This week

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​

Classical maps are known as models with discrete time which enable one to explore a broad range of dynamics (from integrable to strongly chaotic). One famous example in the chaotic case is provided by the cat maps introduced by V. I. Arnold and A. A. Avez. It is a linear map on the two-dimensional torus. Recently a chain of coupled cat maps was introduced to model extremal black holes. We will remind the definition of the model and show how it can be used in a statistical physics perspective. We can define and exactly solve the diffusion problem and derive the diffusion coefficient from the microscopic dynamics. If time allows I will give some elements how to quantise the model.

For sink-free orientations in graphs of minimum degree at least 3, we present a marginal sampling algorithm for the status of a single vertex. With this algorithm, we present polynomial-time algorithms to deterministically approximate and randomly approximate the corresponding partition function. We also remark on connections to the independence polynomial.

We propose a microstructural model for the order flow in financial markets that distinguishes between exogenous core orders and endogenous reaction flow, both modeled as Hawkes processes. This model has natural scaling limits that reconcile a number of robust yet apparently contradictory properties of empirical data: persistent signed order flow, rough trading volume and volatility, and power-law market impact. In our framework, all these quantities are pinned down by a single statistic $H_0$, which measures the persistence of the core flow. The signed flow converges to the sum of a fractional process with Hurst index $H_0$ and a martingale, while traded volume is a rough process with Hurst index $H_0-1/2$. From these results, no-arbitrage constraints enable us to deduce that volatility is rough, with Hurst parameter $2H_0-3/2$, and that the price impact of trades is power law with exponent $2-2H_0$. With $H_0 \approx 3/4$, this model is not only consistent with the square-root law of market impact but matches remarkably well empirical estimates for signed order flow, unsigned volume and volatility.

Collaborative work is widely valued in education, spanning group projects, presentations, and research interactions. While students are often given definitions of what successful collaboration should look like, they rarely receive guidance on how to develop the skills to engage meaningfully. This can lead to superficial or frustrating experiences. In this talk, I present how theory-based approaches, including Boundary Crossing and Dialogical Inquiry, can help us understand the social and cognitive processes underpinning collaboration. Viewing collaborative learning through these lenses highlights how reflective dialogue and knowledge co-construction can be fostered, transforming collaborative activities into meaningful and rewarding mathematical experiences. These skills, including the ability to collaborate across disciplines, are vital for students’ development within and beyond academia.

The linear growth of entanglement after a quench from a state with short-range correlations is a universal feature of many body dynamics. It has been shown to occur in integrable and chaotic systems undergoing either Hamiltonian, Floquet or circuit dynamics and has also been observed in experiments. The entanglement dynamics emerging from long-range correlated states is far less studied, although no less viable using modern quantum simulation experiments. In this talk, I will present the dynamics of the bipartite entanglement entropy in quenches starting from Crosscap States, also knows as Entangled Antipodal Pair States, and one possible extension dubbed Entangled Mutlipodal States.
These are volume law states, constructed by entangling 2 or more equidistant points of a finite and periodic system. I will focus on the evolution of these initial states, in a free fermionic quench and probe the dynamics of bipartite entanglement entropy. In particular, I will show how one can derive an effective description of the entanglement dynamics, that matches the exact results. The quench dynamics is captured by an emergent quasiparticle picture description, which differs from the one that characterizes quenches from lowly entangled states, due to the long-range correlations of the initial states. For small enough subsystems, the entanglement entropy, initially remains to the volume law value and then there is a linear in time decrease, followed by a series of oscillatory revivals which happen around a constant value. For larger subsystems entanglement may increase, showcasing more intricate dynamics, where one can also retrieve physics of lowly entangled state quenches.

I will discuss scattering on the Coulomb branch of planar N=4 SYM at finite ’t Hooft coupling. This describes a family of classical open-string S-matrices that smoothly interpolates between perturbative parton scattering at weak coupling and flat-space string scattering at strong coupling. I will focus on the four-point amplitude and discuss its remarkably rich structure: nonlinear Regge trajectories, dual conformal invariance, an intricate spectrum of bound states with an accumulation point, and a two-particle cut. Using dispersion relations and S-matrix bootstrap techniques, these properties can be incorporated to constrain the amplitude at finite ’t Hooft coupling, and I will discuss bounds on Wilson coefficients, couplings to bound states, and the overall shape of the amplitude.

This is based on https://arxiv.org/abs/2510.19909.

Latent position models are widely used in statistical network analyses, with applications in the literature spanning biology, finance, computing, politics, and social interactions, amongst others. The theory of estimation of the latent positions is well developed in the literature, but uncertainty quantification has largely centred on asymptotics. For applications requiring critical decisions such as cyber-security, reliably quantifying uncertainty in practical, finite-sample settings could open up a range of new, network-wide analytical techniques; for example, in anomaly detection, nodes might be flagged not just for having clearly outlying estimated latent positions, but from apparent inliers actually being difficult to characterise; or in changepoint detection, assessing behavioural shifts from time-varying embedding paths. This talk explores the challenges in taking a Bayesian approach, focusing on a flexible indefinite dot product graph model.