Week 10.11.2024 – 16.11.2024

Scattering amplitudes provide crucial theoretical input in collider and gravitational wave physics, and at the same time exhibit a remarkable mathematical structure. These lectures will introduce essential concepts and modern techniques exploiting this structure so as to efficiently compute amplitudes and their building blocks, Feynman integrals, in perturbation theory. We will start by decomposing gauge theory amplitudes into simpler pieces based on colour and helicity information. Focusing on tree level, we will then show how these may be determined from their analytic properties with the help of Britto-Cachazo-Feng-Witten recursion. Moving on to loop level, we will define the the class of polylogarithmic functions amplitudes and integrals often evaluate to, and explain their properties as well as relate them to the universal framework for predicting their singularities, known as the Landau equations. Time permitting, we will also summarise the state of the art in the calculation of the aforementioned singularities, and their intriguing relation to mathematical objects known as cluster algebras.

Last passage percolation is a model of random planar geometry which captures notions of distances and geodesics. It admits a rich scaling limit, called the directed landscape, that is conjectured to be universal for all geometric models in the so called KPZ universality class. A closely related notion is the KPZ fixed point, which represents the scaling limit of certain growing interfaces. A variational formula links the evolution of the KPZ fixed point to the directed landscape. The optimizer of the variational formula is akin to the polymer endpoint of a point-to-line last passage percolation problem. I will explain how to compute the law of this endpoint using the integrability of the KPZ fixed point. Joint work with Jeremy Quastel and Sourav Sarkar.

Complex networks have emerged as the primary framework for modeling the dynamics of interacting systems. However, networks inherently describe pairwise interactions, while real-world systems often involve interactions among groups of three or more units. In this talk, I will explore social systems as a natural testing ground for higher-order network approaches. I will briefly demonstrate how incorporating higher-order mechanisms can lead to the emergence of novel phenomena, presenting recent results on the influence of structural features and seeding strategies on emergent dynamics. Finally, I will delve into the microscopic dynamics of empirical higher-order structures, examining the mechanisms governing their temporal dynamics at both the individual and group levels. This will involve characterizing how individuals navigate groups and how groups form and dissolve. I will conclude by proposing a dynamical hypergraph model that closely reproduces empirical observations.

The Hilbert projective metric is a generalisation, due to Hilbert, of the classic Cayley—Klein distance in hyperbolic geometry. In 1957 Birkhoff proved that, under the Hilbert metric, positive linear operators on positive convex cones contract. While this result has found applications in other branches of analysis, it has generally been overlooked by the probabilistic community, despite providing, for example, a straightforward approach to the study of the stability of Markov chains. In this talk I will give an overview of the Hilbert metric and its properties, with a particular focus on its merits in applied probability theory. Starting in greater generality by working on locally convex topological vector spaces, I will give a definition of the Hilbert metric using duality, and provide a proof of (a slighter stronger version of) Birkhoff’s contraction result. I will then look specifically at the space of probability measures equipped with the Hilbert metric, and explore its structure, carrying on a comparison with other metrics on the side. Finally, as an application, I will introduce the stochastic filtering problem, and how the Hilbert metric can be effectively used to prove stability of the filtering equations.

This talk is based on joint work with Sam Cohen (Oxford).

HiGHS is open-source optimization software for linear programming (LP), mixed-integer programming (MIP) and quadratic programming (QP). This talk will give a brief insight into the state-of-the-art techniques underlying its solvers, most of which were originally written as “gradware” by PhD students, and an overview of how HiGHS came to be developed. Independent benchmark results will be given to justify the claim that HiGHS has the world’s best open-source linear optimization solvers. The team developing HiGHS was responsible for a 2021 REF Impact Case Study, and has the potential to generate further Impact for the next REF, so observations on the creation of Impact via software development, and how to fund such work, will be given. Current solver developments will be outlined. These include a new interior point solver for LP and QP, and the exploitation of GPU computing.

Gaussian processes are notorious for scaling cubically with the size of the training set, preventing application to very large regression problems. Computation-aware Gaussian processes (CAGPs) tackle this scaling issue by exploiting probabilistic linear solvers to reduce complexity, widening the posterior with additional computational uncertainty due to reduced computation. However, the most commonly used CAGP framework results in (sometimes dramatically) conservative uncertainty quantification, making the posterior unrealistic in practice. In this work, we prove that if the utilised probabilistic linear solver is calibrated, in a rigorous statistical sense, then so too is the induced CAGP. We thus propose a new CAGP framework, CAGP-GS, based on using Gauss-Seidel iterations for the underlying probabilistic linear solver. CAGP-GS performs favourably compared to existing approaches when the test set is low-dimensional and few iterations are performed. We test the calibratedness on a synthetic problem, and compare the performance to existing approaches on a large-scale global temperature regression problem.

Percolation is the study of long-range connectivity in randomly connected systems, and has been studied extensively for the last 60+ years, in fact quite intensely at King’s College (Cyril Domb, Michael Fisher, John Essam, M. F. Sykes) in the early days. In this talk I survey some of important results, such as crossing (John Cardy, Gerard Watts), scaling, universality, and exact results. Determination of exact thresholds over a wide range of lattices, using a generalization of the Sykes-Essam result based upon the star-triangle transformation, will be discussed (work with Chris Scullard). Numerical methods (work with Chris Lorenz, Stephen Mertens, Youjin Deng, Mark Newman) will also be discussed. Some recent work on Simply-Connected Compoennts (work with Peter Grassberger) will also be presented.