The APDE seminar on Monday, 5/10, will be given by Jeffrey Galkowski online via Zoom from 11.10am to 12.00pm PT (note the time change). To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Exponential accuracy for the method of perfectly matched layers.

Abstract: The method of perfectly matched layers (PML) is used to compute solutions to time harmonic wave scattering problems. This method can be seen as a numerical adaptation of the method of complex scaling in which the infinite exterior region is replaced by a Dirichlet condition on a finite region. In this talk, we recall the methods of complex scaling and PML and study the error produced by replacing the genuine scattering problem with the PML truncation. We show that this error decays exponentially as a function of the scaling angle, the scaling width, and the frequency. Based on joint work with E. Spence and D. Lafontaine.

The APDE seminar on Monday, 5/3, will be given by Elena Giorgi online via Zoom from 4.10pm to 5.00pm PT. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: The stability of charged black holes.

Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

The APDE seminar on Monday, 4/19, will be given by Namaluba Malawo online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Resonances for thin barriers on the half-line.

Abstract: The analysis of scattering by thin barriers is important for many physical problems, including quantum corrals. To model such a barrier, we use a delta function potential on the half-line. Our main results compute decay rates for particles confined by this barrier. The decay rates are given by imaginary parts of resonances. We show that they energy dependence of the decay rates is logarithmic when the barrier is weaker and polynomial when the barrier is stronger. To compute them, we derive a formula for resonances in terms of the Lambert W function and apply a series expansion. Joint work with Kiril Datchev.

The APDE seminar on Monday, 4/5, will be given by Maciej Zworski online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Internal waves and homeomorphism of the circle.

Abstract: The connection between the formation of internal waves in fluids and
the dynamics of homeomorphisms of the circle was investigated by
oceanographers in the 90s and resulted in novel experimental
observations (Maas et al, 1997). The specific homeomorphism is given
by a chess billiard” and has been considered by many authors (John
1941, Arnold 1957, Ralston 1973, … , Lenci et al 2021). The relation
between the nonlinear dynamics of this homeomorphism and linearized
internal waves provides a striking example of classical/quantum
correspondence (in a classical and surprising setting of fluids!) and,
using a model of tori and of zeroth order pseudodifferential
operators, it has been a subject of recent research, first by Colin de
Verdière-Saint Raymond 2020 and then by Dyatlov, Galkowski, Wang and
the speaker. In these works, many facets of the relationship between
hyperbolic sources and sinks for the classical dynamics and internal
waves in fluids were explained. I will present some of these results
as well as some numerical discoveries (including those of
Almonacid-Nigam 2020). I will also describe various open problems.

The APDE seminar on Monday, 3/29, will be given by Larry Guth online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Local smoothing for the wave equation.

Abstract: The local smoothing problem asks about how much solutions to the wave equation can focus. It was formulated by Chris Sogge in the early 90s. Hong Wang, Ruixiang Zhang, and I recently proved the conjecture in two dimensions. In this talk, we will build up some intuition about waves to motivate the conjecture, and then discuss some of the obstacles and some ideas from the proof.

The APDE seminar on Monday, 2/22, will be given by Alexis Drouot online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Mathematical aspects of topological insulators.

Abstract: Topological insulators are intriguing materials that block conduction in their interior (the bulk) but support robust asymmetric currents along their edges. I will discuss their analytic, geometric and topological aspects using an adiabatic framework favorable to quantitative predictions.

The APDE seminar on Monday, 11/23, will be given by Grigorios Fournodavlos online via Zoom from 9:10 to 10am (note the time change). To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Asymptotically Kasner-like singularities.

Abstract: The Kasner metric is an exact solution to the Einstein vacuum equations, containing a Big Bang singularity. Examples of more general singularities in the vicinity of Kasner are in short supply, due its complicated dynamics. I will present a recent joint work with Jonathan Luk, which constructs a large class of singular solutions with Kasner-like behavior, without symmetry or analyticity assumptions.

The APDE seminar on Monday, 11/09, will be given by Jared Speck online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Stable Big Bang formation in general relativity: The complete sub-critical regime.

Abstract: The celebrated theorems of Hawking and Penrose show that under appropriate assumptions on the matter model, a large, open set of initial data for Einstein’s equations lead to geodesically incomplete solutions. However, these theorems are “soft” in that they do not yield any information about the nature of the incompleteness, leaving open the possibilities that i) it is tied to the blowup of some invariant quantity (such as curvature) or ii) it is due to a more sinister phenomenon, such as incompleteness due to lack of information for how to uniquely continue the solution (this is roughly known as the formation of a Cauchy horizon). Despite the “general ambiguity,” in the mathematical physics literature, there are heuristic results, going back 50 years, suggesting that whenever a certain “sub-criticality” condition holds, the Hawking–Penrose incompleteness is caused by the formation of a Big Bang singularity, that is, curvature blowup along an entire spacelike hypersurface. In recent joint work with I. Rodnianski and G. Fournodavlos, we have given a rigorous proof of the heuristics. More precisely, our results apply to Sobolev-class perturbations – without symmetry – of generalized Kasner solutions whose exponents satisfy the sub-criticality condition. Our main theorem shows that – like the generalized Kasner solutions – the perturbed solutions develop Big Bang singularities. In this talk, I will provide an overview of our result and explain how it is tied to some of the main themes of investigation by the mathematical general relativity community, including the remarkable work of Dafermos–Luk on the stability of Kerr Cauchy horizons. I will also discuss the new gauge that we used in our work, as well as intriguing connections to other problems concerning stable singularity formation.

The APDE seminar on Monday, 10/19, will be given by Maxime van de Moortel online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Nonlinear interaction of three impulsive gravitational waves for the Einstein equations.

Abstract: An impulsive gravitational wave is a weak solution of the Einstein vacuum equations whose metric admits a curvature delta singularity supported on a null hypersurface; the spacetime is then an idealization of a gravitational wave emanating from a strongly gravitating source. In the presence of multiple sources, their impulsive waves eventually interact and it is interesting to study the spacetime up to and after the interaction. For such singular solutions, the classical well-posedness results (such as the bounded L^2 curvature theorem) are not applicable and it is not even clear a priori whether the initial regularity propagates or a worse singularity occurs from the nonlinear interaction. I will present a local existence result for U(1)-polarized Cauchy data featuring three impulsive gravitational waves of small amplitude propagating towards each other. The proof is achieved with the help of localization techniques inspired from Christodoulou’s short pulse method and new tools in Harmonic Analysis, notably anisotropic estimates that are tailored to the problem. This is joint work with Jonathan Luk.

The APDE seminar on Monday, 11/02, will be given by Bjoern Bringmann online via Zoom from 4:10 to 5pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity.

Abstract. In this talk, we discuss the construction and invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree-nonlinearity. In the first part of the talk, we construct the Gibbs measure and examine its properties. We discuss the mutual singularity of the Gibbs measure and the so-called Gaussian free field. In contrast, the Gibbs measure for one or two-dimensional wave equations is absolutel continuous with respect to the Gaussian free field. In the second part of the talk, we discuss the probabilistic well-posedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. At the moment, this is the only theorem proving the invariance of any singular Gibbs measure under a dispersive equation.