Laplacian growth, Schwarz reflection, and random normal matrices

California Institute Of Technology, Pasadena CA

Investigators

Abstract

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.