University of Glasgow

to follow

AF-embeddability, i.e., the question whether a given C*-algebra can be realised as a subalgebra of an AF-algebra, has been studied for a long time with prominent early results by Pimsner and Voicuescu who constructed such embeddings for irrational rotation algebras in 1980. Since then, many AF-embeddings have been constructed for concrete examples but also many non-constructive AF-embeddability results have been obtained for classes of algebras typically assuming the UCT. 

In this talk by Joachim Zacharias, we will consider a separable unital C*-algebra A of decomposition rank at most 1 and construct from a suitable system of 1-decomposable cpc-approximations an AF-algebra E together with an embedding of A into E and a conditional expectation of E onto A without assuming the UCT. We also consider some extensions of this inclusion and indicate some applications.

An unpublished theorem of Clozel, proven with global techniques, says that the class of essentially discrete series representations of a connected reductive p-adic group is stable under twist by automorphisms of the complex numbers, and hence this class is defined over $\bar{\mathbb{Q}_\ell}$. Recent work of Solleveld, building on work of Kazhdan-Varshavsky-Solleveld, says that the same is true of the class of standard representations. Stefan Dawydiak will give a geometric proof of this result for the principal block, and use this to deduce a local proof of Clozel's theorem for the general linear group. Time permitting, Stefan will also give geometric formulas for certain Harish-Chandra Schwartz functions that help illustrate these results.

Groupoids provide a rich supply of C*-algebras, and there are many results describing the structure of these C*-algebras using properties of the underlying groupoid. For non-Hausdorff groupoids, less is known, largely due to the existence of 'singular' functions in the reduced C*-algebra. This talk will discuss two approaches to studying ideals in non-Hausdorff groupoid C*-algebras. The first uses Timmermann's Hausdorff cover to reduce certain problems to the setting of Hausdorff groupoids. The second will restrict to isotropy groups. For amenable second-countable étale groupoids, these techniques allow us to characterise when the ideal of singular functions has dense intersection with the underlying groupoid *-algebra. This is based on joint work with K. A. Brix, J. B. Hume, and X. Li, as well as upcoming work with J. B. Hume.

Right-angled Artin groups (RAAGs) can be viewed as a generalisation of free groups. To what extent, then, do the techniques used to study automorphisms of free groups generalise to the setting of RAAGs? One significant advance in this direction is the construction of 'untwisted Outer space' for RAAGs, a generalisation of the influential Culler-Vogtmann Outer space for free groups. A consequence of this construction is an upper bound on the virtual cohomological dimension of the 'untwisted subgroup' of outer automorphisms of a RAAG. However, this bound is sometimes larger than one expects; I present work showing that, in fact, it can be arbitrarily so, by forming a new complex as a deformation retraction of the untwisted Outer space. In a different direction, another subgroup of interest is that consisting of symmetric automorphisms. Generalising work in the free groups setting from 1989, I present an Outer space for the symmetric automorphism group of a RAAG. A consequence of the proof is a strong finiteness property for many other subgroups of the outer automorphism group.

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.