I will describe current opportunities for funding in mathematics and statistics at the National Science Foundation, as well as issues that arise in proposal preparation. There will be ample opportunity for questions from the audience.

Stem cells have the ability to renew and to differentiate into progenitor cells that ultimately form all of the tissues in an organism. The current interest in stem cells C both adult and embryonic C is through the promise that they hold for regenerative medicine.
That promise, however, relies on the assumption that stem cells will respond to our modifications of them in ways that we desire. However, experience with interventions in other natural systems C from fishing to antibiotics C shows that acting without thinking about evolutionary consequences is fraught with danger. The time for an evolutionary ecology of stem cells is now. I will show how this can be done, using the hematopoeitic system as an example. To begin, we model the system of stem, progenitor, and fully differentiated cells using a combination of stochastic simulation and associated ordinary differential equations. These models must capture the feedback controls C both positive and negatve C on the stem cell system. Once this framework is organized, it is possible to ask a variety of questions. In this talk, I will focus on two. First, I will use evolutionary invasion analysis to ask if modifications of stem cell parameters will have the effects that we desire. Second, I will use state dependent life history theory, implemented through stochastic dynamic programming, to understand why stem cells are mainly quiesencent and compare this result with the classic experiment of Till and colleagues. This work reminds us that nothing in biology makes sense except in the light of evolution.

We present a few numerical methods for inverse problems related to the
radiative transport equation. We consider two specific applications of those inverse problems: optical tomography and incoherent imaging in random media. Numerical examples based on synthetic and experimental data will be presented.

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.