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.

We will talk about a paper by A. Moreira and J.C. Yoccoz, where they proved a conjecture by Palis according to which the arithmetic sums of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval.