University of Cardiff

Often, one tries to understand the behaviour of non-commutative random variables or of von Neumann algebras through matricial approximations. In some cases, such as when appealing to the determinant conjecture or investigating the soficity of a group, it is important to find approximations by matrices with good algebraic conditions on their entries (e.g., being integers). On the other hand, the most common tool for generating asymptotic independence -- conjugating with random unitaries -- often destroys such delicate structure.

 I will speak on recent joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson, where we investigate graph products (an interpolation between free and tensor products) and conjugation of matrix models by large structured random permutations. We show that with careful control of how the permutation matrices are chosen, we can achieve asymptotic graph independence with amalgamation over the diagonal matrices. We are able to use this fine structure to prove that strong $1$-boundedness for a large class of graph product von Neumann algebras follows from the vanishing of the corresponding first $L^2$-Betti number. The main idea here is to show that a version of the determinant conjecture holds as long as the individual algebras have generators with approximations by matrices with entries in the ring of integers of some finite extension of Q satisfying some conditions strongly reminiscent of soficity for groups.

The study of topologically ordered quantum phases has led to interesting connections with, for example, the study of subfactors. In this talk, I will introduce a new axiomatisation of such quantum models defined on d-dimensional square lattices in terms of nets of projections. These local topological order axioms are satisfied by known 2D models such as the toric code and Levin-Wen models built on a unitary fusion category. We show that these axioms lead to a definition of boundary algebras naturally living on a hyperplane. This boundary algebra encodes information about the excitations in the bulk theory, leading to a bulk-boundary correspondence. I will outline the main points, with an emphasis on interesting connections to operator algebras and fusion categories. Based on joint work with C. Jones, Penneys, and Wallick (arXiv:2307.12552).

The formation of surface meltwater has been linked with the disintegration of many ice shelves in the Antarctic Peninsula over the last several decades. Despite the importance of surface meltwater production and transport to ice shelf stability, knowledge of these processes is still lacking. Understanding the surface hydrology of ice shelves is an essential first step to reliably project future sea level rise from ice-sheet melt.

In order to better understand the processes driving meltwater distribution on ice shelves, we present the first comprehensive model of surface hydrology to be developed for Antarctic ice shelves, enabling us to incorporate key processes such as the lateral transport of surface meltwater. Recent observations suggest that surface hydrology processes on ice shelves are more complex than previously thought, and that processes such as lateral routing of meltwater across ice shelves, ice shelf flexure and surface debris all play a role in the location and influence of meltwater. Our model allows us to account for these and is calibrated and validated through both remote sensing and field observations.

Twisted K-theory is an enrichment of topological K-theory that allows local coefficient systems called twists. For spaces and twists equipped with an action by a group, equivariant twisted K-theory provides an even finer invariant. Equivariant twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the corresponding K-groups to the Verlinde ring of the associated loop group. From the point of view of homotopy theory only a small subgroup of all possible twists is considered in classical treatments. In this talk I will discuss a construction that is joint work with David Evans and produces interesting examples of non-classical twists over the Lie groups SU(n) and over tori constructed from exponential functors. They arise naturally as Fell bundles and are equivariant with respect to the conjugation action of the group on itself. For the determinant functor our construction reproduces the basic gerbe over SU(n) used by Freed, Hopkins and Teleman.

Unitary solutions of the Yang-Baxter equation ("R-matrices") play a prominent role in several fields, such as quantum field theory and topological quantum computing, but are difficult to find directly and remain somewhat mysterious. In this talk I want to explain how one can use subfactor techniques to learn something about unitary R-matrices, and a research programme aiming at the classification of unitary R-matrices up to a natural equivalence relation. This talk is based on joint work with Roberto Conti, Ulrich Pennig, and Simon Wood.

A bundle of C*-algebras is a collection of algebras continuously parametrised by a topological space. There are (at least) two different definitions in operator algebras that make this intuition precise: Continuous C(X)-algebras provide a flexible analytic point of view, while locally trivial C*-algebra bundles allow a classification via homotopy theory. The section algebra of a bundle in the topological sense is a C(X)-algebra, but the converse is not true. In this talk I will compare these two notions using the classical work of Dixmier and Douady on bundles with fibres isomorphic to the compacts  as a guideline. I will then explain joint work with Marius Dadarlat, in which we showed that the theorems of Dixmier and Douady can be generalized to bundles with fibers isomorphic to stabilized strongly self-absorbing C*-algebras. An important feature of the theory is the appearance of higher analogues of the Dixmier-Douady class.