We will introduce the Baum-Connes conjecture without coefficients, in the setting of discrete groups, and try to explain why it is interesting for operator algebraists. We will give some idea of the LHS and the RHS of the conjecture, without being too formal, and rather than trying to define the assembly map, we will explain what it does for finite groups, for the integers, for free groups, and finally for wreath products of a finite group with the integers (the latter result is joint work with R. Flores and S. Pooya; it raises a few open questions about classifying the corresponding group C*-algebras up to isomorphism).

The Cuntz semigroup is a geometric refinement of K-theory that plays an important role in the structure theory of C*-algebras. It is defined analogously to the Murray-von Neumann semigroup by using equivalence classes of positive elements instead of projections.
Starting with the definition of the Cuntz semigroup of a C*-algebra, we will look at some of its classical applications. I will then talk about the recent breakthroughs in the structure theory of Cuntz semigroups and some of the consequences.

After recalling some motivation for studying highly-connected graphs in the context of operator algebras and large-scale geometry, we will introduce the notion of "asymptotic expansion" recently defined by Li, Nowak, Spakula and Zhang. We will explore some applications of this definition, hopefully culminating in joint work with Li, Vigolo and Zhang.

In 1997 Pimsner described how to construct two universal C*-algebras associated with an injective C*-correspondence, now known as the Toeplitz--Pimsner and Cuntz--Pimsner algebras. In this talk I will recall their construction, focusing for simplicity on the case of a finitely generated projective correspondence. I will describe the associated six-term exact sequence in K(K)-theory and explain how these can be used in practice for computational purposes. Finally, I will describe how, in the case of a self-Morita equivalence, these exact sequences can be interpreted as an operator algebraic version of the classical Gysin sequence for circle bundles.

In this talk, I would like to advertise the strict equality between two objects from very different areas of mathematical physics:

- Kahane's Gaussian Multiplicative Chaos (GMC), which uses a log-correlated field as input and plays an important role in certain conformal field theories.

- A reference model in random matrices called the Circular Beta Ensemble (CBE).

The goal is to give a precise theorem whose loose form is GMC = CBE. Although it was known that random matrices exhibit log-correlated features, such an exact correspondence is quite a surprise. 

Ambitwistor strings are a class of holomorphic worldsheet models that directly describe massless quantum field theories, such as supergravity and super Yang-Mills. Their correlators give remarkably compact amplitude representations, known as the CHY formulas: characteristic worldsheet integrals that are fully localized on a set of polynomial constraints known as the scattering equations. Moreover, the ambitwistor string models provide a natural way of extending these formulas to loop level, where the constraints can be used to simplify the formulas (originally on higher genus curves) to 'forward limit-like' constructions on nodal spheres. After reviewing these developments, I will discuss one of the peculiar features of this approach: the worldsheet formulas on nodal spheres result in a non-standard integrand representation that makes it difficult to e.g. apply established integration techniques. While several approaches for addressing this look feasible or have been put forward in the literature, they only work for the simplest toy models. Taking inspiration from these attempts, I want to discuss a novel strategy to overcome this difficulty, and formulate compact worldsheet formulas with standard Feynman propagators.