中科院数学与系统科学研究院

数学研究所

学术报告

复动力系统研讨班

 

报告人LUCAS KAUFMANNIBS Center for Complex Geometry)

 HOLOMORPHIC DYNAMICS AND RANDOM MATRICES

  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.

 

 

第一讲:524日(星期二)上午9——11

 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.

 

第二讲:526日(星期四)上午9——11

 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.

 

第三讲:531日(星期二)上午9——11

 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.

 

第四讲:62 (星期四)上午9——11

 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.

 

第五讲:67 (星期二)下午2——4

 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.

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