University of Glasgow

TBA

Quantized universal enveloping algebras admit an intriguing class of (unbounded) Hilbert space representations obtained via their cluster structure. In these so-called positive representations the standard generators act by (essentially self-adjoint) positive operators. 

The aim of this talk is to discuss some analytical questions arising in this context, and in particular to what extent these representations can be understood using the theory of locally compact quantum groups in the sense of Kustermans and Vaes. I will focus on the simplest case in rank 1, where many of the key features (and difficulties) are already visible. (Based on work in progress with Kenny De Commer, Gus Schrader and Alexander Shapiro). 

If $E/\mathbb{Q}$ is an elliptic curve, and $F/\mathbb{Q}$ is a finite Galois extension, then $E(F)$ is not merely a finitely generated abelian group, but also a Galois module. If we fix a finite group $G$, and let $F$ vary over all $G$-extensions, then what can we say about the statistical behaviour of $E(F)$ as a $\mathbb{Z}[G]$-module? In this talk I will report on joint work with Adam Morgan, in which we investigate the simplest non-trivial special case of this very general question. Our work has surprising connections to questions about frequency of failure of the Hasse principle for genus 1 hyperelliptic curves, and to work of Heath-Brown on 2-Selmer group distributions in quadratic twist families.

Nakajima quiver varieties form an important class of examples of conical symplectic singularities. For example, such varieties of dimension 2 are Kleinian singularities. Starting from this, I will describe a combinatorial approach to classifying the next case, affine quiver varieties of dimension 4. If time permits, I will try to say the implications we obtained and how can one compute the number of crepant symplectic resolutions of these varieties. This is a joint project with Samuel Lewis.

For any symplectic singularity, one can consider the minimal degenerations between symplectic leaves - these are the relative singularities of a pair of adjacent leaves in the closure relation. I will describe a complete classification of these minimal degenerations for Nakajima quiver varieties. It provides an effective algorithm for computing the associated Hesse diagrams. In the physics literature, it is known that this Hasse diagram can be computed using quiver subtraction. Our results appear to recover this process. I will explain applications of our results to the question of normality of leaf closures in quiver varieties. The talk is based on joint work in progress with Travis Schedler.

A C*-correspondence between two C*-algebras is a generalization of a *-homomorphism. Laca and Neshveyev showed that, like a *-homomorphism, there is an induced map between traces of the algebras. Given sufficient regularity conditions, the map defines a bounded operator between the spaces of (bounded) tracial linear functionals. 

This operator can be of independent interest - a special case of correspondence gives Ruelle's operator associated to a non-invertible discrete-time dynamical system, and the study of Ruelle's operator is the basis of his thermodynamic formalism. Moreover, by the work of Laca and Neshveyev, the operator's positive eigenvectors determine the KMS states of the gauge action on the Cuntz-Pimsner algebra of the correspondence.

Given a C*-correspondence from a C*-algebra to itself, we will present a sufficient condition on the C*-correspondence that implies the operator on traces has a unique positive eigenvector, and moreover a spectral gap. This result recovers the Perron-Frobenius theorem, aspects of Ruelle's thermodynamic formalism, and unique KMS state results for a variety of constructions of Cuntz-Pimsner algebras, including the C*-algebras associated to self-similar groupoids. The talk is based on work in progress.

Associated to every shift space, the Cuntz-Krieger algebra (C-K algebra for abbreviation) is an invariant of conjugacy defined and developed by K. Matsumoto, S. Eilers, T. Carlsen, and many of their collaborators in the last decade. In particular, Carlsen defined the C-K algebra to be the full groupoid C*-algebra of the “cover”, which is a topological system consisting of a surjective local homeomorphism on a zero-dimensional space induced by the shift space. 

In 2022, K. Brix proved that the C-K algebra of the Sturmian shift has finite nuclear dimension, where the Sturmian shift is the (unique) minimal shift space with the smallest complexity function: p_X(n)=n+1. In recent results (joint with Z. He), we show that for any minimal shift space with finitely many left special elements, its C-K algebra always have finite nuclear dimension. In fact, this can be further applied to the class of aperiodic shift spaces with non-superlinear growth complexity. 

Choose your favourite connected graph $\Delta$ and shade a subset $J$ of its vertices. The intersection arrangement associated to the data $(\Delta, J)$ is a collection of real hyperplanes in dimension $|Jc|$, first defined by Iyama and Wemyss. This construction involves taking the classical Coxeter arrangement coming from $\Delta$ and then setting all variables indexed by $J$ to be zero. It turns out that for many choices of $J$ the chambers of the intersection arrangement admit a nice combinatorial description, along with a wall crossing rule to pass between them. I will start by making all this precise before discussing my work to classify tilings of the hyperbolic plane arising as intersection arrangements. This has applications to local notions of stability conditions using the tilting theory of contracted preprojective algebras.