In this lecture, I focus on ergodic theory of group actions. We consider the translation action of a discrete group G on the product space {0,1}^G equipped with the product of probability measures \mu_g on {0,1}. When all \mu_g are equal, these are the classical Bernoulli actions, which are probability measure preserving. When the \mu_g are distinct, non measure preserving actions of Krieger type III may appear. I will explain an intricate connection to L^2-cohomology. In particular, I will show that a group G admits a Bernoulli action of type III_1 if and only if G has nonzero first L^2-cohomology. I will also explain why the group of integers does not admit a Bernoulli action of type II_\infty and why type III_\lambda only arises when G has more than one end. This is joint work with J. Wahl, and with M. Björklund and Z. Kosloff.

Stefaan Vaes教授是比利时鲁汶大学的教授兼分析系主任。他的研究重点是算子代数及其与群理论和遍历理论的联系。2010年，他受邀在国际数学家大会和2016年欧洲数学大会上发表演讲。2015年，他获得了法朗基奖，这是比利时科学界的最高荣誉。2017年春天，他成为剑桥牛顿研究所的罗斯柴尔德研究员。Stefaan Vaes是比利时皇家科学院(KVAB)的成员之一，同时也是《泛函分析》的主编之一。

课件：beijing-chern-lecture-4