In-Person Talk 

After reviewing recent progress in random Hermitian and non-Hermitian matrix theory, we prove that eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation Hypothesis (ETH), which is a strong form of Quantum Unique ergodicity (QUE) with optimal speed of convergence. This requires proving certain multi-resolvent local laws established via a novel bootstrap scheme. Additionally, we also prove that the fluctuations in QUE are universal, relying on energy methods for multi-particle Dyson Brownian motion.