HOLOMORPHIC DYNAMICS AND RANDOM MATRICES

地  点：zoom会议号: 980 1259 4407    密码: minicourse
链接：https://zoom.us/j/98012594407?pwd=b3g3SXdLdEh5QVF6VDdOc3VCWHl6Zz09

要：

The aim of this mini-course is to overview fundamental results of holomorphic dynamics in one and several variables and illustrate a recent connection with the theory of products of random matrices.

The mini-course will consist of five lectures.

Lecture 1: Overview and background. In this first talk, I will overview the theory of holomorphic dynamics in one and several complex variables, introduce its main objects and highlight the main questions and results. I will also introduce basic tools from pluripotential theory, which are indispensable in the higher dimensional setting.

Lecture 2: Dynamics of endomorphisms of P^k. The aim of this talk is to present the main results and tools in higher dimensional holomorphic dynamics. For simplicity, I will focus in the case of holomorphic endomorphisms of the complex projective space. In particular, I will discuss Green currents and measures, statistical properties of the associated dynamical system, equidistribution results and spectral properties of transfer

operators.

Lecture 3: Dynamics of correspondences. This talk will focus on holomorphic and meromorphic correspondences, that is, multivalued maps. I will introduce these objects and state their basic properties. The dynamics of correspondences with large topological degree will be discussed.

Lecture 4: From correspondences to random matrices. In this talk I will introduce a class of correspondences on Riemann surfaces called “weakly modular”. In particular, I will show that they satisfy a spectral gap theorem. I will discuss relations with group actions on the Riemann sphere and random matrices.

Lecture 5: Random walks on SL2(C). This talk is about products of random matrices in SL2(C). This is a classical research topic but, as we will see, it can be studied from the point of view of holomorphic dynamics. In particular, I will show how methods from complex analysis yield spectral gap results for Markov operators and lead to the proof of several limit theorems.