Week 16.02.2026 – 22.02.2026

Energy conditions were originally formulated as pointwise bounds on contractions of the stress–energy tensor and have played a central role as assumptions in many foundational results of classical general relativity, most notably the singularity theorems. However, these conditions are generically violated by quantum fields, which admit states with locally negative energy density. Such violations are nevertheless constrained: quantum energy inequalities impose bounds on the magnitude and duration of negative energy.

In this course, I will first introduce the classical energy conditions and review their physical motivation and known violations. Then I will provide a brief introduction to quantum field theory on curved spacetimes and demonstrate how quantum energy inequalities can be derived. Finally, I will discuss in detail the average null energy condition and the limitations it imposes to causality violating spacetimes.

Course plan:
Lecture 1: Classical energy conditions and their violations
Lecture 2: Quantum field theory on curved spacetimes
Lecture 3: A derivation of a quantum energy inequality
Lecture 4: The average null energy condition​

Consider the two-dimensional lattice Z^2 as a graph, where edges connect neighbouring vertices. A six-vertex configuration is an orientation of the edges satisfying the ice rule: at each vertex, exactly two edges point in and two point out. This terminology originates from the interpretation of the six-vertex model as a statistical model of ice formation.

In this talk, we will study random six-vertex configurations sampled from a natural probability distribution. The main result of this work is a new determinantal formula for correlation functions of the model. The proof relies on a bijection between six-vertex configurations and ensembles of non-intersecting lattice paths, which allows the correlations to be expressed in terms of determinants.

This is joint work with Samuel G. G. Johnston.

We revisit the problem of portfolio selection, where an investor maximizes utility subject to a risk constraint. Our framework is very general and accommodates a wide range of utility and risk functionals, including non-concave utilities such as S-shaped utilities from prospect theory and non-convex risk measures such as Value at Risk.
Our main contribution is a novel and complete characterization of well-posedness for utility-risk portfolio selection in one period that takes the interplay between the utility and the risk objectives fully into account. We show that under mild regularity conditions the minimal necessary and sufficient condition for well-posedness is given by a very simple either-or criterion: either the utility functional or the risk functional need to satisfy the axiom of sensitivity to large losses. This allows to easily describe well-posedness or ill-posedness for many utility/risk pairs, which we illustrate by a large number of examples. The talk is based on joint work with Leonardo Baggiani and Nazem Khan.

The act of assessing others' work against a markscheme can be a useful process in developing one's own understanding of a topic. For this reason, we might consider introducing peer assessment as a component of some of our modules. Last term I trialled the use of the tool "FeedbackFruits", which is available within Moodle, for the assessment of two written courseworks on the first year module MATH0008 (Applied Mathematics 1). I will show what I have learnt from the process, and hopefully give helpful tips to anyone else considering following in my footsteps.

Gene regulatory networks can be modelled as nonlinear dynamical systems whose trajectories encode sequences of gene expression directing biological programmes. Function is determined by the geometry of these trajectories in phase space, which specifies the ordered progression of gene states required for correct developmental outcomes (such as patterning or limb formation). Closely related networks may share similar geometric structure while exhibiting different temporal behaviour, giving rise to distinct functional timescales defined by transient dynamics such as oscillation periods or relaxation times rather than steady states. Parameter perturbations typically affect geometry and timing in coupled and unpredictable ways, complicating comparison between systems and targeted modulation of tempo. We introduce a framework that characterises functionality via equivalence classes of orbits, inducing a distance between parametrised systems that separates temporal reparameterisation from geometric deformation. This enables identification of parameter directions that modulate functional timescale while preserving the underlying dynamical landscape.

I will discuss thermal correlators in holographic CFTs. Using the operator product expansion, one can isolate a sector of the correlator which exhibits singularities. Some of these singularities are associated with the singularities of dual asymptotically AdS black holes, providing a useful window into the black hole interior.

Normal Computing is developing hardware to accelerate stochastic differential equations (SDEs). In this talk, I'll focus on a particular novel SDE discretisation that is hardware-friendly in the sense that each iteration uses 1-2 bit random increments rather than full precision. This lattice random walk discretisation differs significantly from traditional methods such as Euler-Maruyama and in particular offers favourable mathematical properties (e.g. robustness to gradient noise) as well as avenues for hardware co-design.

The paper link is https://arxiv.org/abs/2508.20883


If time permits I may also spend 10 minutes or so on an additional related paper on a complete characterisation of SDEs (https://arxiv.org/abs/2601.07834).