Mathematical Sciences: Harmonic Analysis Algorithms

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An REU site project will focus on the use of advanced computational experimentation to expose undergraduate researchers to a variety of current mathematical research frontiers. A program to carry out numerical experiments on the qualitative behavior of solutions of nonlinear evolution equations and elliptic systems will be available. Studies of asymptotic behavior of solutions to the porous medium equation represents one specific area. Numerical data obtained will amplify recent theoretical results on uniqueness for extremal functions in the Sobolev inequality. Other work will concentrate on the phi-transform, a new technique developed for decomposing general functions into sums of simple terms. It is based on important advances in harmonic analysis in the last twenty years. In addition to being relatively easy to understand, this transform exhibits certain speed characteristics of the fast Fourier transform. Properties of the transform make it useful in numerical work; understanding these properties is a task well suited for undergraduate research. Student interns will also be involved in the creation of numerical models for calculation of automorphism groups for domains in the space of several complex variables. This highly theoretical field lends itself to numerical interpretation through certain boundary differential quantities known as the Chern-Moser-Tanaka invariants. Tedious calculations necessary to put the boundary of a domain in normal form can now be carried out. Students will develop the basic algebra and symbol manipulation algorithms which will be needed, learning about the complex mathematical structures and their analysis simultaneously.