Edinburgh University

In many machine learning tasks, it is crucial to extract low-dimensional and descriptive features from a data set. In this talk, I present a method to extract features from multi-dimensional space-time signals which is motivated, on the one hand, by the success of path signatures in machine learning, and on the other hand, by the success of models from the theory of regularity structures in the analysis of PDEs. I will present a flexible definition of a model feature vector along with numerical experiments in which we combine these features with basic supervised linear regression to predict solutions to parabolic and dispersive PDEs with a given forcing and boundary conditions. Interestingly, in the dispersive case, the prediction power relies heavily on whether the boundary conditions are appropriately included in the model. The talk is based on the following joint work with Andris Gerasimovics and Hendrik Weber: https://arxiv.org/abs/2108.05879

 In this talk, we will present the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant differential operators. We describe in particular the renormalized equation for a very large class of spacetime dependent renormalization schemes. Our approach bypasses the previous approaches in the translation-invariant setting. This is joint work with Ismael Bailleul.

Let W be the Witt algebra of vector fields on the punctured complex plane, and let Vir be the Virasoro algebra, the unique nontrivial central extension of W.  We discuss work in progress with Alexey Petukhov to analyse Poisson ideals of the symmetric algebra of Vir.

We focus on understanding maximal Poisson ideals, which can be given as the Poisson cores of maximal ideals of Sym(Vir) and of Sym(W).  We give a complete classification of maximal ideals of Sym(W) which have nontrivial Poisson cores.  We then lift this classification to Sym(Vir), and use it to show that if $\lambda \neq 0$, then $(z-\lambda)$ is a maximal Poisson ideal of Sym(Vir).

The FEniCS system [1] allows the description of finite element discretisations of partial differential equations using a high-level syntax, and the automated conversion of these representations to working code via automated code generation. In previous work described in [2] the high-level representation is processed automatically to derive discrete tangent-linear and adjoint models. The processing of the model code at a high level eases the technical difficulty associated with management of data in adjoint calculations, allowing the use of optimal data management strategies [3].

This previous methodology is extended to enable the calculation of higher order partial differential equation constrained derivative information. The key additional step is to treat tangent-linear
equations on an equal footing with originating forward equations, and in particular to treat these in a manner which can themselves be further processed to enable the derivation of associated adjoint information, and the derivation of higher order tangent-linear equations, to arbitrary order. This enables the calculation of higher order derivative information -- specifically the contraction of a Kth order derivative against (K - 1) directions -- while still making use of optimal data management strategies. Specific applications making use of Hessian information associated with models written using the FEniCS system are presented.

[1] "Automated solution of differential equations by the finite element method: The FEniCS book", A. Logg, K.-A. Mardal, and  G. N. Wells (editors), Springer, 2012
[2] P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes, "Automated derivation of the adjoint of high-level transient finite element programs", SIAM Journal on Scientific Computing 35(4), C369--C393, 2013
[3] A. Griewank, and A. Walther, "Algorithm 799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation", ACM Transactions on Mathematical Software 26(1), 19--45, 2000

I will discuss recent developments in the study of scattering amplitudes in Einstein-Yang-Mills theory. At tree level we find new structures at higher order collinear limits and novel connections with amplitudes in Yang-Mills theory using the CHY formalism. Finally I will comment on unitarity based observations regarding one-loop amplitudes in the theory. 

In 1997, Maxim Kontsevich gave a universal formula for the
quantization of Poisson brackets.  It can be viewed as a perturbative
expansion in a certain two-dimensional topological field theory.  While the
formula is explicit, it is currently impossible to compute in all but the
simplest cases, not least because the values of the relevant Feynman
integrals are unknown.  In forthcoming joint work with Peter Banks and Erik
Panzer, we use Francis Brown's approach to the periods of the moduli space
of genus zero curves to give an algorithm for the computation of these
integrals in terms of multiple zeta values.  It allows us to calculate the
terms in the expansion on a computer for the first time, giving tantalizing
evidence for several open conjectures concerning the convergence and sum of
the series, and the action of the Grothendieck-Teichmuller group by gauge
transformations.