Six undergraduates will receive support from this Research Experiences for Undergraduates award. They will work under the direction of several faculty advisors all of whom have strong research programs under way. This program continues a successful program of undergraduate research at Washington University. It will be augmented during the next two years by a series of visits by leading mathematical researchers. The environment for this activity is particularly fecund at Washington University which has produced many of the top finalists and leading teams in the Putnam examinations over the past ten years. Included among the many topics which will be available for student projects is the study of the phi-transform. This concept is a very recent variation of the wavelet transform allowing one to decompose general functions into sums of simple terms. It has interesting computational characteristics which allow for simple but important questions to be undertaken early without need for extensive theoretical training. Work will also be done on the boundary behavior of conformal mappings and harmonic measure. This work will fit well into the development of students who have recently completed an undergraduate complex variables course. Another area of interest includes problems arising in the theory of several complex variables. Here, researchers can construct numerical models for automorphism groups and perform calculations of invariants associated with boundary differential invariants. The work only requires basic algebra and an understanding of symbol manipulation programs. In the process, students will develop concrete understanding of geometric structures and the analysis related to them. A more applied program developing mathematical models of molecular evolution will also be available. Students will consider questions concerning the distribution of nucleotides in the coding regions of individuals chosen from a random-mating population. Connections with algorithms for estimating phylogenies of creatures whose traits are known in the present will be investigated.
