Mathematical Sciences: Harmonic Measure Analytic Capacity and Rectifiable Sets

在这个项目中，主要研究人员将研究调和测度的几何性质，特别是平面区域上的调和测度与一维Hausdorff测度之间的关系。主要研究人员还将研究有关调和测量和分析能力、可纠集和调和测量之间的关系以及高维球面子域上拉普拉斯算子的特征值的估计等问题。主要研究人员感兴趣的是研究一个集合的一个解析量，即调和度量和这个集合的几何之间的关系。调和测度是概率论和复变函数论中的一个重要而有用的概念。在这个项目中，主要的研究人员将寻求根据集合的几何性质来刻画集合的调和测度的性质，例如它的Hausdorff维度。

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.