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，它们 本文主要讨论整环上调和测度之间的关系 在平面和一维Hausdorff测度中。 的 主要调查人员还将调查涉及以下问题的问题： 调和测度与解析容量，调和测度与解析容量的关系 可求长集与谐波测度，以及 球面子域上拉普拉斯算子的特征值 在更高的维度。 主要研究人员有兴趣研究 被称为谐波的分析量之间的关系 一个集合的测度和这个集合的几何。 谐波测量 是一个重要而有用的概念， 理论和复变函数理论。 本工程 主要研究人员将寻求表征 一个集合在几何性质方面的调和测度， 例如它的Hausdorff维数。

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.