The so-called moonshine phenomenon relates modular forms and finite group representations. After the celebrated monstrous moonshine, various new examples of moonshine connection have been discovered in recent years. The study of these new moonshine examples has revealed interesting connections to K3 surfaces, arithmetic geometry, and string theory. In this colloquium I will give an overview of these recent developments.

# 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.

In this talk we will discuss non-Abelian T-duality as a solution generating technique in type II Supergravity, briefly reviewing its potential to motivate, probe or challenge classifications of supersymmetric solutions, and focusing on the open problem of providing the newly generated AdS brackgrounds with consistent dual superconformal field theories. These can be seen as renormalization fixed points of linear quivers of increasing rank. As illustrative examples, we consider the non-Abelian T-duals of AdS5xS5, the Klebanov-Witten background, and the IIA reduction of AdS4xS7, whose proposed quivers are, respectively, the four dimensional N=2 Gaiotto-Maldacena theories describing the worldvolume dynamics of D4-NS5 brane intersections, its N=1 mass deformations realized as D4-NS5-NS5’, and the three dimensional N=4 Gaiotto-Witten theories, corresponding to D3-D5-NS5. Based on 1705.09661 and 1609.09061.

Consider dY(t)=f(X(t))dX(t), where X(t) is a pure jump Levy process with finite p-variation norm, 1<= p < 2, and f is a Lipchitz continuous function. Following the geometric solution construction of Levy-driven stochastic differential equations in (Williams 2001), we develop a class of epsilon-strong simulation algorithms that allows us to construct a probability space, supporting both the geometric solution Y and a fully simulatable process Y_epsilon, such that Y_epsilon is within epsilon distance from Y under the uniform metric on compact time intervals with probability 1. Moreover, the users can adaptively choose epsilon’ < epsilon, so that Y_epsilon’ can be constructed conditional on Y_epsilon. This tolerance-enforcement feature allows us to easily combine our algorithm with Multilevel Monte Carlo for efficient estimation of expectations, and adding as a benefit a straightforward analysis of rates of convergence. This is joint with Jose Blanchet, Fei He and Offer Kella.

Symplectic reduction is the natural quotient construction for symplectic manifolds. Given a free and proper action of a Lie group G on a symplectic manifold M, this process produces a new symplectic manifold of dimension dim(M) - 2 dim(G). For non-free actions, however, the result is usually fairly singular. But Sjamaar-Lerman (1991) showed that the singularities can be understood quite precisely: symplectic reductions by non-free actions are partitioned into smooth symplectic manifolds, and these manifolds fit nicely together in the sense that they form a stratification.

Symplectic reduction has an analogue in hyperkähler geometry, which has been a very important tool for constructing new examples of these special manifolds. In this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified

spaces whose strata are hyperkähler.

McKean-Vlasov SDEs are SDEs where the coefficients depend on the law of the solution to the SDE. Their interest is in the links with nonlinear PDEs on one side (the SDE-related Fokker-Planck equation is nonlinear) and with interacting particles on the other side: the McKean-Vlasov SDE be approximated by a system of weakly coupled SDEs. In this talk we consider McKean-Vlasov SDEs with irregular drift: though well-posedness for this SDE is not known, we show a large deviation principle for the corresponding interacting particle system. This implies, in particular, that any limit point of the particle system solves the McKean-Vlasov SDE. The proof combines rough paths techniques and an extended Vanrdhan lemma.

This is a joint work with Thomas Holding.

A well known result by Schaeffer (1973) shows that generic solutions to a scalar conservation law are piecewise smooth, containing a finite family of shock curves.

In this direction, it is of interest to find other classes of nonlinear hyperbolic equations where nearly all solutions (in a Baire category sense) are piecewise smooth, and classify their singularities.

The talk will mainly focus on conservative solutions to the nonlinear variational wave equation $u_{tt} - c(u)(c(u) u_x)_x = 0$. For an open dense set of $C^3$ initial data, it is proved that the conservative solution is piecewise smooth in the $t - x$ plane, while the gradient $u_x$ can blow up along finitely many characteristic curves. The analysis relies on a variable transformation which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.

A detailed description of the solution profile can be given, in a neighborhood of every singular point and every singular curve.

Some results on structurally stable singularities have been obtained also for dissipative solutions, of the above wave equation. Recent progress on the Burgers-Hilbert equation, and related open problems, will also be discussed.

These results are in collaboration with Geng Chen, Tao Huang, Fang Yu, and Tianyou Zhang.

There have been enormous advances in both our ability to represent scattering amplitudes at the integrand-level (for an increasingly wide variety of quantum field theories), and also in our integration technology (and our understanding of the functions that result). In this talk, I review both sides of these recent developments. At the integrand-level, I describe the "prescriptive" refinement of generalized unitarity, and show how closed, integrand-level formulae can be given for all leading-weight contributions to any amplitude in any quantum field theory. Regarding integration, I describe some new results that could be summarized as "dual-conformal sufficiency": that all planar, ultraviolet-finite integrands can be regulated and computed directly in terms of manifestly dual-conformal integrals. I illustrate the power of having such representations, and discuss the role played by a (conjectural) cluster-algebraic structure for kinematic dependence.

The latest CPUs by Intel and ARM support vectorised operations, where a single set of instructions (e.g. add, multiple, bit shift, XOR, etc.) are performed in parallel for small batches of data. This can provide great performance improvements if each parallel instruction performs the same operation, but carries the risk of performance loss if each needs to perform different tasks (e.g. if else conditions). I will present the work I have done so far looking into how to recover the full performance of the hardware, and some of the challenges faced when trading off between ever larger parallel tasks, risks of tasks diverging, and how certain coding styles might be modified for memory bandwidth limited applications. Examples will be taken from finance and Monte Carlo applications, inspecting some standard maths library functions and possibly random number generation.