This talk is concerned with several eigenvalue problems from a linear stability analysis for the steady state morphogen gradients in Drosophila wing imaginal discs. These problems share several common difficulties including the followings: 1) The steady state solution which occurs in the coefficients of the relevant differential equations of the stability analysis is only know qualitatively and numerically. 2) Though the governing differential equations are linear, the eigenvalue parameter appears nonlinearly in the differential equations as well as in the boundary conditions. 3) The eigenvalues are determined not only as solutions of a homogeneous boundary value problem in differential equations with homogeneous boundary conditions of the Dirichlet type, they also come from a condition arising from a boundary condition of the original problem complementary to the Dirichlet condition.

This talk reports on two principal results for these nonlinear eigenvalue problems. Regarding the stability of the steady state morphogen gradients, we prove that the eigenvalues must all be positive and hence the steady state morphogen gradients are asymptotically stable. In addition, we will report on a novel result pertain to the determination of the smallest (positive) eigenvalue that determines the decay time of transients and the time needed to reach steady state. Here we prove that the smallest eigenvalue does not come from the nonlinear Dirichlet eigenvalue problem and we need only to find the smallest root of a relevant polynomial. Keeping in mind that even the steady state solution needed for the stability analysis is only known numerically, not having to solve the nonlinear Dirichlet eigenvalue problem is both an attractive theoretical outcome and a huge computational simplification.

The work reported was done jointly the speakers colleague, Professor Qing Nie, at UC Irvine and Professor Yuan Lou of the Department of Mathematics at the Ohio State University. The work of Professor Nie and the speaker was supported by NIH Grant R01- GM67247.

In this talk, he will discuss the regularization effect of the noise in ordinary and partial differential equations. The main results are the existence and uniqueness of strong solutions for nonlinear equations when the drift coefficient is not Lipschitz. The proofs of these results are based on the Girsanov transformation of measure. Some recent regularization results by fractional noise will be also presented.

Our recent progress in mathematical analysis and computation of time
harmonic Maxwell's equations in complicated media will be discussed.
For the direct problems, recent regularity results will be introduced.
Various types of boundary conditions will be discussed to reduce
the scattering problem into a bounded domain. The first convergence
analysis of the recent Perfect Matched Layer (PML) approach for Maxwell's
equations will be presented. For the inverse medium scattering,
a continuation approach based on uncertainty principle will be
presented for both multiple and fixed frequency boundary data.
Issues on convergence will be addressed. Our on-going research on
related topics and multiscale modeling of nano optics will be
highlighted.

We outline several methods for obstacle reconstruction from
far field scattering data, comparing and contrasting the strengths of
each. The first method
we discuss is a ``single wave" technique for determining the general
size and location of
scattering obstacles. Once we know where to look, we focus our
computational energy
into a local region, using multi-static/multi-frequency data to tease
out the shape of the scatterers. We
will compare several different ideas for accomplishing this ranging
from generalized
filtered backprojection to linear sampling. All of these methods are
so-called ``direct methods" -
no iterative refinement proceedure is used. We will discuss
advantages and disadvantages to these approaches.