Probabilistic and Analytic Aspects of the Loewner Energy

勒纳能量的概率和分析方面

基本信息

批准号：
1953945
负责人：
Scott Sheffield
金额：
$ 17.27万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953945&HistoricalAwards=false
关键词：
Probabilistic Analytic Aspects Loewner Energy

项目摘要

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.

该项目涉及概率和复杂分析领域的研究。Loewner 能量是测量简单平面环的圆度的量，它源自 Schramm-Loewner 演化 (SLE)（一种随机分形曲线模型）的渐近行为。 SLE 在随机共形几何和二维统计力学中发挥着核心作用，这些力学利用微观层面的给定信息来研究系统的宏观几何，令人惊讶的是，这种概率驱动的 Loewner 能量。可以使用数学和数学物理学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与这些分支之间存在深刻的联系。揭示这些联系并探索围绕洛纳能量的各种观点如何为概率论和其他领域带来新的见解。结果预计还将揭示理论物理学基础数学架构的新方面。乔丹曲线的洛纳能量是定义为因此，通过 Loewner 微分方程得出的其驱动函数的狄利克雷能量可以被视为 SLE 的卡梅伦-马丁空间，其具有多个布朗运动作为驱动函数。Loewner 能量和 SLE 的定义都取决于此。然而，使用拉普拉斯行列式发现了 Loewner 能量的等效且内在的描述，并且被称为 Weil-Petersson 度量的 Kähler 势。该研究项目首先旨在通过焊接同胚的规范测量来提供 SLE 环的类似内在描述，然后研究将 Loewner 能量涉及到其他场景多弦或更高属面、受分析恒等式的启发。随机共形几何的结果，以及双曲 3 空间中最小曲面的关系。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。