Diffusion processes are widely used to model the evolution of random values over time. In many applications, the diffusion process is constrained to a finite domain. We consider the estimation problem of a diffusion process constrained by a polytope, i.e. intersection of finitely many (hyper-)planes, given a discretely observed time series data. Since the boundary behaviours of a diffusion process are characterised by its drift and diffusion functions, we derive sufficient conditions on the drift and diffusion functions for the nonattainablity of a polytope. We use deep learning to estimate the drift and diffusion, and ensure that their constraints are satisfied throughout the training.

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. This flow is also related to the classical continuous Heisenberg model in ferromagnetism and to the 1-D cubic Schrödinger equation. In this lecture I will first talk about the state of the art of the binormal flow conjecture, as well as about mathematical methods and results for the binormal flow. Then I will introduce a class of solutions at the critical level of regularity that generate singularities in finite time and describe some of their properties. These results are joint work with Luis Vega.

One of the main questions in the theory of the linear transport equation is whether uniqueness of solutions to the Cauchy problem holds in the case the given vector field is not smooth. We will show that even for incompressible, Sobolev (thus quite “well-behaved”) vector fields, uniqueness of solutions can drastically fail. This result can be seen as a counterpart to DiPerna and Lions’ well-posedness theorem, and, more generally, it can be interpreted as an instance of the “flexibility vs. rigidity” duality, which is a common feature of PDEs appearing in completely different fields, such as differential geometry and fluid dynamics (joint with G. Sattig and L. Székelyhidi). 

The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric.

We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show—in a quantitative sense—that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.

Spacetimes with compact directions play an important role in supergravity and string theory. The simplest such example is the Kaluza-Klein spacetime, where the compact space is a flat torus. An interesting question to ask is whether this spacetime, when viewed as an initial value problem, is stable to small perturbations of initial data. In this talk I will discuss the global, non-linear stability of the Kaluza-Klein spacetime to toroidal-independent perturbations and the particular nonlinear structure appearing in the associated PDE system.

The propagation of high-frequency electromagnetic waves can be analyzed using the geometrical optics approximation. In the case of large but finite frequencies, the geometrical optics approximation is no longer accurate, and polarization-dependent corrections at first order in wavelength modify the propagation of light in an inhomogenous medium via a spin-orbit coupling mechanism. This effect, known as the spin Hall effect of light, has been experimentally observed. In this talk I will discuss recent work which generalizes the spin Hall effect to the propagation of light and gravitational waves in inhomogenous spacetimes. This is based on joint work with Marius Oancea and Jeremie Joudioux.

Precision predictions for binary mergers are essential for the nascent field of gravitational wave astronomy. The initial inspiral part can be treated perturbatively. We present results for the post-Minkowskian expansion of conservative binary dynamics, previously available only at the 2nd order for several decades, at the 3rd and 4th orders in the expansion. Our calculations are based on quantum field theory and use powerful methods developed in the modern scattering amplitudes program, as well as loop integration techniques developed for precision collider physics. Furthermore, we take initial steps in calculating radiative binary dynamics and obtain analytically the total radiated energy in hyperbolic black hole scattering, at the lowest order in G but all orders in velocity.