Group theory meets Hilbert spaces

10.07.2024, 14:00 - 16:00  –  Campus Golm, Building 9, Room 2.22 and via Zoom
Institutskolloquium

Piotr Nowak (Warsaw), Adam Skalski (Warsaw)

14:00  Adam Skalski (Academy of Science, Warsaw):  Haagerup property: groups and operator algebras.
14:45  Tea and Coffee Break
15:15  Piotr Nowak (Academy of Science, Warsaw):  Rigidity results for groups, sums of squares and applications.

Adam Skalski (Academy of Science, Warsaw):  Haagerup property: groups and operator algebras.

Abstract:     Amenability of a discrete group is a concept due to John von Neumann, describing the existence of `nice' averages, and prohibiting the existence of paradoxical decompositions. The Haagerup property is a certain analytic softening of the notion of amenability, discovered in 1977 by Uffe Haagerup. We will discuss the history and importance of this notion and its analogues, presenting various applications and relations with operator algebras and noncommutative geometry.
     Finally we will mention recent results regarding concrete examples of maximal Haagerup subgroups and subalgebras, obtained in joint work with Yongle Jiang.

Piotr Nowak (Academy of Science, Warsaw):  Rigidity results for groups, sums of squares and applications.

Abstract:     Vanishing theorems for group cohomology with unitary coefficients represent a form of rigidity of the group. For instance Kazhdan's property (T) which roughly says that if the group acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero invariant vector, is characterized by such vanishing for all unitary representation of the first cohomology group. I will give a brief introduction to group cohomology and discuss how vanishing of cohomology with unitary coefficients can be described in terms of sums of squares in certain matrix rings. These methods can be used to obtain vanishing of cohomology for particular groups, and I will explain how they led to proving property (T) for Aut(F_n), the automorphism group of the free group on n generators, for n at least 5.
    

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