can reach nearly linear complexity. Here, we show a framework that can be used to unify dense and sparse structured direct solvers, which are traditionally thought to be very distinct subjects. Such a unification makes it feasible to design new multi-dimensional structures that can conveniently handle sophisticated structures in dense 2D and 3D discretized problems. More specifically, we propose multi-layer hierarchically semiseparable (MHS) structures that integrate multiple layers of rank and tree structures in a recursive sparsification-localization strategy. We lay out theoretical foundations for MHS structures and justify the feasibility of MHS approximations for these dense matrices. Rigorous rank bounds for the rank structures are given. Representative subsets of mesh points are used to illustrate the multi-layer structures as well as the fast structured factorization. The framework makes it natural and convenient to 
(1) share ideas between dense and sparse direct solvers;
(2) perform stability and error analysis and reuse algorithm design based on simple hierarchical structures;
(3) establish intrinsic connections to other methods such as eigenvalue solvers and even multigrid methods.

We will discuss the mathematics of portfolio optimization, assuming asset returns have a fat-tailed, alpha-stable distribution. A PDF-file of the slides is available for download.

http://www.math.uci.edu/sites/math.uci.edu/files/Fat Tailed Kelly.pdf

In this talk we present a novel boundary integral equation method for the numerical solution of problems of scattering by obstacles and defects in the presence of layered media. This new approach, that we refer to as the windowed Green function method (WGFM), is based on use of smooth windowing functions and integral kernels that can be expressed directly in terms of the free-space Green function. The WGFM is fast, accurate, flexible and easy to implement. In particular straightforward modifications of existing (accelerated or unaccelerated) solvers suffice to incorporate the WGF capability. The mathematical basis of the method is simple: the method relies on a certain integral equation that is smoothly windowed by means of a low-rise windowing function, and is thus supported on the union of the obstacle and a small flat section of the interface between the two penetrable media. Various numerical experiments demonstrate that both the near- and far-field errors resulting from the proposed approach, decrease faster than any negative power of the window size. In some of those examples the proposed method is up to thousands of times faster, for a given accuracy, than an integral equation method based on use of the layer Green function and the numerical approximation of Sommerfeld integrals. Generalizations of the WGFM to problems of scattering by obstacles in layered media composed by any finite number of layers as well as wave propagation and radiation in open dielectric waveguides are also included in this presentation.

The overconvergent modular symbols of Stevens provide a natural framework for computing p-adic L-functions of newforms, but the modular symbols (and p-adic L-functions) attached to ordinary Eisenstein series are essentially trivial. Working with a larger space of pseudo-distributions, we construct non-trivial Eisenstein symbols and compute their p-adic L-functions. As a corollary, we compute the p-adic L-function of the "evil twin" Eisenstein series of critical slope. If time permits, I'll discuss work in progress on computing the symmetric square p-adic L-function at Eisenstein points on the eigencurve, as well as applications.