Let $G$ be a group acting transitively on a finite set $\Omega$. Then $G$ acts on $\Omega \times \Omega$ componentwise. Define the orbitals to be the orbits of $G$ on $\Omega \times \Omega$. The diagonal orbital is the orbital of the form $\Delta = \{(\alpha, \alpha) \mid \alpha \in \Omega \}$. The others are called non-diagonal orbitals. Let $\Gamma$ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set $\Omega$ and edge set $(\alpha,\beta) \in \Gamma$ with $\alpha, \beta \in \Omega$. If the action of $G$ on $\Omega$ is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding specific bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups. 

I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.  The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos--Holzegel--Rodnianski on the linear stability of the Schwarzschild family.  This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

The graph C*-algebra construction associates a unital C*-algebra to any directed graph with finitely many vertices and countably many edges in a way which generalizes the fundamental construction by Cuntz and Krieger. We say that two such graphs are C*-equivalent when they define isomorphic C*-algebras, and give a description of this relation as the smallest equivalence relation generated by a number of "moves" on the graph that leave the C*-algebras unchanged. The talk is based on recent work with Arklint and Ruiz, but most of these moves have a long history that I intend to present in some detail.

In this talk I will explain how a subfactor (ie an inclusion of type II_1 factors) give rise to a diagrammatic algebra called the Temperley-Lieb-Jones algebra. We will observe the connection between the index of the subfactor, and the TLJ algebra. In the TLJ algebra setting, we will observe that indices below four are discrete, while any number above four can be an index.

A well-behaved field theory living on a fixed background has a causality structure defined by the background metric. In holography, however, signals can travel through the bulk, and some bulk metrics would allow a signal to travel faster than the speed of light as seen on the boundary. These are called boundary causality violating metrics. Holographers usually work with a classical bulk metric, in which case they declare that boundary causality violating metrics are forbidden. However, in a full quantum gravity path integral, these metrics do contribute. The question is then: how to avoid causality violation in this context? In this talk I will give a prescription that achieves this.

I will discuss a string theory perspective on the Bethe/Gauge correspondence for the XXX superspin chain. I explain how to realize 4d Chern-Simons theory with gauge supergroup using branes, and how the brane configurations for the superspin chain get mapped to 2d N = (2,2) quiver gauge theories proposed by Nekrasov. This is based on my ongoing work with Nafiz Ishtiaque, Faroogh Moosavian and Surya Raghavendran.