Probabilistic and Analytic Aspects of the Loewner Energy

这个项目涉及概率和复杂分析领域的研究。洛夫纳能量是测量简单平面环的圆度的一个量。它源于随机分形曲线模型Schramm-Loewner演化(SLE)的渐近行为。SLE在随机共形几何和二维统计力学中起着核心作用，它们在微观水平上研究具有给定信息的系统的宏观几何。令人惊讶的是，这种概率驱动的洛夫纳能量可以用来自似乎完全不同的数学和数学物理分支的基本概念来描述，包括几何函数论、泰希米勒理论、保形场理论和弦理论。这些联系暗示了随机共形几何和这些分支之间的深层联系。这项研究项目旨在揭示这些联系，并探索围绕洛夫纳能量的各种视角如何为概率论和其他领域带来新的见解。这些结果还有望揭示理论物理背后的数学结构的新方面。乔丹曲线的洛夫纳能量通过洛夫纳微分方程式定义为其驱动函数的狄里克莱特能量。因此，有限能量曲线可以看作是以多重布朗运动为驱动函数的SLE的卡梅隆-马丁空间。勒夫纳能量和SLE的这种定义强烈依赖于曲线的参数化。然而，利用拉普拉斯行列式发现了Loewner能量的一个等价和内在的描述，即泛Teichmüler空间上Weil-Petersson度规的Kähler势。这项研究项目首先旨在通过焊接同胚上的正则度量来提供SLE回路的类似的内在描述，然后研究Loewner能量在涉及多弦或更高亏格曲面的其他场景中的推广，从随机共形几何得到的结果启发的解析恒等式，以及与双曲3-空间中极小曲面的关系。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。

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.