Date of Award

2019

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

McDonald, Patrick

Area of Concentration

Mathematics

Abstract

There is a rich literature on the relationship between geometry, the heat equation, and Brownian motion. An early result in this direction is the Saint-Venant Theorem, which states that amongst planar domains of equal area the average expected first exit time of Brownian motion is maximized on a disk. This result was extended by Kinateder, McDonald and Miller to show that amongst domains of equal volume in Rn, the L1-norms of all moments of the first exit time of Brownian motion are maximized on an n-ball. In this thesis, we use comparison results for elliptic and parabolic PDEs under Steiner symmetrization to show that amongst triangles and quadrilaterals of equal area, the L1-norms of all exit time moments of Brownian motion are maximized on the equilateral triangle and square, respectively. We also present a preliminary study of applications of the L1-moment spectrum to image clustering.

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