Classical real algebriac geometry studies the zero sets of polynomial functions.The complexity of such zero sets(topologically, geometrically or analytically) can be controlled by the degrees of the polynomials.One of the natural questions is to what extent such results would be also true for solutions of partial differential equations.One immediate issue comes up is the "analyticity" which, for solutions of PDEs often require equations to be of elliptic and parabolic types ,and may be with analytic coffecients etc.The purpose of this lecture is to present various progress towarding such a general goal. We shall describe a global analytical quantity called frequency,which plays the similar rule as the degree for the polynomials. We shall illustrate how the frequency measures the quantative analyticity and controls the complexity of solutions in various senses.

Lecture 2: Wednesday April 18 4:10 pm A 304 WH
Optimal Partitions of Eigenvalues.

Here we shall discuss the following optimization problem for eigenvalues.Consider a bounded domain in the Euclidian Space, and let m be a positive integer.One wants to divide the domain into m subdomains in such a way that the sum of the first eigenvalues of the Laplacian on each subdomains with the Dirichelet boundary conditions to be minimun with respect to all reasonable partitions of the domain. It appears that this lecture would have nothing to do with the previous one. However, we shall see that the notion of frequency introduced in the first lecture would play again an important rule in the understanding of this optimal partition problem. We shall describe classical solutions of the problem.

Lecture 3: Thursday April 19 4:10 pm A 304 WH
Singular perturbations and complexity of moduli spaces of solutions.

The existence of solutions and their behavior of the singularly perturbed semilinear elliptic problems have been studied by many authors. There are ,however,some general and seemingly important issues that have not been examined carefully.In this lecture, I shall discuss a few of such issues, in particular, the energy spectrums and a nonlinear quantum ergodicity and, to present some preliminary results. These studies can be viewed as extension of whose discussed in the first lecture, which aimed preliminary at solutions of linear equations.

For additional information:

Department of Mathematics
Michigan State University
East Lansing, MI 48824-1027
(517)355-9680
ginther@math.msu.edu
www.math.msu.edu/Lecture_Series