Conformal Stochastic Geometry, Dyson Gas, Potential Theory and Conformal Field Theory

该奖项支持随机（或随机）共形几何及其在物理和复分析中的应用的新兴领域的研究和教育。近年来，物理和数学中的几个学科通过关注随机共形几何问题而获得了巨大的推动-随机形状，回路，路径和表面的统计，其概率度量在很大程度上由共形对称性决定。这些问题是感兴趣的物理和数学的各个领域，特别是现代复杂的分析，其中各自的见解和方法往往是共享。提案中描述的这类问题本质上是概率性的，使用概率方法至关重要。另一方面，共形随机几何可以被看作是经典共形分析的“量子”扩展，因此现代势理论和复分析方法是另一个重要的部分。Dyson气体和Dyson扩散是该提议的一个统一主题。在物理学中，戴森气体出现在量子霍尔效应、描述具有分数电荷和统计的粒子的Calogero模型、量子混沌和随机矩阵理论中。在数学中，戴森气体出现在塞尔伯格积分理论、复分析、正交多项式理论和统计力学的可积模型中。近年来，人们逐渐清楚戴森气体与共形随机几何和Virasoro代数的表示有着内在的联系。由于Dyson气体可以用经典分析的方法来研究，因此提出用Dyson气体作为一个平台来严格地解决共形几何的难题是目标之一。随机几何的概念起源于研究边界上开始和结束的随机界面的临界现象领域。但是最近对随机几何对象（例如临界波动簇，临界表面，随机自避免行走和随机增长的模式表明，保形几何的方法和概念是更广泛的一类发展中的物理学领域的核心，数学其中包括不稳定的流体流动，随机高斯场，量子引力，随机矩阵，非平衡增长过程，包括扩散限制聚集，淬火无序系统中的临界现象和分数霍尔效应。这些问题共同构成了一个新兴的共形随机几何领域，共形随机几何领域的一个重要特征是数学和物理方法、方法和直觉的自然相互作用。该建议建立在这样的统一的势头，专注于当前感兴趣的跨学科问题，落入共形随机几何领域。其中一些有精确的数学公式;其他人还没有到目前为止制定了严格的方式。我们的目标之一是发展数学公式的重要物理问题的域共形随机几何。拟议研究的更广泛影响在于汇集来自物理和数学各个领域的想法，包括复分析，概率论，随机矩阵，共形场论，非平衡增长现象和无序系统。拟议的研究将通过将结果传达给各个研究团体并促进不同领域从业者之间的合作，使这些领域更加紧密。该提案的教育和推广部分将把研究纳入从高中到研究生的物理教学。该项目将为几名研究生提供研究和培训机会。

The award supports research and education in the emerging area of stochastic (or random) conformal geometry and its applications to physics and complex analysis.In recent years several subjects in physics and mathematics have received a tremendous boost by focusing on problems of stochastic conformal geometry - statistics of random shapes, loops, paths and surfaces whose probabilistic measure is largely determined by conformal symmetry. These problems are of interest to various fields in physics and mathematics, especially modern complex analysis where respective insights and methods are often shared. The class of problems described in the proposal is probabilistic in nature, and the use of probabilistic methods is essential. On the other hand, conformal stochastic geometry can be seen as "quantum" extension of classical conformal analysis, therefore methods of modern potential theory and complex analysis are yet another essential part.A unifying theme of the proposal is Dyson gas and Dyson diffusion. In physics Dyson gas appears in Quantum Hall Effect, in Calogero models describing particles with fractional charge and statistics, quantum chaos and random matrix theory. In mathematics Dyson gas appears in the theory of Selberg integrals, complex analysis, the theory of orthogonal polynomials and integrable models of statistical mechanics. In recent years it became clear that Dyson gas has an intrinsic relation with conformal stochastic geometry and representations of Virasoro algebra. Since Dyson's gas can be studied by methods of classical analysis one of the aspect of the target of the proposal to use Dyson gas as a plat to form to attack difficult problems of conformal geometry rigorously.The notion of stochastic geometry has originated in the field of critical phenomena in studies of random interfaces starting and ending on boundaries, but recent quest for statistics of random geometrical objects such as critical fluctuating clusters, critical surfaces, random self-avoiding walks and randomly growing patterns showed that methods and concepts of conformal geometry are at the core of a broader class of developing fields of physics and mathematics. Among them unstable fluid flows, random Gaussian fields, quantum gravity, random matrices, non-equilibrium growth processes including the diffusion limited aggregation, critical phenomena in systems with quenched disorder and Fractional Hall effect. Together, these problems constitute a newly emerging field of conformal stochastic geometry.An important feature of the field of conformal stochastic geometry is a natural interaction of mathematical and physical approaches, methods, and intuition. The proposal builds on the momentum of such unification by focusing on interdisciplinary problems of current interest that fall into the realm of conformal stochastic geometry. Some of them have accurate mathematical formulations; others have not been so far formulated in a rigorous way. One of our goals is to develop mathematical formulations of important physical problems in the domain of conformal stochastic geometry. The broader impact of the proposed research lies in bringing together ideas from various fields of physics and mathematics including complex analysis, probability theory, random matrices, conformal field theory, non-equilibrium growth phenomena, and disordered systems. The proposed research will result in bringing these fields closer by communicating the results to various research communities and promoting collaborations between practitioners in diverse areas. The education and outreach component of the proposal will integrate research into teaching of physics from high school to graduate level. The project will provide research and training opportunities for several graduate students.