Dyson's gas in 2D and conformal field theory

研究者计划研究戴森气体的数学理论，特别是与共形场理论有关的问题。拟进行的主要研究课题如下。(a)二维戴森气体：经典和量子拉普拉斯增长、随机正态矩阵和一般单组分等离子体系综、等离子体液滴的分分支和不同奇点对应的普适律、Polyakov-Alvarez型公式和配分函数的渐近性。(b)共形场理论：共形场的Dyson气体近似；随机Loewner演化的共形场论；戴森集合中的SLE曲线。戴森气体是一个经典的模型，它涉及到许多有趣的物理现象，如流体力学中的Hele-Shaw流和量子力学中的分数霍尔效应。在数学中，戴森气体是随机矩阵理论的重要组成部分。这个模型在物理层面上得到了深入的研究，但在数学方面仍有许多基本问题有待解决。研究者将专注于几个这样的问题，试图将数学（复杂分析，概率论，统计力学）和理论物理（共形场论，非平衡增长现象，无序系统）的各个领域的思想结合起来。该提案的教育部分包括开发新的研究生课程和组织两个长期研究项目。该项目将为一些研究生和博士后提供研究和培训的机会。

The investigator plans to work on the mathematical theory of Dyson's gas, in particular on problems related to conformal field theory. The main topics of the proposed research will be the following. (a) Dyson's gas in two dimensions: classical and quantum Laplacian growth, random normal matrices and general one-component plasma ensembles, bifurcations of the plasma droplets and the universality laws corresponding to different types of singularities, the Polyakov-Alvarez type formulas and the asymptotics of the partition function. (b) Conformal field theory: Dyson's gas approximation of conformal fields; conformal field theory of stochastic Loewner evolutions; SLE curves in Dyson's ensembles.Dyson's gas is a classical model that relates to a great variety of interesting physical phenomena such as Hele-Shaw flows in hydrodynamics and fractional Hall effect in quantum mechanics. In mathematics, Dyson's gas is an important part of random matrix theory. The model has been intensively studied on the physical level but many fundamental problems remain open on the mathematical side. The investigator will focus on several such problems trying to bring together ideas from various areas of mathematics (complex analysis, probability theory, statistical mechanics) and theoretical physics (conformal field theory, non-equilibrium growth phenomena, disordered systems). The educational component of the proposal consists in the development of new graduate courses and in the organization of two long term research programs. The project will provide research and training opportunities for several graduate students and postdocs.