Cambridge University

Schwarzian Theory is a quantum field theory which has attracted a lot of attention in the physics literature in the context of two-dimensional quantum gravity, black holes and AdS/CFT correspondence. It is predicted to be universal and arise in many systems with emerging conformal symmetry, most notably in Sachdev--Ye--Kitaev random matrix model and Jackie--Teitelboim gravity.

In this talk we will discuss our recent progress on developing rigorous mathematical foundations of the Schwarzian Field Theory, including rigorous construction of the corresponding measure, calculation of both the partition function and a natural class of correlation functions, and a large deviation principle.

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd (integral) distance from each other. We will also discuss some further results with Rose McCarty and Michal Pilipczuk concerning prime and polynomial distances.

I will discuss joint work with S. Galatius and A. Kupers in which we investigate the homology of general linear groups over a ring $A$ by considering the collection of all their classifying spaces as a graded $E_\infty$-algebra. I will first explain diverse results that we obtained in this investigation, which can be understood without reference to $E_\infty$-algebras but which seem unrelated to each other: I will then explain how the point of view of cellular $E_\infty$-algebras unites them.

One of the main characterisations of word-hyperbolic groups is that they are the groups with a linear isoperimetric function. That is, for a compact 2-complex X, the hyperbolicity of its fundamental group is equivalent to the existence of a linear isoperimetric function for disc diagrams D -->X.
It is likewise known that hyperbolic groups have a linear annular isoperimetric function and a linear homological isoperimetric function. I will talk about these isoperimetric functions, and about a (previously unexplored)  generalisation to all homotopy types of surface diagrams. This is joint work with Dani Wise.

I will talk about supersymmetric index of 4d N=1 supersymmetric theories on S^1xM_3 which counts supersymmetric states.  
In the first part, I will discuss a general formula to describe an asymptotic behaviour of the index in the limit of shrinking S^1
which we refer to as 4d (refined) supersymmetric Cardy formula. This part is based on arXiv:1611.00380 with Lorenzo Di Pietro.
In the second part, I will apply this formula to black hole physics. I will mainly focus on superconformal index of SU(N) N=4 super Yang-Mills theory
which is expected to be dual to type IIB superstring theory on AdS_5 x S^5. We will see that the index in the large-N limit reproduces the Bekenstein-Hawking entropy
of rotating charged BPS black hole on the gravity side. Our result for finite N makes a prediction to the black hole entropy with full quantum corrections.
The second part is based on arXiv:1901.08091.

It is often fruitful to study an infinite discrete group via its finite quotients.  For this reason, conditions that guarantee many finite quotients can be useful.  One such notion is residual finiteness.
A group is residually finite if for any non-identity element g there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and I’ll show that in this case it is enough to consider homomorphisms onto alternating groups.

When groups may be built up as graphs of 'simpler' groups, it is often 
of interest to study how good residual finiteness properties of simpler 
groups can imply residual properties of the whole. The essential case of 
this theory is the study of residual properties of finite groups. In 
this talk I will discuss the question of when a graph of finite 
$p$-groups is residually $p$-finite, for $p$ a prime. I describe the 
previous theorems in this area for one-edge and finite graphs of groups, 
and their method of proof. I will then state my recent generalisation of 
these theorems to potentially infinite graphs of groups, together with 
an alternative and more natural method of proof. Finally I will briefly 
describe a usage of these results in the study of accessibility -- 
namely the existence of a finitely generated inaccessible group which is 
residually $p$-finite.

In this talk, we will explain how to construct embedded closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toric degeneration technique. As an example, we will illustrate the construction for tropical curves that contribute to the Gromov–Witten invariant of the line class of the quintic threefold. The construction will in turn provide many homologous and non-Hamiltonian isotopic Lagrangian
rational homology spheres, and a geometric interpretation of the multiplicity of a tropical curve as the weight of a Lagrangian. This is a joint work with Helge Ruddat.

We will consider the following deceptively simple question, formulated recently by Po Shen Loh who connected it to an open problem in Ramsey Theory. Define the '2-less than' relation on the set of triples of integers by saying that a triple x is 2-less than a triple y if x is less than y in at least two coordinates. What is the maximal length of a sequence of triples taking values in {1,...,n} which is totally ordered by the '2-less than' relation?

In his paper, Loh uses the triangle removal lemma to improve slightly on the trivial upper bound of n^2, and conjectures that the truth should be of order n^(3/2). The gap between these bounds has proved to be surprisingly resistant. We shall discuss joint work with Tim Gowers, giving some developments towards this conjecture and a wide array of natural extensions of the problem. Many of these extensions remain open.
 

Let G be a split reductive group over a finite extension k of Q_p. Reeder and Yu have given a new construction of supercuspidal representations of G(k) using geometric invariant theory. Their construction is uniform for all p but requires as input stable vectors in certain representations coming from Moy-Prasad filtrations. In joint work, Jessica Fintzen and I have classified the representations of this kind which contain stable vectors; as a corollary, the construction of Reeder-Yu gives new representations when p is small. In my talk, I will give an overview of this work, as well as explicit examples for the case when G = G_2. For these examples, I will explicitly describe the locus of all stable vectors, as well as the Langlands parameters which correspond under the local Langlands correspondence to the representations of G(k).