Laplacian growth, Schwarz reflection, and random normal matrices

拉普拉斯增长（LG）是两个物体之间边界处最重要的动力学类型之一。例如，LG自然出现在研究涉及移动边界的某些问题。作为一种生长过程，LG被广泛认为是物理学和自然科学中观察到的各种普遍模式形成的模型，例如细菌菌落的生长。 在这个经典的数学物理领域的一个新的发展是最近发现的事实，即在微观水平上，LG的机制是非常密切相关的等离子体系综中的粒子的行为。等离子体是物质的四种基本状态之一。等离子体的一种常见形式是在氖灯上看到的，事实上等离子体是宇宙中最常见的物质状态。这些等离子体系综被描述为数学对象称为随机矩阵，具有复杂的本征值。LG和随机矩阵的关系已经在物理层面上得到了深入的研究，但许多基本问题在数学方面仍然是开放的。PI尼古拉·马卡罗夫将重点关注几个这样的问题。本研究的主要内容和目标是：证明重标点过程的收敛性和描述随机正规矩阵模型（RNM）中各类边界点的普适性规律，证明Hele-Shaw流的层状性并严格推导其可积性; RNM模型中与代数液滴相关的施瓦茨反射的动力学分析。为了实现这些目标，PI将汇集来自数学（复分析，共形动力学，概率论）和理论物理（共形场论，非平衡增长现象，无序系统）各个领域的工具和想法。该项目的教育部分将是编制新的研究生课程和组织科学讲习班和会议。该项目将为研究生和博士后提供研究和培训机会。

Laplacian growth (LG) is one of the most important types of dynamics at the boundary between two objects. For example, LG arises naturally in the study of certain problems involving a moving boundary. As a growth process, LG is widely recognized as a model for the formation of various universal patterns observed in physics and natural sciences such as the growth of bacterial colonies. A new exiting development in this classical area of mathematical physics was the recent discovery of the fact that on the microscopic level, the mechanism of LG is very closely related to the behavior of particles in plasma ensembles. Plasma is one of the four fundamental states of matter. A common form of plasma is seen in neon signs and in fact plasma is the most common state of matter in the universe. These plasma ensembles are described in terms of mathematical objects call random matrices that have complex eigenvalues. The relation of LG and random matrices has been intensively studied on the physical level but many fundamental problems remain open on the mathematical side. The PI, Nikolai Makarov, will focus on several such problems. The main topics and goals of the study will be the following: the proof of the convergence of rescaled point processes and the description of the universality laws for various types of boundary points in the random normal matrix model (RNM); the proof of the laminarity of Hele-Shaw flows and the rigorous derivation of their integrability properties;the analysis of the dynamics of the Schwarz reflections associated with algebraic droplets in the RNM model. To achieve these goals, the PI will bring together tools and ideas from various areas of mathematics (complex analysis, conformal dynamics, probability theory) and theoretical physics (conformal field theory, non-equilibrium growth phenomenon, disordered systems). The educational component of the project will be the development of new graduate courses and organization of scientific workshops and conferences. The project will provide research and training opportunities for graduate students and postdocs.