Fast and accurate computation of electromagnetic phenomena plays an important role in understanding the underlying physics for many complex physical and biological systems, such as lasing in optical fiber lasers, electrostatics forces in solvation model of biomolecules, and irradiation damage in materials under extreme conditions. In this talk, we will present two new algorithm developments with applications in these areas.

Image Charge Approximations of Reaction Fields and FMM for Charges inside a Dielectric Sphere

The reaction field of a charge inside a dielectric sphere, induced by a surrounding dissimilar dielectric medium, has applications in the study of electrostatic forces in the defect evolutions in material under extreme neutron irradiation, and hybrid explicit/implicit solvation models for biomolecules. In both cases, the long range Coulomb interactions have been identified as of primary influence in materials resistance to amorphorization under extreme conditions in the first case, and the free energy and the solvation study of biomolecules for the second. We have developed new discrete image charge approximations for the reaction field of a charge inside a dielectric sphere at high accuracy with only 2-3 image charges. Based on this result, we have extended the Fast Multipole Method to calculate the electrostatic interactions of charges inside or outside a dielectric sphere. The resulting O(N) algorithm has applications in computational materials and biology.

A Generalized Discontinuous Galerkin (GDG) Method based on Split Distributions for PDE with Nonsmooth Solutions

To model optical wave propagations in inhomogenous waveguides under the paraxial approximation, we need to solve time dependent Schrödinger equations with nonsmooth solutions as a result of field discontinuities at material interfaces. We will present a new type of discontinuous Galerkin method based on split distributions and their incorporations into the PDEs to account for jumps in solutions and derivatives. Special integration by parts formula for the split distributions is developed. The resulting generalized discontinuous Galerkin (GDG) method will be flexible to handle various types of interface jump conditions (time dependent and nonlinear) with high accuracy and easy to extend to multi-dimensional and other type PDEs with nonsmooth solutions. A full vector GDG-BPM (beam propagation method) will be developed to study gain guided fiber laser for efficient generations of high energy power sources.

The geometries of sub-cellular structures in cardiac myocytes play a critical role in regulating the cells' functions. We present a chain of image and geometric processing approaches for constructing multi-scale and realistic models of 3D cardiac cells. Two types of imaging data are considered: one is the confocal light microscopy images of whole cells and the other is the electron microscopy images of individual calcium release units (CRUs). Images are pre-processed by anisotropic filtering and contrast enhancement techniques. Sub-cellular structures are extracted by automatic image analysis approaches including 3D image segmentation and skeletonization. High quality surface and volumetric meshes are generated for finite element simulation of excitation-contraction (E-C) coupling in cardiac myocytes.

This talk discusses some fast numerical methods using certain rank
structured matrices: sequentially semiseparable (SSS) matrices
and hierarchically semiseparable (HSS) matrices. I will first
briefly show an example using semiseparable matrices: to find
polynomial roots and to estimate their conditions in $O(n^2)$ flops
instead of classical $O(n^3)$, where $n$ is the degree of the
polynomial. Then I will discuss in detail a superfast multifrontal
type direct method for large discretized linear systems. Rank
structures in the multifrontal method for certain discretized
matrices are exploited. Then semiseparable matrices are used
to approximate dense frontal and update matrices in the elimination.
A generalized semiseparable type factorization is obtained in linear
time. The overall sover has nearly linear complexity, linear storage,
and good potential for parallelization. It can also work as an
effective preconditioner.

I will spend most of the time formulating Serre's conjecture and explaining some of its applications: for instance, it implies Artin's conjecture for 2-dimensional odd complex representations of the absolute Galois group of Q. I will sketch some of the main ideas in the recent proof of the conjecture in joint work with Wintenberger, as completed by Kisin. I will also explain, if time permits, how our work offers a fresh perspective on Wiles' proof of Fermat's Last Theorem (FLT). For instance it gives a diiferent, in a sense more elementary, aproach to Ribet's level lowering results, a key ingredient in the proof of FLT.

It has been known since Euler that the values of the Riemann zeta function at negative integers are certain rational numbers, namely the Bernoulli numbers Bk. Similarly, the values of Dirichlet L-functions at s=0 are related to class numbers of certain number fields. These are simple instances of a common phenomenon, namely that the values of L-functions at critical points are algebraic, up to a simple factor, and that these algebraic numbers are related to algebraic quantities such as class numbers and Selmer groups. The present talk will be a survey talk on the algebraicity of special values of L-functions and their divisibility properties modulo primes.