This award continues an Research Experiences for Undergraduates site at Rice University for a research project for 8 students. The project involves the detailed study of the heat kernel on the n-dimensional unit sphere in (n+1)-dimensional Euclidean space. The special functions involved in such problems are trigonometric for n = 1, Legendre polynomials for n = 2, and ultraspherical polynomials for larger n. Unsolved problems for research for the students include the monotonic properties of the heat kernel. For fixed time, the monotonicity with respect to the location on the sphere was solved, essentially by the Research Experiences for Undergraduates site at Rice last year. For fixed location interesting questions abound in investigating its behavior as a function or time.
