Fine Properties of Harmonic Measure and Rotation of Planar Domains

摘要：本研究的主要目的是更深入地了解平面结构域的共形几何、谐波几何和度量几何。更具体地说，研究将侧重于在绿线的旋转上建立尖锐的度量界限，并阐明它们与谐波测度和保角映射性质的关系。该研究应提供对单连通域和非单连通域边界之间联系的更清晰的理解。Binder还将探索这些思想在复杂动力学中的可能应用。保角映射和调和测度是经典的对象，在数学、工程和物理的不同领域广泛使用了一百多年。相比之下，具有混沌（或“分形”）行为的系统理论是数学中最新和最活跃的领域之一，在生物学，物理学，计算机科学和工程中有着广泛的应用。宾德的项目旨在运用现代混沌系统理论的技术和思想来研究有关保角映射和调和测度的古老问题，希望为这些古老的学科提供新的启示，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.