C3

In this talk from Alexander Ravnanger, he provides a dynamical description of the largest AF-ideal in certain crossed products by the integers. In the case of the uniform Roe algebra of the integers, this reveals an interesting connection to a well-studied object in topological semigroup theory. On the way, he gives an overview of what is known about the abundance of projections in such crossed products, the structure of the simple quotients, and concepts of low-dimensionality for uniform Roe algebras.

Graph products of groups were introduced by Green as a construction that encompasses both direct products and free products. Likewise, the notion of graph product of von Neumann algebras, introduced by Caspers and Fima, recovers both tensor products and free products. Camille Horbez will present rigidity theorems for graph products of tracial von Neumann algebras, and discuss the computation of their symmetries, drawing parallels with the case of groups. This is a joint work with Adrian Ioana. 

AF-embeddability, i.e., the question whether a given C*-algebra can be realised as a subalgebra of an AF-algebra, has been studied for a long time with prominent early results by Pimsner and Voicuescu who constructed such embeddings for irrational rotation algebras in 1980. Since then, many AF-embeddings have been constructed for concrete examples but also many non-constructive AF-embeddability results have been obtained for classes of algebras typically assuming the UCT. 

In this talk by Joachim Zacharias, we will consider a separable unital C*-algebra A of decomposition rank at most 1 and construct from a suitable system of 1-decomposable cpc-approximations an AF-algebra E together with an embedding of A into E and a conditional expectation of E onto A without assuming the UCT. We also consider some extensions of this inclusion and indicate some applications.

Convex geometry has long been influenced by the study of dualities and extremal inequalities, with origins in classical affine geometry and functional analysis. In this talk, Kasia Wyczesany will explore an abstract concept of duality, focusing on the classical idea of the polar set, which captures the duality of finite-dimensional normed spaces. This notion leads to fundamental questions about volume products, inspiring some of the most famous inequalities in the field. Whilst Mahler’s influential 1939 conjecture regarding the minimiser of the volume product will be mentioned, the emphasis will be on the Blaschke–Santaló inequality, which identifies the maximiser, along with its modern extensions. Main new results are joint work with S. Artstein-Avidan and S. Sadovsky, and S. Artstein-Avidan and M. Fradelizi. 

We introduce equivariant bivariant K-theory for bornological algebras by taking a presentable refinement of the bivariant K-theory of Lafforgue and Paravicini. An upshot of this refinement is that we may purely formally define a Bost-Connes assembly map via localisation in the sense of Meyer-Nest. Another feature built into the refinement is a large UCT-class; on this UCT-class, we show that the rationalised Chern-Connes character from KK-theory to local cyclic homology is an equivalence. This is joint work with Anupam Datta.

to follow

In 1962, Horn raised the following problem: Let A and B be n-by-n Hermitian matrices with respective eigenvalues a_1,...,a_n and b_1,...,b_n. What can we say about the possible eigenvalues c_1,...,c_n of A + B?

The deterministic perspective is that the set of possible values for c_1,...,c_n are described by a collection of inequalities known as the Horn inequalities.

Free probability offers the following alternative perspective on the problem: if (A_n) and (B_n) are independent sequences of n-by-n random matrices with empirical spectra converging to probability measures mu and nu respectively, then the random empirical spectrum of A_n + B_n converges to the free convolution of mu and nu.

But how are these two perspectives related?

In this talk Samuel Johnston will discuss approaches to free probability that bridge between the two perspectives. More broadly, Samuel will discuss how the fundamental operations of free probability (such as free convolution and free compression) arise out of statistical physics mechanics of corresponding finite representation theory objects (hives, Gelfand-Tsetlin patterns, characteristic polynomials, Horn inequalities, permutations etc).

This talk is based on joint work with Octavio Arizmendi (CIMAT, Mexico), Colin McSWiggen (Academia Sinica, Taiwan) and Joscha Prochno (Passau, Germany).

This talk by Tim Austin, at the University of Warwick, will be an introduction to "almost periodic entropy".  This quantity is defined for positive definite functions on a countable group, or more generally for positive functionals on a separable C*-algebra.  It is an analog of Lewis Bowen's "sofic entropy" from ergodic theory.  This analogy extends to many of its properties, but some important differences also emerge.  Tim will not assume any prior knowledge about sofic entropy.

After setting up the basic definition, Tim will focus on the special case of finitely generated free groups, about which the most is known.  For free groups, results include a large deviations principle in a fairly strong topology for uniformly random representations.  This, in turn, offers a new proof of the Collins—Male theorem on strong convergence of independent tuples of random unitary matrices, and a large deviations principle for operator norms to accompany that theorem.

In this talk, Konstantin Recke, University of Oxford,  will report on some results pertaining to the interplay between geometric group theory, operator algebras and probability theory. Konstantin will introduce so-called invariant percolation models from probability theory and discuss their relation to geometric and analytic properties of groups such as amenability, the Haagerup property (a-T-menability), $L^p$-compression and Kazhdan's property (T). Based on joint work with Chiranjib Mukherjee (Münster).