We present a new dynamical way of establishing local laws for sparse random matrices, the Bernoulli flow method. It is based on a Markovian jump process, where the entries of the matrix jump independently from 0 to 1 at rate one. As an application, we show optimal (up to a constant) isotropic delocalisation for bulk eigenvectors of Erdös-Rényi graphs with edge probability p \geq (log N)^2/N. In the same regime, we obtain a local law with optimal (up to a constant) error bounds. Joint work with Antti Knowles.

The characterisation of the physics of the M5-brane remains an important open problem in string theory. While the superconformal field theory that resides on a planar M5-brane in flat space is poorly understood, other configurations involving M5-branes wrapped on certain manifolds have well-known superconformal field theory descriptions, including N=2 class S field theories. In this talk, I will use new methods based on exceptional generalised geometry to describe the gravity duals of class S field theories, compute a universal sector of their light-operator spectrum, and provide, for the first time, a holographic match of their superconformal index.

Drug discovery is slow and expensive, and a growing body of AI work tackles this by training generative models that propose new candidate molecules directly, searching chemical space far faster than a human chemist could. Most of this work has focused on standard small molecules, leaving more specialized but valuable classes underexplored.

Macrocycles are ring-shaped molecules that offer a promising alternative to small-molecule drugs due to their enhanced selectivity and binding affinity against difficult targets. Despite their chemical value, they remain underexplored in generative modeling, likely owing to their scarcity in public datasets and the challenges of enforcing topological constraints in standard deep generative models.

We introduce MacroGuide: Topological Guidance for Macrocycle Generation, a diffusion guidance mechanism that uses Persistent Homology to steer the sampling of pretrained molecular generative models toward the generation of macrocycles, in both unconditional and conditional (protein pocket) settings. At each denoising step, MacroGuide constructs a Vietoris-Rips complex from atomic positions and promotes ring formation by optimizing persistent homology features. Empirically, applying MacroGuide to pretrained diffusion models increases macrocycle generation rates from 1% to 99%, while matching or exceeding state-of-the-art performance on key quality metrics such as chemical validity, diversity, and PoseBusters checks.

Accepted to ICML 2026. Paper: https://arxiv.org/abs/2602.14977