Proposal: DMS 9501611 PI: Elizabeth Housworth Institution: U. of Oregon Title: Isoperimetric-Type Inequalities Arising from the Study of Brownian Motion in Domains Normalized by their Inradius ABSTRACT: This Career award will fund research on two main mathematical questions as well as an innovative education plan. The first mathematical problem concerns an integral mean inequality which would constitute a sharpening of a known lifetime result for Brownian motion in convex, simply connected planar domains normalized so that the largest disk they contain has radius one (or asymptotically one.) The second problem concerns a packing measure analysis of harmonic measure which would provide an analogue to the results of Makarov which relate a lower bound on the growth of the derivative of a univalent function to the proper Hausdorff measure function for the support of harmonic measure on the boundary of a domain. The education plan includes redesigning the senior/beginning graduate mathematical statistics courses to give the students a feeling for data and teaching a course for undergraduate mathematics majors by a modified Moore method, in order to give our undergraduates insight into the process of doing mathematical and statistical research. This award will fund both research in mathematics and course development in the mathematical sciences. During the award period, it is proposed to redesign an upper level statistics course to include work with actual data instead of focusing entirely on the mathematical theory behind statistics. There are also plans to teach a course for mathematics majors using self-discovery techniques in order to give undergraduates insight into the process of doing mathematical research. Two mathematical questions will be researched during the award period. One problem involves studying the pathological behavior of a certain class of mathematical functions. The techniques employed are related to those used for the study of fractals. The second pr oblem is derived from considering the lifetime of a particle moving randomly inside a region and dying when it touches the boundary of the region. Different regions allow the particle to live different lengths of time. Applications of this research are relevant in applied contexts, such as in studying stress on beams.
