Complex Analysis and Random Geometry

该项目探索了自相似几何的概率和确定性方面。自相似集的特征在于它们在不同的尺度上看起来是相同的。这样的集合出现在动力系统的研究中，例如，在复杂动力学和Mandelbrot集合的研究中。另一方面，在概率论和统计物理中，人们经常遇到随机自相似集。这些对象只是在不同的尺度上具有相同的统计特性。概率论与确定性理论之间有惊人的相似之处。该奖项支持的研究探索了这些类比，并使用复杂分析的方法解决了关于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提供培训和指导的机会。PI将通过组织会议、研讨会和暑期学校来促进数学知识的传播。在该奖项下进行的研究涉及随机和确定性环境下的共形自相似结构的几何。复映射迭代的Julia集说明了后一种设置，而前者包括Schramm-Loewner演化等主题。该项目旨在回答共形自相似对象的基本正则性问题，包括有限Loewner能量的Jordan曲线。还将研究穿孔球面的Teichmueller空间的一种新的参数化法。项目的其他动机来自确定性框架和随机框架之间的相互作用，特别是这两个领域之间的方法和结果的转移。例如，复杂动力学中多项式的共形交配的概念与谢菲尔德关于随机球面的树的交配有着密切的相似之处。PI的研究使用了复杂动力学中发展起来的方法，为随机结构提供解析构造。相反，来自概率理论的见解将转化为复杂动力学的新研究途径。保形焊接是这两种理论中最重要的工具，该提案旨在解决关于Weil-Petersson曲线、Liouville量子重力盘的焊接以及Werner在约旦曲线上的保形限制措施的几个基本问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.