Fine Properties of Harmonic Measure and Rotation of Planar Domains

建议：DMS-9970283首席研究员：Ilia Binder摘要：建议研究的主要目的是更深入地了解平面区域的共形几何、调和几何和度量几何。更具体地说，这项研究将集中于建立格林直线旋转的精确度量界限，并阐明它们与调和测度和共形映射的性质的关系。这项研究应该能更清楚地理解单连通和非单连通域的界限之间的联系。Binder还将探索这些思想在复杂动力学中的可能应用。保角映射和调和测量是经典对象，在不同的数学领域以及在工程和物理中广泛使用了一百多年。相比之下，具有混沌(或“分形”)行为的系统理论是最新和最活跃的数学领域之一，在生物学、物理学、计算机科学和工程中有着广泛的应用。宾德的项目试图利用现代混沌系统理论的技术和思想来研究与共形映射和调和测量有关的历史悠久的问题，希望为这些古老的学科带来新的曙光，并为数学和应用科学中潜在的令人兴奋的应用打开大门。

Proposal: DMS-9970283Principal Investigator: Ilia BinderAbstract: The main objective of the proposed research is to understand more deeply the conformal, harmonic, and metric geometry of planar domains. More specifically, the research will focus on establishing sharp metrical bounds on the rotation of Green lines and elucidating their relationship with the properties of harmonic measure and conformal mappings. The research should provide a clearer understanding of the connection between the bounds for simply connected and nonsimply connected domains. Binder will also explore possible applications of these ideas to complex dynamics.Conformal mappings and harmonic measure are classical objects, extensively used for more than a hundred years in different areas of mathematics, as well as in engineering and physics. By contrast, the theory of systems with chaotic (or "fractal") behavior is one of the newest and most active areas of mathematics, with a multitude of applications to biology, physics, computer science and engineering. Binder's project seeks to employ the techniques and ideas of the modern theory of chaotic systems in order to investigate time-honored questions concerning conformal mappings and harmonic measure, in the hope of shedding new light on these venerable subjects and opening doors to potentially exciting applications in both mathematics and the applied sciences.