Complex Analysis and Random Geometry

复杂分析和随机几何

• 批准号: 2350481
• 负责人: Steffen Rohde
• 金额: $ 29.98万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-15 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350481&HistoricalAwards=false
• 关键词: Complex; Analysis; Random; Geometry

The project explores probabilistic and deterministic aspects of self-similar geometry. Self-similar sets are characterized by the property that they look the same at different scales. Such sets arise in the study of dynamical systems, for instance, in complex dynamics and the study of the Mandelbrot set. On the other hand, in probability theory and statistical physics one often encounters stochastically self-similar sets. Such objects only have the same statistical properties at different scales. There are surprising analogies between the probabilistic theory and its deterministic counterpart. The research supported by this award explores these analogies and addresses foundational questions regarding self-similar objects, using methods from complex analysis. The project also provides opportunities for the training and mentoring of junior researchers, including graduate students and postdoctoral researchers. The PI will contribute to the dissemination of mathematical knowledge through the organization of conferences, workshops, and summer schools.Research to be conducted under this award involves the geometry of conformally self-similar structures, both in stochastic and deterministic settings. Julia sets for the iteration of complex mappings illustrate the latter setting, while the former includes topics such as Schramm-Loewner evolution. The project aims to answer fundamental regularity questions for conformally self-similar objects, including Jordan curves of finite Loewner energy. A new parametrization of the Teichmueller spaces of punctured spheres will also be studied. Additional motivation for the project arises from the interaction between the deterministic and stochastic frameworks, notably, the transfer of methods and results between these two areas. For instance, the concept of conformal mating of polynomials in complex dynamics bears close similarity to Sheffield's mating of trees construction for random spheres. The PI’s research uses methods developed in complex dynamics to provide analytic constructions for random structures. Conversely, insights from the probabilistic theory translate to new research avenues in complex dynamics. Conformal welding is a tool of central importance in both theories, and the proposal aims to resolve several fundamental questions regarding Weil-Petersson curves, welding of Liouville Quantum Gravity discs, and Werner's conformal restriction measure on Jordan curves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目探讨了自相似几何形状的概率和确定性方面。自相似集的特征是它们在不同尺度上看起来相同的属性。这种集合在动态系统的研究中出现，例如，复杂的动力学和曼德布罗特集合的研究。另一方面，在概率理论和统计物理学中，人们经常遇到随机上的自相似集合。此类对象仅在不同尺度上具有相同的统计属性。概率理论与其确定性对应物之间存在惊喜。该奖项支持的研究探讨了这些类比，并使用复杂分析中的方法解决了有关自相似对象的基本问题。该项目还为初级研究人员（包括研究生和博士后研究人员）的培训和心理化提供了机会。 PI将通过组织，研讨会和暑期学校的组织来促进数学知识的传播。在该奖项下进行的研究涉及在随机和确定性的环境中的保同性自相似结构的几何形状。朱莉娅（Julia）进行复杂映射的迭代设置说明了以后的设置，而前者包括诸如Schramm-Loewner Evolution之类的主题。该项目旨在回答与共同类似物体的基本规律性问题，包括结局Loewner Energy的Jordan曲线。穿刺球体的Teichmueller空间的新参数也将是研究。该项目的其他动机来自确定性和随机框架之间的相互作用，尤其是方法和结果之间的转移和结果之间的相互作用。例如，复杂动力学中多项式的保形交配的概念与谢菲尔德的树木构造的随机球体非常相似。 PI的研究使用复杂动力学开发的方法来为随机结构提供分析结构。相反，概率理论的见解转化为复杂动力学的新研究途径。保形焊接在这两种理论中都是至关重要的工具，该提案旨在解决有关Weil-Petersson曲线，Liouville量子重力盘焊接的几个基本问​​题，以及Werner对Jordan曲线对NSF的法定任务和众多的支持，这表明了NSF的众多范围，这表明了NSF的众多范围，这表明了NSF的众多奖励，这表明了NSF的合法范围。 标准。