Southampton

After reviewing different aspects of thermalization and chaos in holographic quantum systems, I will argue that universal aspects can be captured using an effective field theory framework that shares similarities with hydrodynamics. Focusing on the quantum butterfly effect, I will explain how to develop a simple effective theory of the 'scramblon' from path integral considerations. I will also discuss applications of this formalism to shockwave scattering in black hole backgrounds in AdS/CFT.

We will review some open questions about automorphisms of free groups, give some partial answers, and explain the deformation spaces of trees that they act on, as well as the geometry of these spaces arising from the Lipschitz metric. This will be a gentle introduction to the topic, focused on introducing the concepts.

 Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellin transformed to correlation functions on the celestial sphere at null infinity called celestial amplitudes. We study various properties of massless four-point scalar and gluon celestial amplitudes such as conformal partial wave decomposition, crossing relations and optical theorem. As a byproduct, we derive the analog of the single and double soft limits for all gluon celestial amplitudes.

Cluster Adjacency is a geometric principle which defines a subclass of multiple polylogarithms with analytic properties compatible with that of scattering amplitudes and Feynman loop integrals. We use this principle to a priori remove the redundances in the perturbative bootstrap approach and efficiently compute the four-loop NMHV heptagon. Moreover, cluster adjacency is naturally applied to the space of $A_n$ polylogarithms and generates numerous structures therein to be explored further.

I will review the fishnet model, which is an integrable scalar QFT, obtained by an extreme gamma deformation of N=4 super Yang-Mills. The theory has a peculiar perturbative expansion in which many quantities at a fixed loop order are given by a single Feynman diagram. This feature allows the reinterpretation of Feynman loop integrals as integrable systems.

I will describe how to recast perturbative quantum gravity using non-perturbative techniques from conformal field theory, focussing on the case of N=4 super Yang-Mills theory. By resolving the degeneracy among double trace operators at large N we are able to bootstrap one-loop supergravity corrections from the OPE of the CFT.
 

A group is called residually finite if every non-trivial element can be homomorphically mapped to a finite group such that the image is again non-trivial. Residually finite groups are interesting because quite a lot of information about them can be reconstructed from their finite quotients. Baumslag showed that if G is a finitely generated residually finite group then Aut(G) is also residually finite. Using a similar method Grossman showed that if G is a finitely generated conjugacy separable group with "nice" automorphisms then Out(G) is residually finite. The graph product is a group theoretic construction naturally generalising free and direct products in the category of groups. We show that if G is a finite graph product of finitely generated residually finite groups then Out(G) is residually finite (modulo some technical conditions)

Deciding whether or not two elements of a group are conjugate might seem like a trivial problem. However, there exist finitely presented groups where this problem is undecidable: there is no algorithm to output yes or no for any two elements chosen. In this talk Houghton groups (a family of groups all having solvable conjugacy problem) will be introduced as will the idea of twisted conjugacy: a generalisation of the conjugacy problem where an automorphism is also given. This will be our main tool in answering whether finite extensions and finite index subgroups of any Houghton group have solvable conjugacy problem.