In this talk I will review some of the key ideas behind the study of entanglement measures in 1+1D quantum field theories employing the so-called branch point twist field approach. This method is based on the existence of a one-to-one correspondence between different entanglement measures and different multi-point functions of a particular type of symmetry field. It is then possible to employ standard methods for the evaluation of correlation functions to understand properties of entanglement in bipartite systems. Time permitting, I will then present a recent application of this approach to the study of a new entanglement measure: the symmetry resolved entanglement entropy.

The spread of a matrix is defined as the diameter of its spectrum. In this talk, we consider the problem of maximizing the spread of a symmetric non-negative matrix with bounded entries and discuss a number of recent results. This optimization problem is closely related to a pair of conjectures in spectral graph theory made by Gregory, Kirkland, and Hershkowitz in 2001, which were recently resolved by Breen, Riasanovsky, Tait, and Urschel. This talk will give a light overview of the approach used in this work, with a strong focus on ideas, many of which can be abstracted to more general matrix optimization problems.

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

The possibility of `tacit collusion', in which interactions across market-making algorithms lead to an outcome similar to collusion among market makers, has increasingly received regulatory scrutiny. 
    We model the interaction of market makers in a dealer market as a stochastic differential game of intensity control with partial information and study the resulting dynamics of bid-ask spreads. Competition among dealers is modeled as a Nash equilibrium, which we characterise in terms of a system of coupled Hamilton-Jacobi-Bellman (HJB) equations, while Pareto optima correspond to collusion. 
    Using a decentralized multi-agent deep reinforcement learning algorithm to model how competing market makers learn to adjust their quotes, we show how the interaction of market-making algorithms may lead to tacit collusion with spread levels strictly above the competitive equilibrium level, without any explicit sharing of information.
 

The applied, numerical and asymptotic analysis of acoustic, electromagnetic and elastic scattering by smooth scatterers (e.g. a cylinder or a sphere) is a classical topic in applied mathematics. However, many real-world applications involve highly non-smooth scatterers with geometric structure on multiple length scales. Examples include acoustic scattering by trees and other vegetation in the modelling of urban noise propagation, electromagnetic scattering by snowflakes and ice crystal aggregates in climate modelling and weather prediction, and elastic scattering by cracks and other interfaces in seismic imaging and hydrocarbon exploration. In such situations it may be more appropriate to model the scatterer not by a smooth surface but by a fractal, a geometric object with self-similarity properties and detail on every length scale. Well-known examples include the Cantor set, Sierpinski triangle and the Koch snowflake. In this talk I will give an overview of our recent research into acoustic scattering by such fractal structures. So far our work has focussed on establishing well-posedness of the scattering problem and integral equation reformulations of it, and developing and analysing numerical methods for obtaining approximate solutions. However, there remain interesting open questions about the high frequency (short wavelength) asymptotic behaviour of solutions, and whether the self-similarity of the scatterer can be exploited to derive more efficient approximation techniques.

Many microorganisms must navigate strange biological environments whose physics are unique and counter-intuitive, with wide-ranging consequences for evolutionary biology and human health. Mucus, for instance, behaves like both a fluid and an elastic solid. This can affect locomotion dramatically, which can be highly beneficial (e.g. for mammalian spermatozoa swimming through cervical fluid) or extremely problematic (e.g. the Lyme disease spirochete B. burgdorferi swimming through the extracellular matrix of human skin). Mathematical modeling and numerical simulations continue to provide new fundamental insights about the biological world in and around us and point toward new possibilities in biomedical engineering. These complex fluid phenomena can either enhance or retard a microorganism's swimming speed, and can even change the direction of swimming, depending on the body geometry and the properties of the fluid. We will discuss analytical and numerical insights into swimming through model viscoelastic (Oldroyd-B) and liquid-crystalline (Ericksen-Leslie) fluids, with a special focus on the important and in some cases dominant roles played by the presence of nearby boundaries.