Southampton University

Conformal defects can be characterised by their contributions to the Weyl anomaly. The coefficients of these terms, often called defect central charges, depend on the particular defect insertion in a given conformal field theory. I will review what is currently known about defect central charges across dimensions, and present novel results. I will discuss many examples where they can be computed exactly without requiring any approximations or limits. Particular emphasis will be placed on recently developed tools for superconformal defects as well as defects in free theories.

Free-by-cyclic groups are easy to define – all you need is an automorphism of F_n. Their properties (for example hyperbolicity, or relative hyperbolicity) depend on this defining automorphism, but not always transparently. I will introduce these groups and some of their properties, and connect some to properties of the defining automorphism. I'll then discuss some ideas and techniques we can use to understand their automorphisms, including finding useful actions on trees and relationships with certain subgroups of Out(F_n). (This is joint work with Armando Martino.)

Property (FA) is one of the `rigidity properties’ defined for groups, concerning the way a group can act on trees. We’ll take a look at why you might be interested in an action on a tree, what the property is, and then investigate which automorphism groups of free products have it.

Let $G$ be a group which splits as $G = F_n * G_1 *...*G_k$, where every $G_i$ is freely indecomposable and not isomorphic to the group of integers.  Guirardel and Levitt generalised the Culler- Vogtmann Outer space of a free group by introducing an Outer space for $G$ as above, on which $\text{Out}(G)$ acts by isometries. Francaviglia and Martino introduced the Lipschitz metric for the Culler- Vogtmann space and later for the general Outer space. In a joint paper with Francaviglia and Martino, we prove that the group of isometries of the Outer space corresponding to $G$ , with respect to the Lipschitz metric, is exactly $\text{Out}(G)$. In this talk, we will describe the construction of the general Outer space and the corresponding Lipschitz metric in order to present the result about the isometries.

A recent calculation of the multi-Higgs boson production in scalar theories
with spontaneous symmetry breaking has demonstrated the fast growth of the
cross section with the Higgs multiplicity at sufficiently large energies,
called “Higgsplosion”. It was argued that “Higgsplosion” solves the Higgs
hierarchy and fine-tuning problems. The phenomena looks quite exciting,
however in my talk I will present arguments that: a) the formula for
“Higgsplosion” has a limited applicability and inconsistent with unitarity
of the Standard Model; b) that the contribution from “Higgsplosion” to the
imaginary part of the Higgs boson propagator cannot be re-summed in order to
furnish a solution of the Higgs hierarchy and fine-tuning problems.

Based on our recent paper https://arxiv.org/abs/1808.05641 (A. Belyaev, F. Bezrukov, D. Ross)

Graph products are a class of groups that 'interpolate' between direct and free products, and generalise the notion of right-angled Artin groups. Given a property that free products (and maybe direct products) are known to satisfy, a natural question arises: do graph products satisfy this property? For instance, it is known that graph products act on tree-like spaces (quasi-trees) in a nice way (acylindrically), just like free products. In the talk we will discuss a construction of such an action and, if time permits, its relation to solving systems of equations over graph products.

Scattering amplitudes computed at a fixed loop order, along with any other object computed in perturbative QFT, can be expressed as a linear combination of a finite basis of loop integrals. To compute loop amplitudes in practise, such a basis of integrals must be determined. In this talk I introduce a new algorithm for finding bases of loop integrals and discuss its implementation in the publically available package Azurite.

The common convention when dealing with hyperbolic groups is that such groups are finitely

generated and equipped with the word length metric relative to a finite symmetric generating

subset. Gromov's original work on hyperbolicity already contained ideas that extend beyond the

finitely generated setting. We study the class of locally compact hyperbolic groups and elaborate

on the similarities and differences between the discrete and non-discrete setting.

Coarse geometry provides a very useful organising point of view on the study
of geometry and analysis of discrete metric spaces, and has been very
successful in the context of geometric group theory and its applications. On
the other hand, the work of Carlsson, Ghrist and others on persistent
homology has paved the way for applications of topological methods to the
study of broadly understood data sets. This talk will provide an
introduction to this fascinating topic and will give an overview of possible
interactions between the two.