For centuries, we have explored the Universe and discovered novel astrophysical sources and phenomena through the observation of electromagnetic waves. In 2015, we observed the first gravitational wave passing through the Earth produced by the merger of two black holes. Since then, a few hundred gravitational waves have been detected, including the gravitational signal produced by the coalescence of two neutron stars, accompanied by a plethora of electromagnetic counterparts observed around the world. In this talk, I will highlight the theoretical work underpinning these observations and review the main results in astrophysics, cosmology and fundamental physics. I will then discuss the discovery potential of future observatories in space and on the ground.
Following an introduction to general relativity, the Einstein field equations, and an overview of recent results on the stability of black holes, I will discuss work in progress towards the construction of singular perturbations of spacetimes via the insertion of small black holes. This aims to provide the first rigorous examples of spacetimes describing the merger of two black holes with extreme mass ratios.
Stochastic analysis makes precise the intuition that paths of Markov diffusions look like Brownian motion. The struggle for similar precision applied to a larger class of random fields led to the invention of controlled paths and regularity structures and opened the way to a new approach to Euclidean quantum field theory. However, despite a lot of progress, it feels that we are still at the infancy of this rich field where analysis, geometry and algebra meet with probability theory in new ways. I will try to name few problems which I think are important to our deeper understanding of these connections.
