Mini Course: Topological Phenomena in the Cuntz semigroup 

Mathematical Institute, University of Oxford

10-12 Sept 2025

This short mini course aims to introduce participants to the interplay between algebraic and differential topology and the  Cuntz semigroup of C*-algebras. It will describe the use of the Cuntz semigroup to build C*-algebras outside the scope of the Elliott classification programme.  There will be opportunities for participants to offer contributed talks.

It is well known to mathematicians that there is a deep relationship between the arithmetic of algebraic varieties and their geometry.  

In recent years, there has been increasing evidence for a geometric representation of quantum chaos within Einstein's theory of general relativity. Despite the lack of a complete theoretical framework, this overview will explore various examples of this phenomenon. It will also discuss the lessons we have learned from it to address several existing puzzles in quantum gravity, such as the black hole information paradox and off-shell wormhole geometries.

Symmetry Topological Field Theories (SymTFTs) are topological field theories that encode the symmetry structure of global symmetries in terms of a theory in one higher dimension. While SymTFTs for internal (global) symmetries have been highly successful in characterizing symmetry aspects in the last few years, a corresponding framework for spacetime symmetries remains unexplored. We propose an extension of the SymTFT framework to include spacetime symmetries. In particular, we propose a SymTFT for the conformal symmetry in various spacetime dimensions. We demonstrate that certain BF-type theories, closely related to topological gravity theories, possess the correct topological operator content and boundary conditions to realize the conformal algebra of conformal field theories living on boundaries. As an application, we show how effective theories with spontaneously broken conformal symmetry can be derived from the SymTFT, and we elucidate how conformal anomalies can be reproduced in the presence of even-dimensional boundaries.
 

I will discuss a uniform waist inequality in codimension 2 for the family of finite covers of a Riemannian manifold whose fundamental group has Kazhdan‘s property T. I will describe a general strategy to prove waist inequalities based on a higher property T for Banach spaces. The general strategy can be implemented in codimension 2 but is conjectural in higher codimension. We speculate about the situation for lattices in semisimple Lie groups. Based on joint work with Uri Bader

The spectral gap (or bass note) of a closed hyperbolic surface is the smallest non-zero eigenvalue of its Laplacian. This invariant plays an important role in many parts of hyperbolic geometry. In this talk, I will speak about joint work with Will Hide on the question of which numbers can appear as spectral gaps of closed arithmetic hyperbolic surfaces.

I will relate two notorious open questions in low-dimensional topology.  The first asks whether every hyperbolic group is residually finite. The second, the congruence subgroup property, relates the finite-index subgroups of mapping class groups to the topology of the underlying surface. I will explain why, if every hyperbolic group is residually finite, then mapping class groups enjoy the congruence subgroup property. Time permitting, I may give some further applications to the question of whether hyperbolic 3-manifolds are determined by the finite quotients of their fundamental groups.

In the 1990s, Goodwillie developed a theory of calculus for homotopical functors. His idea was to approximate a functor by a tower of ‘polynomial functors’, similar to how one approximates a function by its Taylor series. The role of linear polynomials is played by functors that behave like homology theories, in the sense that there is a Mayer-Vietoris sequence computing their homotopy groups. As such, the Goodwillie tower interpolates between stable and unstable homotopy theory. The theory has application to the computation of the homotopy groups of spheres, higher algebra, and algebraic K-theory. In my talk, I will give an introduction to this topic. In particular, I will explain that Goodwillie's calculus reveals a deep connection between the homotopy theory of spaces and Lie algebras and how this is related to a chain rule for the derivatives of functors.