University of Glasgow

Together with Alex Degtyarev and Vincent Florence we introduced a new link invariant, called slope, of a colored link in an integral homology sphere. In this talk I will define the invariant, highlight some of its most interesting properties as well as its relationship to Conway polynomials and to the  Kojima–Yamasaki eta-function. The stress in this talk will be on our latest computational progress: a formula to calculate the slope from a C-complex.

The recent work on nc convex sets of Davidson, Kennedy, and Shamovich show that there is a rich interplay between the category of operator systems and the category of compact nc convex sets, leading to new insights even in the case of C*-algebras. The category of nc convex sets are a generalization of the usual notion of a compact convex set that provides meaningful connections between convex theoretic notions and notions in operator system theory. In this talk, we present a duality theorem for norm closed self-adjoint subspaces of B(H) that do not necessarily contain the unit. Using this duality, we will describe various C*-algebraic and operator system theoretic notions such as simplicity and subkernels in terms of their convex structure. This is joint work with Matthew Kennedy and Nicholas Manor.

I will give an introduction to semigroup C*-algebras of ax+b-semigroups over rings of algebraic integers in algebraic number fields, a class of C*-algebras that was introduced by Cuntz, Deninger, and Laca. After explaining the construction, I will briefly discuss the state-of-the-art for this example class: These C*-algebras are unital, separable, nuclear, strongly purely infinite, and have computable primitive ideal spaces. In many cases, e.g., for Galois extensions, they completely characterise the underlying algebraic number field.