L5

Since their introduction by Gromov in the 80s, a wealth of tools have been developed to study hyperbolic groups. Thus, when studying a class of groups, a characterisation of those that are hyperbolic can be very useful. In this talk, we will turn to the class of one-relator groups. In previous work, we showed that a one-relator group not containing any Baumslag--Solitar subgroups is hyperbolic, provided it has a Magnus hierarchy in which no one-relator group with a so called `exceptional intersection' appears. I will define one-relator groups with exceptional intersection, discuss the aforementioned result and will then provide a characterisation of the hyperbolic one-relator groups with exceptional intersection. Finally, I will then discuss how this characterisation can be used to establish properties for all one-relator groups.

Various graphs associated to surfaces have proved to be important tools for studying the large scale geometry of mapping class groups of surfaces, among other applications. A seminal paper of Masur and Minsky proved that perhaps the most well known example, the curve graph, is Gromov hyperbolic. However, this is not the case for every naturally defined graph associated to a surface. We will present joint work with Jacob Russell classifying a wide family of graphs associated to surfaces according to whether the graph is Gromov hyperbolic, relatively hyperbolic or not relatively hyperbolic.
 

I will discuss some new invariants of 2-complexes. They are inspired by recent developments in the theory of one-relator groups, but also have the potential to unify the theories of many well-studied families including small-cancellation presentation complexes, CAT(0) 2-complexes and 3-manifold spines, in addition to the motivating examples of one-relator presentation complexes. The fundamental result is that these invariants are the extrema of explicit linear-programming problems, and in particular are rational, computable and realised. The definitions suggest a conjectural “map” of 2-complexes, which I will attempt to describe.
 

Consider a random group $\Gamma$ with $k$ generators and $r$ random relators of large length $N$. We study the geometry of the character variety of $\Gamma$ with values in $\SL(2,\C)$ or more generally any semisimple Lie group $G$. This is the moduli space of group homomorphisms from $\Gamma$ to $G$ up to conjugation. We are in particular able to determine its dimension, number of components and Galois group, with an excellent control on the probability of exceptions. The proofs use effective Chebotarev type theorems as well as new spectral gap bounds  for Cayley graphs of finite simple groups. They are also conditional on GRH. Joint work with Peter Varju and Oren Becker.

When considering compatifications of heterotic string theory down to 3D, the heterotic $G_2$ system arises naturally. It is a system for both geometric fields and gauge fields over a manifold with a $G_2$-structure. In particular, it asks for the $G_2$-structure to be coclosed. We will begin this talk defining this system and giving a description of the geometry of cohomogeneity one manifolds. Then, we will look for coclosed $G_2$-structures in the cohomogeneity one setting. We will end up by proving the existence of a family of coclosed $G_2$-structures which are invariant under a cohomogeneity one action of $\text{SU}(2)^2$ on certain seven-dimensional simply connected manifolds.

In this talk I will review some of the key ideas behind
the study of entanglement measures in 1+1D quantum field theories employing
the so-called branch point twist field approach. This method is based on the
existence of a one-to-one correspondence between different entanglement
measures and different multi-point functions of a particular type of
symmetry field. It is then possible to employ standard methods for the
evaluation of correlation functions to understand properties of entanglement
in bipartite systems. Time permitting, I will then present a recent
application of this approach to the study of a new entanglement measure: the
symmetry resolved entanglement entropy.

Modular forms of slow growth admit a decomposition in terms of the eigenfunctions of the Laplacian operator in the Upper Half Plane. Whilst this technology has been used for many years in the context of Number Theory, it has only recently been used to further understand the partition function and the spectrum of Conformal Field Theories in 2d. In this talk, we’ll review the technology and how it has been applied to CFTs by several authors, as well as present a few new results.

Hydrodynamics allow for efficient computation of many-body dynamics and have been successfully used in the study of black hole horizons, collective behaviour of QCD matter in heavy ion collisions, and non-equilibrium behaviour in strongly-interacting condensed matter systems.
In this talk, I will present the application of hydrodynamics to quantum field theory with an infinite number of local conservation laws. Such an integrable system can be described within the recently developed framework of generalised hydrodynamics. I will present the key assumptions of generalised hydrodynamics as well as summarise some recent developments in this field. In particular, I will concentrate on the study of the SU(3)_2-Homogeneous sine-Gordon model. Thanks to the hydrodynamic approach, we were able to identify the key dynamical signatures of unstable excitations in this integrable quantum field theory and simulate the real time RG-flow of the theory between interacting and free conformal regimes.
The talk is based on joint work with Olalla Castro-Alvaredo, Cecilia De Fazio and Benjamin Doyon.

The notion of topological order was introduced by Xiao-Gang Wen in order to capture the features of the exotic phases of matter given by fractional quantum Hall phases. I will motivate why the corresponding mathematical structures are higher categories with additional properties. In 2+1-dimensions, I will explain in details how the definition of fusion category arises from physical and geometrical intuitions about topological orders. Finally, I will sketch how the notion of higher fusion category emerges in higher dimensions.

Representations of finite reductive groups have a rich, well-understood structure, first explored by Deligne--Lusztig. In joint work with Anne-Marie Aubert and Dan Ciubotaru, we show a way to lift some of this structure to representations of p-adic groups. In particular, we consider the relation between Lusztig's nonabelian Fourier transform and a certain involution we define on the level of p-adic groups. This talk will be an introduction to these ideas with a focus on examples.