Week 23.02.2026 – 01.03.2026

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​

The Theory of Dynamical Systems studies how systems evolve over time,
often through differential equations or iterated transformations. Even
deterministic systems governed by simple laws can display chaotic
behaviour, marked by extreme sensitivity to initial conditions — tiny
changes in the starting state may lead to dramatically different
outcomes. Using the Lorenz attractor as an example, we illustrate this
unpredictability, popularly known as the butterfly effect. We then
show how a probabilistic approach provides the appropriate framework
for analysing chaotic systems, offering the right tools to make
meaningful predictions precisely where a purely deterministic
description proves inadequate.

In the area of combinatorics known as ‘permutation patterns’, seemingly innocuous questions can conceal a surprising degree of difficulty, giving rise to combinatorial problems that range from trivial to unsolved (despite decades of work). In this talk, we will explore several ways in which permutation pattern questions interface with probability — sometimes giving rise to probabilistic processes, sometimes illuminated by probabilistic reasoning. Some of these will be generalizations of known processes, others new, and still others conjectural.



Based on forthcoming joint work with Slim Kammoun and Einar Steingrimsson.

In the area of combinatorics known as ‘permutation patterns’, seemingly innocuous questions can conceal a surprising degree of difficulty, giving rise to combinatorial problems that range from trivial to unsolved (despite decades of work). In this talk, we will explore several ways in which permutation pattern questions interface with probability — sometimes giving rise to probabilistic processes, sometimes illuminated by probabilistic reasoning. Some of these will be generalizations of known processes, others new, and others still conjectural.

Based on forthcoming joint work with Slim Kammoun and Einar Steingrimsson.

Integrability in planar N=4 SYM has led to the development of the cutting-edge Quantum Spectral Curve (QSC), a Riemann-Hilbert problem for a handful of Q-functions which encode the spectrum of conformal dimensions. With the QSC the planar N=4 SYM spectral problem is solved - anyone with a laptop can compute the dimension of any operator at any coupling. In a handful of examples, structure constants have also been
shown to simplify enormously when expressed in terms of the QSC Q-functions, but no systematic derivation is available even at tree level.
Using recent advancements in the Separation of Variables program for high-rank integrable systems I will explain how to systematically obtain
the Q-function representation for tree-level structure constants in the SU(4) sector. Based on upcoming work with T. Bargheer, C. Bercini, and G. Lefundes.

Near-extremal black holes with an AdS2 throat are of great interest in string theory and in holography due to their ubiquity as classical gravitational saddles. The emergent SL(2,R) symmetry associated to the throat plays an important role in their low-temperature physics. In this talk, I will describe recent progress on analytically computing holographic correlators in black hole spacetimes with a near-extremal AdS2xR2 near-horizon geometry, focusing on current-current and shear correlators. By improving on previous matching calculations, I will show how it is possible to obtain analytical approximations to the correlators that interpolate between the hydrodynamic (frequencies small compared to the temperature) and the non-hydrodynamic low-temperature regimes. The expressions we obtain capture the hydrodynamic poles, the gapped poles controlled by the SL(2,R) symmetry of the AdS2 throat and the successive collisions with them. I will also comment on the appearance of zero temperature gapless poles in the spectrum and their relation to the SL(2,R) spectrum.

See: https://www.londonmathfinance.org.uk/seminar

See: https://www.londonmathfinance.org.uk/seminar

In this talk we shall discuss the status of local-global principles for semi-integral points on orbifold pairs of Markoff type. If time permits, we will discuss a way to count Markoff orbifold pairs which satisfy the semi-integral Hasse principle while the corresponding Markoff surface lacks integral points. This talk is based on a joint work with Justin Uhlemann.

We provide a new perspective on Takens' Theorem from the theory of dynamical systems to investigate the importance of path dependent modelling in Finance and Machine Learning. We also provide some insight on the construction of architectures from the point
of view of invariant theory.