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​

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.