In 2008, Toms constructed a counterexample to the Elliott conjecture: a pair of simple, separable, nuclear and unital C*-algebras which are indistinguishable by the Elliott invariant, but are not isomorphic. The key to distinguishing this pair of carefully crafted C*-algebras lies with a rather refined invariant called the Cuntz semigroup. Consequently, Toms’s counterexample highlighted the importance of the Cuntz semigroup to the classification of C*-algebras.

In this talk, we will discuss the Cuntz semigroup in the context of graph C*-algebras, a highly diverse class of mostly non-simple C*-algebras. In particular, we will accentuate how the highly organised structure of a unital graph C*-algebra is reflected in its Cuntz semigroup and if enough time permits, mention properties of unital graph C*-algebras that are revealed by these Cuntz semigroups.

Much recent research has gone into understanding the first order theory of II1 factors. Very recently, Peterson released a preprint which develops deformation rigidity in the ultrapower setting. His techniques give many explicit examples of non-isomorphic ultrapowers for natural families of II1 factors. In this talk, I will introduce some of Peterson's techniques and results, including an analogue of amenability in the ultrapower setting and the interplay between property T and malleable deformations.

One of the problems posed by Kadison in 1967 asks whether every separably acting von Neumann algebra is generated by a single element. The problem remains open in its full generality but significant progress has been made since. One can of course ask the same question in the C*-algebraic setting where, however, counterexamples are abundant even among commutative C*-algebras. I will give an overview of the history of the problem and then discuss some recent results on single generation of C*-algebras associated to graphs and C*-algebras with Cartan subalgebras.

We consider the question of whether almost-homomorphisms are close to honest homomorphisms. I’ll survey a few historical results, with different source/target collections of algebras, and also consider what to take as the definition of “almost-homomorphisms”. If we end up having time, I will sketch an elementary proof that almost-characters from commutative C*-algebras are close to honest characters.

Descriptive set theory provides a useful framework for studying the complexity of classification problems in operator algebras. In this talk I will discuss how C*-algebras can be encoded as points in a Borel space, and introduce several equivalent parametrizations, including a new one in terms of ideals of a universal C*-algebra. I will then discuss examples of natural classes of C*-algebras that form Borel sets, as well as a parametrization of *-homomorphisms and recent results on the Borelness of certain functors. Time permitting, I will introduce KK-theory and the Kasparov product, and explain a new result showing that the Kasparov product is Borel in a certain appropriate parametrized setting.

We provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider on the discretized circle the rate equation for the process where pairs of particles sitting on the same side jump simultaneously to the two neighboring sites, and the reverse jump where a pair of particles sitting on a common site jump simultaneously to the side in the middle. Depending on the rates, in the vanishing mesh size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. We identify the limiting gradient structure to be driven by entropy with respect to a generalization of the diffusive transport type with nonlinear mobility via EDP convergence. Furthermore, the DLSS equation with nonlinear mobility of the power type shares qualitative similarities with the fast diffusion and porous medium equations, since we find traveling wave solutions with algebraic tails and polynomial compact support, respectively.    
       

Joint work with Alexander Mielke and Artur Stephan arXiv:2510.07149. The DLSS part is based on joints works with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré.