C3

The landmark completion of the Elliott classification program for unital separable simple nuclear C*-algebras saw three regularity properties rise to prominence: Z-stability, a C*-algebraic analogue of von Neumann algebras' McDuffness; finite nuclear dimension, an operator algebraic version of having finite Lebesgue dimension; and strict comparison, a generalization of tracial comparison in II_1 factors. Given their relevance to classification, most of the investigations into their interplay have focused on the simple nuclear case.

 The purpose of this talk is to advertise the general study of these properties and discuss their applications both within and outside operator algebras. Concretely, I will explain how characterizing when certain twisted group C*-algebras are Z-stable can provide new partial solutions to a well-known problem in generalized time-frequency analysis; this is joint work with U. Enstad. If time allows, I will also briefly discuss how a different incarnation of tracial comparison (finite radius of comparison) for non-commutative tori relates to the existence of smooth Gabor frames; this last part is joint work with U. Enstad and also H. Thiel.

Guentner, Willet and Yu defined a notion of dynamic asymptotic dimension for an étale groupoid that can be used to bound the nuclear dimension of its groupoid C*-algebra.  To have finite dynamic asymptotic dimension, the isotropy subgroups of the groupoid must be locally finite.  I will discuss 1) how to use similar ideas to bound the nuclear dimension of the C*-algebra of a groupoid with `large' isotropy subgroups and 2) the limitations of that approach. In an application to the C*-algebra of a directed graph,  if the C*-algebra is stably finite, then its nuclear dimension is at most 1.  This is joint work with Dana Williams. 

Kasparov's bivariant K-theory (or KK-theory) is an extremely powerful invariant for both C*-algebras and C*-dynamical systems, which was originally motivated for a tool to solve classical problems coming from topology and geometry. Its paramount importance for classification theory was discovered soon after, impressively demonstrated within the Kirchberg-Phillips theorem to classify simple nuclear and purely infinite C*-algebras. Since then, it can be said that every methodological novelty about extracting information from KK-theory brought along some new breakthrough in classification theory. Perhaps the most important example of this is the Lin-Dadarlat-Eilers stable uniqueness theorem, which forms the technical basis behind many of the most important articles written over the past decade. In the recent landmark paper of Carrion et al, it was demonstrated how the stable uniqueness theorem can be upgraded to a uniqueness theorem of sorts under extra assumptions. It was then posed as an open problem whether the statement of a desired "KK-uniqueness theorem" always holds.

In this talk I want to present the affirmative answer to this question: If A and B are separable C*-algebras and (f,g) is a Cuntz pair of absorbing representations whose induced class in KK(A,B) vanishes, then f and g are strongly asymptotically unitarily equivalent. The talk shall focus on the main conceptual ideas towards this theorem, and I plan to discuss variants of the theorem if time permits. It turns out that the analogous KK-uniqueness theorem is true in a much more general context, which covers equivariant and/or ideal-related and/or nuclear KK-theory.

Two-dimensional chiral conformal field theories (CFTs) admit three distinct mathematical formulations: vertex operator algebras (VOAs), conformal nets, and Segal (functorial) chiral CFTs. With the broader aim to build fully extended Segal chiral CFTs, we start with the input of a conformal net. 

In this talk, we focus on presenting three equivalent constructions of the category of solitons, i.e. the category of solitonic representations of the net, which we propose is what theory (chiral CFT) assigns to a point. Solitonic representations of the net are one of the primary class of examples of bicommutant categories (a categorified analogue of a von Neumann algebras). The Drinfel’d centre of solitonic representations is the representation category of the conformal net which has been studied before, particularly in the context of rational CFTs (finite-index nets). If time permits, we will briefly outline ongoing work on bicommutant category modules (which are the structures assigned by the Segal Chiral CFT at the level of 1-manifolds), hinting towards a categorified analogue of Connes fusion of von Neumann algebra modules.

(Bicommutant categories act on W*-categories analogous to von Neumann algebras acting on Hilbert spaces)

In subfactor theory, it has been observed that operator algebras often admit symmetries beyond mere groups, sometimes called quantum symmetries. Besides recent substantial progress on the classification programs of simple amenable C*-algebras and group actions on them, there has been increasing interest in their quantum symmetries. This talk is devoted to an attempt to ensure the existence of various quantum symmetries on simple amenable C*-algebras, at least in the purely infinite case, by providing a systematic way to produce them. As a technical ingredient, a simplicity criterion for certain Pimsner algebras is given.

Networks provide a powerful language to model and analyse interconnected systems. Their building blocks are  edges, which can  then be combined to form walks and paths, and thus define indirect relations between distant nodes and model flows across the system. In a traditional setting, network models are first-order, in the sense that flow across nodes is made of independent sequences of transitions. However, real-world systems often exhibit higher-order dependencies, requiring more sophisticated models. Here, we propose a variable-order network model that captures memory effects by interpolating between first- and second-order representations. Our method identifies latent modes that explain second-order behaviors, avoiding overfitting through a Bayesian prior. We introduce an interpretable measure to balance model size and description quality, allowing for efficient, scalable processing of large sequence data. We demonstrate that our model captures key memory effects with minimal state nodes, providing new insights beyond traditional first-order models and avoiding the computational costs of existing higher-order models.

A classical result of Dixmier and Douady enables us to classify locally trivial bundles of C*-algebras with compact operators as fibres via methods in homotopy theory. Dadarlat and Pennig have shown that this generalises to the much larger family of bundles of stabilised strongly self-absorbing C*-algebras, which are classified by the first group of the cohomology theory associated to the units of complex topological K-theory. Building on work of Evans and Pennig I consider Z/pZ-equivariant C*-algebra bundles over Z/pZ-spaces. The fibres of these bundles are infinite tensor products of the endomorphism algebra of a Z/pZ-representation. In joint work with Pennig, we show that the theory refines completely to this equivariant setting. In particular, we prove a full classification of the C*-algebra bundles via equivariant stable homotopy theory.

A successful strategy to handle problems involving primes is to approximate them by a more 'simple' function. Two aspects need to be balanced. On the one hand, the approximant should be simple enough so that the considered problem can be solved for it. On the other hand, it needs to be close enough to the primes in order to make it an admissible to replacement. In this talk I will present how one can construct general approximants in the context of the Circle Method and will use this to give a different perspective on Goldbach type applications.

Self-similar groups are groups of automorphisms of infinite rooted trees obeying a simple but powerful rule. Under this rule, groups with exotic properties can be generated from very basic starting data, most famously the Grigorchuk group which was the first example of a group with intermediate growth.

Nekrashevych introduced a groupoid and a C*-algebra for a self-similar group action on a tree as models for some underlying noncommutative space for the system. Our goal is to compute the K-theory of the C*-algebra and the homology of the groupoid. Our main theorem provides long exact sequences which reduce the problems to group theory. I will demonstrate how to apply this theorem to fully compute homology and K-theory through the example of the Grigorchuk group.

This is joint work with Benjamin Steinberg.

Given two von Neumann algebras A,B with an action by a locally compact (quantum) group G, one can consider its associated equivariant correspondences, which are usual A-B-correspondences (in the sense of Connes) with a compatible unitary G-representation. We show how the category of such equivariant A-B-correspondences carries an analogue of the Fell topology, which is preserved under natural operations (such as crossed products or equivariant Morita equivalence). If time permits, we will discuss one particular interesting example of such a category of equivariant correspondences, which quantizes the representation category of SL(2,R). This is based on joint works with Joeri De Ro and Joel Dzokou Talla.