C1

It is well known that if a group von Neumann algebra of a (nontrivial) discrete group is a factor, then it is a factor of type II_1. During the talk, I will answer the following question: which types appear as types of injective factors being group von Neumann algebras of discrete quantum groups (or looking from the dual perspective - von Neumann algebras of bounded functions on compact quantum groups)? An important object in our work is the subgroup of real numbers t for which the scaling automorphism tau_t is inner. This is joint work with Piotr Sołtan.

We are surrounded by systems that involve many elements and the interactions between them: the air we breathe, the galaxies we watch, herds of animals roaming the African planes and even us – trying to decide on whom to vote for.

As common as such systems are, their mathematical investigation is far from simple. Motivated by the realisation that in most cases we are not truly interested in the individual behaviour of each and every element of the system but in the average behaviour of the ensemble and its elements, a new approach emerged in the late 1950s - the so-called mean field limits approach. The idea behind this approach is fairly intuitive: most systems we encounter in real life have some underlying pattern – a correlation relation between its elements. Giving a mathematical interpretation to a given phenomenon and its emerging pattern with an appropriate master/Liouville equation, together with such correlation relation, and taking into account the large number of elements in the system results in a limit equation that describes the behaviour of an average limit element of the system. With such equation, one hopes, we could try and understand better the original ensemble.

In our talk we will give the background to the formation of the ideas governing the mean field limit approach and focus on one of the original models that motivated the birth of the field – Kac’s particle system. We intend to introduce Kac’s model and its associated (asymptotic) correlation relation, chaos, and explore attempts to infer information from it to its mean field limit – The Boltzmann-Kac equation.

 I will start this talk with a brief introduction and summary of the outcome of a joint work with James Gabe. An important special case of the main result is that for any countable discrete amenable group G, any two outer G-actions on stable Kirchberg algebras are cocycle conjugate precisely when they are equivariantly KK-equivalent. In the main body of the talk, I will outline the key arguments toward a special case of the 'uniqueness theorem', which is one of the fundamental ingredients in our theory: Suppose we have two G-actions on A and B such that B is a stable Kirchberg algebra and the action on B is outer and equivariantly O_2-absorbing. Then any two cocycle embeddings from A to B are approximately unitarily equivalent. If time permits, I will provide a (very rough) sketch of how this leads to the dynamical O_2-embedding theorem, which implies that such cocycle embeddings always exist in the first place.

In this talk, I will discuss the notion of quantum limits from different viewpoints: Cordes' work on the Gelfand theory for pseudo-differential operators dating from the 70’s as well as the micro-local defect measures and semi-classical measures of the 90’s. I will also explain my motivation and strategy to obtain similar notions in subRiemannian or subelliptic settings. 

It is well known that two non-isomorphic groups (groupoids) can produce isomorphic C*-algebras. That is, group (groupoid) C*-algebras are not rigid. This is not the case of the L^p-operator algebras associated to locally compact groups ( effective groupoids) where the isomorphic class of the group (groupoid) uniquely determines up to isometric isomorphism the associated L^p-algebras. Thus, L^p-operator algebras are rigid.  Liao and Yu introduced a class of Banach *-algebras associated to locally compact groups. We will see that this family of Banach *-algebras are also rigid.  

In this talk, I will present a noncommutative analogue of Margulis’ factor theorem for higher rank lattices. More precisely, I will give a complete description of all intermediate von Neumann subalgebras sitting between the von Neumann algebra of the lattice and the von Neumann algebra of the action of the lattice on the Furstenberg-Poisson boundary. As an application, we infer that the rank of the semisimple Lie group is an invariant of the pair of von Neumann algebras. I will explain the relevance of this result regarding Connes’ rigidity conjecture.

The classification by K-theory and traces of the category of simple, separable, nuclear, Z-stable C*-algebras satisfying the UCT is an extraordinary feat of mathematics. What's more, it provides powerful machinery for the analysis of the internal structure of these regular C*-algebras. In this talk, I will explain one such application of classification: In the subclass of classifiable C*-algebras consisting of those for which the simplex of tracial states is nonempty, with extremal boundary that is compact and has the structure of a connected topological manifold, automorphisms can be shown to be generically tracially chaotic. Using similar ideas, I will also show how certain stably projectionless C*-algebras can be described as crossed products.

Equivariant Jiang-Su stability is an important regularity property for group actions on C*-algebras.  In this talk, I will explain this property and how it arises naturally in the context of the classification of C*-algebras and their actions. Depending on the time, I will then explain a bit more about the nature of equivariant Jiang- Su stability and the kind of techniques that are used to study it, including a recent result of Gábor Szabó and myself establishing an equivalence with equivariant property Gamma under certain conditions.
 

I'll introduce the classical arithmetic topology dictionary of Mumford-Manin-Mazur-Morishita-etc. I'll then present an interesting instance of parallel phenomena related to symplectic structures on moduli spaces of certain bundles. The arithmetic side turns out to be an application of Poitou-Tate duality. Depending on time, I'll delve into the delicate details which make the analogy useful for Diophantine geometers.