Probabilistic and Analytic Aspects of the Loewner Energy

该项目涉及概率和复分析领域的研究。Loewner能量是测量简单平面环的圆度的量。它源于Schramm-Loewner演化（SLE）的渐近行为，SLE是随机分形曲线的模型。SLE在随机共形几何和二维统计力学中起着核心作用，这些力学研究具有给定微观信息的系统的宏观几何。令人惊讶的是，这种以概率为动机的洛温纳能量可以用数学和数学物理中看似不同的分支的基本概念来描述，包括几何函数论、泰希米勒理论、共形场论和弦理论。这些联系表明随机共形几何和这些分支之间有着深刻的联系。该研究项目旨在揭示这些联系，并探索围绕Loewner能量的各种观点如何为概率论和其他领域带来新的见解。这些结果也有望揭示理论物理学基础数学结构的新方面。通过Loewner微分方程将Jordan曲线的Loewner能量定义为其驱动函数的Dirichlet能量。因此，有限能量曲线可以被看作SLE的Cameron-Martin空间，它具有布朗运动的倍数作为驱动函数。 Loewner能量和SLE的定义强烈依赖于曲线的参数化。然而，使用拉普拉斯行列式发现了Loewner能量的等价和内在描述，并且已知是普适Teichmüller空间上Weil-Petersson度量的Kähler势。本研究计划的目的首先是通过焊接同胚上的规范测度来提供SLE环的类似内在描述，然后研究Loewner能量到其他涉及多弦或更高亏格曲面的场景的推广，随机共形几何结果启发的解析恒等式，和极小曲面的关系，在双曲3-该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的评估支持影响审查标准。

This project concerns research in the areas of probability and complex analysis. The Loewner energy is a quantity measuring the roundness of a simple planar loop. It arises from the asymptotic behaviors of the Schramm-Loewner evolution (SLE), a model of random fractal curves. SLE plays a central role in random conformal geometry and two-dimensional statistical mechanics that study the macroscopic geometry of systems with given information on the microscopic level. Surprisingly, this probabilistically motivated Loewner energy can be described using fundamental concepts from seemingly disparate branches of mathematics and mathematical physics, including geometric function theory, Teichmüller theory, conformal field theory, and string theory. These links suggest deep connections between random conformal geometry and those branches. This research project aims at revealing these connections and exploring how the variety of perspectives around the Loewner energy can bring new insights to probability theory and other fields. The results are expected also to reveal new facets of the mathematical architecture underlying theoretical physics.The Loewner energy of a Jordan curve is defined as the Dirichlet energy of its driving function via the Loewner differential equation. Finite energy curves can, therefore, be viewed as the Cameron-Martin space of SLE, which has a multiple of Brownian motion as driving function. This definition of both Loewner energy and SLE depends strongly on the parametrization of the curves. However, an equivalent and intrinsic description of the Loewner energy was discovered using determinants of Laplacians and is known to be the Kähler potential of the Weil-Petersson metric on the universal Teichmüller space. This research project first aims to provide similar intrinsic descriptions of SLE loops via the canonical measures on the welding homeomorphisms, then studies generalizations of the Loewner energy to other scenarios involving multi-chords or higher genus surfaces, analytic identities inspired by results from random conformal geometry, and the relation to minimal surfaces in the hyperbolic 3-space.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.