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.

There are no analogs of Navier-Stokes equations for solid mechanics. One reason is that information at the atomic scale seems to play a much more important role for solids than for fluids. A satisfactory mathematical theory for solids has to taken into account the behavior of solids at different scales, from electronic to atomic, to macroscopic scales.

I will discuss some of the fundamental problems that we have to resolve in order to build such a theory. I will start by reviewing the geometry of crystal lattices, the quantum as well as classical atomistic models of solids. I will then focus on a few selected problems:

(1) The crystallization problem -- why the ground states of solids are crystals and which crystal structure do they select?

(2) the microscopic foundation of elasticity theory;

(3) stability and instability of crystals;

(4) the electronic structure and density functional theory.