Triangle Seminars

Abstract:
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​

Abstract:
The era of high-precision gravitational data requires a demanding new level of precision in gravitational theory. In this talk I will discuss the use of methods from quantum field theory to address the challenge. These methods reframe our theoretical approach to gravity, making gravity look like two copies of electrodynamics. We will see that the Schwarzschild metric is a "double copy" of the Coulomb charge, and that gravitational waveforms can be determined without ever using the Einstein equation.

Abstract:
This 14-week course explores how modern AI techniques can support mathematical research and problem-solving. It is not only an introduction to AI, but also a survey of its applications in math discovery, illustrated through research papers and coding tasks.

Course topics include:

- AI-assisted coding and reasoning. Running massive math experiments with the help of coding assistants. Using LLMs as research assistants.
- Training and using AI models. From linear models to neural networks: when AI models are useful and when they are not; data collection; model training.
- Transformers and LLMs. Mathematical data as sequences: from individual formulas to statements and proofs. The use of transformers and LLMs.
- Searching for examples and counterexamples. Speeding up search in huge search spaces. Algorithm optimization with AlphaEvolve and its open source analogs. Reinforcement Learning.

The course will be taught at the London Institute for Mathematical Sciences over 14 weeks, starting on February 25, with sessions held on Wednesdays from 10:00 am to 1:00 pm.

The programme is open to mathematicians and theoretical physicists, with proficiency in Python as a strong prerequisite.
For registration and more information on the course and Nebius Academy, visit: https://academy.nebius.com/ai-for-math?utm_source=lims&utm_medium=email&utm_campaign=limsnewsletter

Abstract:
TBA

Abstract:
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.

Abstract:
I will present two recent results about supersymmetric quantum mechanics with 2 supercharges (1d N=2). First, I will explain new Seiberg-like dualities between 1d SQCD-like theories, which are distinct 1d gauge theories with the same supersymmetric ground states. Then I will discuss the coupling of 1d N=2 quivers to 3d N=2 gauge theories in the context of the GLSM/Quantum K-theory correspondence, explaining how such line defects engineer interesting objects (the so-called Schubert classes) in the quantum K-theory of partial flag varieties.