报告一

报告题目：200 Years of the Poisson Integral Formula 泊松积分公式两百年 

报告人：Vadim Kaimanovich

报告时间：9:30 - 10:20 

报告地点：五教5106

摘要：

The first talk of will be devoted to the development of the idea of a boundary representation that first appeared in the work of Poisson, Fourier, Cauchy and Dirichlet in the 19th century. In the modern form the classical Poisson formula states an isomorphism between the Banach spaces of bounded harmonic functions on the hyperbolic plane (i.e., on the interior of the Poincaré disk) and of bounded measurable functions on the boundary circle.

The contributions of Feller, Blackwell, Doob, Dynkin and Furstenberg in the 1950's - 1960's led to a crystallization of the notion of the Poisson boundary of a Markov chain as a measure space that describes the stochastically significant behaviour of its sample paths at infinity. In dynamical terms the Poisson boundary is the space of ergodic components of the time shift on the space of sample paths.

报告二

报告题目：In the Search of Completeness: the Identification Problem 边界等同问题的进一步探索 

报告时间：10:40 - 11:30

摘要：

In the second talk, I will discuss the attempts to describe the Poisson boundary in terms of the observed topological or combinatorial limit behaviour of sample paths, or — if no such behaviour is observed — to prove triviality of the Poisson boundary (i.e., the absence of non-constant bounded harmonic functions known as the Liouville property). This is a particular case of the central problem of ergodic theory: to describe the ergodic components of a given dynamical system. In what concerns the Poisson boundary, the key example is provided by random walks on a finitely generated free group. Its Cayley graph is a homogeneous tree endowed with a natural boundary. Does the convergence of sample paths to the natural boundary fully describe the Poisson boundary? For a large class of  reasonable random walks the answer yes follows from the entropy theory developed by Vershik and the author. However, a very recent breakthrough of Frisch and Chawla not only shows that the answer may be no for certain random walks on free groups, but also in a sense closes this question at all.

报告人简介

Dr.Vadim Kaimanovich is a professor at University of Ottawa, where he served as a Tier I Canada Research Chair. He received a Ph.D.degree in mathematics from St. Peterburg State University under Anatoly Vershik in 1983. Professor Kaimanovich is a world-leading mathematician in geometric group theory, ergodic theory and probability theory. His foundational contributions to the study of lamplighter groups have deeply shaped the modern research on random walks on discrete groups and their boundary theories. His various achievements include the celebrated Kaimanovich-Vershik entropy criterion, a fundamental tool for characterizing the triviality of Poisson boundaries in random walk dynamics.