University of Oxford

In this talk I will discuss a number of topics at the interface between C* algebras and Geometric Group Theory, with an emphasis on Kazhdan projections, various versions of amenability and their connection to the geometry of groups. This is based on joint work with P. Nowak and J. Mackay.

The goal of this talk is to present some elementary examples of fusion 2-categories whilst doing as little higher category theory as possible. More precisely, it turns out that up to a canonical completion operation, certain higher fusion categories are entirely described by their maximal subspaces. I will briefly motivate this completion operation in the 1-categorical case, and go on to explain why working with spaces is good enough in this particular case. Then, we will review some fact about $E_n$-algebras, and why they come into the picture. Finally, we will have a look at some small examples arising from finite groups.

Abstract: We study optimal investment, pricing and hedging problems under model uncertainty, when the reference model is a non-Markovian stochastic factor model, comprising a single stock whose drift and volatility are adapted to the filtration generated by a Brownian motion correlated with that driving the stock. We derive explicit characterisations of the robust value processes and optimal solutions (based on a so-called distortion solution for the investment problem under one of the models) and conduct large-scale simulation studies to test the efficacy of robust strategies versus classical ones (with no model uncertainty assumed) in the face of parameter estimation error.

 

Abstract: We study optimal investment, pricing and hedging problems under model uncertainty, when the reference model is a non-Markovian stochastic factor model, comprising a single stock whose drift and volatility are adapted to the filtration generated by a Brownian motion correlated with that driving the stock. We derive explicit characterisations of the robust value processes and optimal solutions (based on a so-called distortion solution for the investment problem under one of the models) and conduct large-scale simulation studies to test the efficacy of robust strategies versus classical ones (with no model uncertainty assumed) in the face of parameter estimation error.

 

How to calculate the minimal number of summands of a nonzero polynomial in a given polynomial ideal? In this talk, we first discuss the roots of this question in computational algebra. Afterwards, we switch to the viewpoint of a commutative algebraist. In particular, we see that classical tools from this field, such as primary decomposition or the Castelnuovo–Mumford regularity, fail to provide a solution to this problem. Finally, we discuss a concrete example: A standard determinantal ideal generated by $t$-minors does not contain any polynomials with fewer than $t!/2$ terms.

The classification of finite simple groups shows that many (those of Lie type) are obtained as (projectivisations of) subgroups of some $GL_{n}(\mathbb{F}_{q})$.

Green first determined the character table of any $GL_{n}(\mathbb{F}_{q})$ in his influential 1955 paper, while others have since given more explicit constructions of certain `cuspidal' representations.

In this talk, I will introduce parabolic induction as a means of obtaining representations of $GL_{n}(\mathbb{F}_{q})$ from those of $GL_{m}(\mathbb{F}_{q})$ where $m<n$.

Finding the irreducible representations of any $GL_{n}(\mathbb{F}_{q})$ then becomes inductive on $n$ for fixed $q$, with the cuspidal representations serving as atoms for this process.

Harish-Chandra's philosophy of cusp forms reduces the problem to the following two steps:

•  Find the cuspidal representations of any $GL_{n}(\mathbb{F}_{q})$
•  Determine the irreducible components of any representation $\sigma_{1}\circ\dots\circ\sigma_{k}$ parabolically induced from cuspidals $\sigma_{i}$

The majority of my talk will then aim to address each of these points.

Line patterns in free groups are collections of lines in the Cayley graph of a non-abelian free group F, which correspond to finite sets of words in F. Following Cashen and Macura, we will discuss line patterns by looking at the topology of Decomposition Spaces, which are quotients of the boundary of F that correspond to the different line patterns. Given a line pattern, we will also construct a cube complex whose isometry group is isomorphic to the group of quasi isometries of F which (coarsely) preserve the line pattern. This is a useful tool for studying the quasi isometric rigidity of related groups.

For a group G and a finite dimensional linear representation σ : G → GLn(D) over a skew field (division ring) D, the agrarian invariants with respect to σ are the homological invariants of G with coefficient module Dn. In this talk I will discuss the relationship between agrarian invariants, L 2 -invariants, Thurston norm and twisted Alexander polynomials. I will also discuss an ongoing work with Dawid Kielak.

The conformal dimension of a hyperbolic group is a powerful numeric quasi-isometry invariant associated to its boundary.

As an invariant it is finer than the topological dimension and allows us to distinguish between groups with homeomorphic boundaries.

I will start by talking about what conformal geometry even is, before discussing how this connects to studying the boundaries of hyperbolic groups.

I will probably end by saying how jolly hard it is to compute.

The graph complex is a remarkable object with very rich structure and many, sometimes mysterious, connections to topology. To illustrate one such connection, I will attempt to construct a “self-linking” invariant of knots and expand on the ideas behind it.