REU SITE: Interactions Between Algebra, Computation, and Mathematical Physics

Temple University, Philadelphia PA

Investigators

Abstract

This REU features simultaneous subprograms in algebra and applied mathematics. The unifying theme is the interplay between algebraic methods, computational techniques, and mathematical physics. Throughout, there is an emphasis on computer-aided experimentation in a collaborative environment. Students in the algebra subprogram learn to develop -- and prove --mathematical conjectures based on extensive studies of specific examples. The primary mathematical focus is on (finite dimensional) algebraic representation theory -- that is, the study of matrix solutions to systems of polynomial equations in noncommuting variables. Investigations along these lines have played, and continue to play, a key role in modern physics. Moreover, solutions to these systems correspond to objects in ``noncommutative geometric spaces.'' The particular systems considered in this subprogram are mostly composed of quadratic and cubic polynomials in two or three (noncommuting) variables. These systems typically define more abstract algebraic objects whose structures can form the basis for deeper investigations. The use of advanced symbolic computation software plays an important role throughout. Students in the applied mathematics subprogram are introduced to the three components of applied mathematics research: modeling, analysis, and computation. The focus is on two topics arising in materials science: complex electro-magnetic permittivity of dielectrics, and exact relations for fiber-reinforced composites. Students working on complex electro-magnetic permittivity develop mathematical stratgies for predicting how much electro-magnetic energy a material will absorb, in a given frequency, if data for other frequencies is already known; such prediction techniques have the potential to reduce the cost of experimental measurements drastically. Students working on exact relations mathematically analyze the influence of composite microstructure on macroscopic properties; understanding this influence is crucial to developing new materials with various desired properties -- potentially applicable industrial products include skis, golf clubs, and aircraft parts.