华罗庚青年数学论坛学术报告

中科院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

华罗庚青年数学论坛

学术报告

报告人： 王斯萌 博士（哈工大数学研究院）
题  目：Partitions, quantum group actions and rigidity
时  间：2022.03.30（星期三），09:30-11:30
地  点：腾讯会议：722-635-763
摘  要：In this talk, I will present a new combinatorial approach to the study of ergodic actions of compact quantum groups. The connection between compact quantum groups and the combinatorics of partitions goes back to Banica's founding work on the representation theory of free orthogonal quantum groups, and was later formalized in the seminal paper of Banica and Speicher under the theory of "easy quantum groups". Based on some new alternative version of the Tannaka-Krein reconstruction procedure for ergodic actions, we extend Banica and Speicher's combinatorial approach to the setting of ergodic actions of compact quantum groups. Our examples in particular recovers actions on finite spaces, on embedded homogeneous spaces and on quotient spaces. Moreover, we use this categorical point of view to study the quantum rigidity of ergodic actions on classical spaces, and show that the free quantum groups cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang.  The talk is based on the recent preprint arXiv:2112.07506 jointly with Amaury Freslon and Frank Taipe.

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报告人： 王斯萌 博士（哈工大数学研究院）
题  目：Noncommutative individual ergodic theorem for amenable groups
时  间：2022.04.01（星期五），09:30-11:30
地  点：腾讯会议：794-210-220
摘  要：Birkhoff’s celebrated individual ergodic theorem asserts that for a measure-preserving ergodic transformation on a measure space, the time average is equal to the space average almost everywhere. Since the theory of von Neumann algebras is a quantum analogue of the classical measure theory, it is natural to study similar individual ergodic theorems in the setting of von Neumann algebras. The study was exactly initiated by Lance in 1970s, and witnessed fruitful progress in recent decades with the help of modern tools from the operator space theory, such as the noncommutative vector-valued Lp-spaces studied by Pisier, Junge and Xu. This talk aims to give a gentle introduction to the aforementioned topic, and present some recent results on ergodic theorems for actions on von Neumann algebras by amenable groups. In particular, we established a quantum analogue of Lindenstrauss’s pointwise ergodic theorem. Our methods rely essentially on geometric constructions of martingales based on the Ornstein-Weiss quasi-tilings and harmonic analytic estimates coming from noncommutative Calderón-Zygmund theory. The talk is based on joint papers with Guixiang Hong, Ben Liao and Léonard Cadilhac.