Mathematical Sciences: Harmonic Measure Analytic Capacity and Rectifiable Sets

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In this project the principal investigators will study the geometric properties of harmonic measure; in particular, they will focus on the relations between harmonic measure on domains in the plane and one-dimensional Hausdorff measure. The principal investigators will also look into problems involving harmonic measure and analytic capacity, the relationship between rectifiable sets and harmonic measure, and the estimation of eigenvalues of the Laplace operator on subdomains of the sphere in higher dimensions. The principal investigators are interested in studying the relationship between an analytical quantity known as the harmonic measure of a set and the geometry of this set. Harmonic measure is an important and useful concept that arises in probability theory and complex function theory. In this project the principal investigators will seek to characterize properties of the harmonic measure of a set in terms of geometric properties of the set such as its Hausdorff dimension.