Abstract: I will begin the functional analysis seminar series with the first talk. First will be an organizational component wherein I will explain the details of the seminar, describe up a rough plan of the semester's invited speakers and their research directions. Then I will demonstrate the notion of mixed identity freeness in certain categories, and explain applications to problems in C*-algebras, and continuous model theory.
Abstract: Connes embeddability of a group is a finite dimensional approximation property. Turns out this property depends only on the so-called group von Neumann algebra. The property can be extended to all von Neumann algebras. The fact that there is a von Neumann algebra without this property was proved in 2020 using a quantum complexity result MIP*=RE. It is still open for groups. I will discuss the best-known partial result, which is that there is a group action without this property. In particular, this implies the negation to the Aldous-Lyons conjecture, a big problem in probability theory.
Abstract: For a second countable locally compact group, Kazhdan’s property (T) is equivalent to the existence of a global fixed point for every continuous action by affine isometries on a Hilbert space. Equivalently, any coycle (i.e., the translation part of such an action) is bounded. Of course, such a property behaves well with (continuous) morphisms between groups. However, it does not provide tools for studying quasi-homomorphisms, that is, maps which only approximately respect the group structure. In order to study this class of maps, Ozawa introduced wq-cocycles, which are cocycles up to a bounded error. A group is said to have property (TTT) if all wq-cocycles are bounded. In this talk, I will discuss the relationship between this property and other forms of almost property (T). I will also explain how to prove that a group possesses this property, with an emphasis on semisimple groups and their lattices.
Abstract: A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure complexity of Borel equivalence relations. Recently, a lot of research has been done on hyperfiniteness of the orbit equivalence relation on the Gromov boundary induced by various group actions on hyperbolic spaces. In this talk, I will explain my attempt to explore this connection of Borel complexity and geometric group theory for another intensively studied geometric object, which is CAT(0) cube complexes. More precisely, we prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite.
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