I will review some recent progress on D=4, N=2 superconformal field theories in what has come to be known as "Class-S". This is a huge class of (mostly non-Lagrangian) SCFTs, whose properties are encoded in the data of a punctured Riemann surface and a collection (one per puncture) of nilpotent orbits in an ADE Lie algebra.

# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

This week is a women's only week in which we are joined by the Mirzakhani society, the society for undergraduate women in maths, for lunch.

We consider the problem of optimally hedging a portfolio of derivatives in a scenario based discrete-time market with transaction costs. Risk-preferences are specified in terms of a convex risk-measure. Such a framework has suffered from numerical intractability up until recently, but this has changed thanks to technological advances: using hedging strategies built from neural networks and machine learning optimization techniques, optimal hedging strategies can be approximated efficiently, as shown by the numerical study and some theoretical results presented in this talk (based on joint work with Hans Bühler, Ben Wood and Josef Teichmann). We also discuss alternative choices of architectures (based on joint work with Juan-Pablo Ortega).

The goal of the talk will be to introduce a class of functions that answer a question in topology, can be computed via analytic methods more common in the theory of dynamical systems, and in the end turn out to enjoy beautiful modular properties of the type first observed by Ramanujan. If time permits, we will discuss connections with vertex algebras and physics of BPS states which play an important role, but will be hidden "under the hood" in much of the talk.

In this talk I will discuss recent joint work with Dominik Gruber where

we find a reasonable model for random (infinite) Burnside groups,

building on earlier tools developed by Coulon and Coulon-Gruber.

The free Burnside group with rank r and exponent n is defined to be the

quotient of a free group of rank r by the normal subgroup generated by

all elements of the form g^n; quotients of such groups are called

Burnside groups. In 1902, Burnside asked whether any such groups could

be infinite, but it wasn't until the 1960s that Novikov and Adian showed

that indeed this was the case for all large enough odd n, with later

important developments by Ol'shanski, Ivanov, Lysenok and others.

In a different direction, when Gromov developed the theory of hyperbolic

groups in the 1980s and 90s, he observed that random quotients of free

groups have interesting properties: depending on exactly how one chooses

the number and length of relations one can typically gets hyperbolic

groups, and these groups are infinite as long as not too many relations

are chosen, and exhibit other interesting behaviour. But one could

equally well consider what happens if one takes random quotients of

other free objects, such as free Burnside groups, and that is what we

will discuss.

I will talk about R^n valued random processes driven by a "noise", which is generated by a deterministic dynamical system, randomness coming from the choice of the initial condition.

Such processes were considered by D.Kelly and I.Melbourne.I will present our joint work with I.Chevyrev, P.Friz, I.Melbourne and H.Zhang, where we consider the noise with long term memory. We prove convergence to solution of a stochastic differential equation which is, depending on the noise, driven by either a Brownian motion (optimizing the assumptions of Kelly-Melbourne) or a Lévy process.Our work is made possible by recent progress in rough path theory for càdlàg paths in p-variation topology.

I will present a construction of large families of singularity-free stationary solutions of Einstein equations, for a large class of matter models including vacuum, with a negative cosmological constant. The solutions, which are of course real-valued Lorentzian metrics, are determined by a set of free data at conformal infinity, and the construction proceeds through elliptic equations for complex-valued tensor fields. One thus obtains infinite dimensional families of both strictly stationary spacetimes and black hole spacetimes.

In recent years it has been discovered that also non-linear, degenerate equations like the $p$-Poisson equation $$ -\mathrm{div}(A(\nabla u))= - \mathrm{div} (|\nabla u|^{{p-2}}\nabla u)= -{\rm div} F$$ allow for optimal regularity. This equation has similarities to the one of power-law fluids. In particular, the non-linear mapping $F \mapsto A(\nabla u)$ satisfies surprisingly the linear, optimal estimate $\|A(\nabla u)\|_X \le c\, \|F\|_X$ for several choices of spaces $X$. In particular, this estimate holds for Lebesgue spaces $L^q$ (with $q \geq p'$), spaces of bounded mean oscillations and Holder spaces$C^{0,\alpha}$ (for some $\alpha>0$).

In this talk we show that we can extend this theory to Sobolev and Besov spaces of (almost) one derivative. Our result are restricted to the case of the plane, since we use complex analysis in our proof. Moreover, we are restricted to the super-linear case $p \geq 2$, since the result fails $p < 2$. Joint work with Anna Kh. Balci, Markus Weimar.

We present a self-contained construction of the Euclidean $\Phi^4$ quantum

field theory on $\mathbb{R}^3$ based on PDE arguments. More precisely, we

consider an approximation of the stochastic quantization equation on

$\mathbb{R}^3$ defined on a periodic lattice of mesh size $\varepsilon$ and

side length $M$. We introduce an energy method and prove tightness of the

corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow

\infty$. We show that every limit point satisfies reflection positivity,

translation invariance and nontriviality (i.e. non-Gaussianity). Our

argument applies to arbitrary positive coupling constant and also to

multicomponent models with $O(N)$ symmetry. Joint work with Massimiliano

Gubinelli.

Disease transmission and rumour spreading are ubiquitous in social and technological networks. In this talk, we will present our last results on the modelling of rumour and disease spreading in multilayer networks. We will derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks in a multiplex network. Moreover, we will introduce a model of epidemic spreading with awareness, where the disease and information are propagated in different layers with different time scales. We will show that the time scale determines whether the information awareness is beneficial or not to the disease spreading.