**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.

**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.

**Title:** Oscillations in wave map systems and an application to General Relativity

**Abstract: **Due to their nonlinear nature, the Einstein equations are not closed under weak convergence. Compactness singulaties associated to highly oscillatory solutions may be identified with some non-trivial matter. In 1989, Burnett conjectured that, for vacuum sequences, this matter produced in the limit is captured by the Einstein-massless Vlasov model.

In this talk, we give a proof of Burnett’s conjecture under some gauge and symmetry assumptions, improving previous work by Huneau—Luk from 2019. Our methods are more general, and apply to oscillating sequences of solutions to the wave maps equation in (1+2)-dimensions.

This is joint work with André Guerra (University of Oxford).

**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.

**Title:** A stochastic fluid-structure interaction model given by a stochastic viscous wave equation

**Abstract: **We consider a stochastic fluid-structure interaction (FSI) model, given by a stochastic viscous wave equation perturbed by spacetime white noise. The wave equation part of the model describes the elastodynamics of a thin structure, such as an elastic membrane, while the viscous part, which is in the form of the Dirichlet-to-Neumann operator, describes the impact of a viscous, incompressible fluid in a two-way coupled fluid-structure interaction problem. The stochastic perturbation describes random deviations observed in real-life data. We prove that this stochastic viscous wave equation has a mild solution in dimension one, and also in dimension two, which is the physical dimension of the FSI problem (thin 2D membrane). This behavior contrasts that of the stochastic heat and the stochastic wave equations, which do not have function valued mild solutions in dimensions two and higher. This means that in the two dimensional model, unlike the heat and wave equations, dissipation due to fluid viscosity in the viscous wave equation, keeps the stochastically perturbed solution “in control”. We also consider Hölder continuity path properties of solutions and show that the solution is Hölder continuous up to Hölder exponent 1/2 in both space and time, after stochastic modification. This is joint work with Suncica Canic.

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.

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.

Title: Stability results for anisotropic systems of wave equations

Abstract: In this talk, I will describe a global stability result for a nonlinear anisotropic system of wave equations. This is motivated by studying phenomena involving characteristics with multiple sheets. For the proof, I will describe a strategy for controlling the solution based on bilinear energy estimates. Through a duality argument, this will allow us to prove decay in physical space using decay estimates for the homogeneous wave equation as a black box. The final proof will also require us to exploit a certain null condition that is present when the anisotropic system of wave equations satisfies a structural property involving the light cones of the equations.

]]>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.

]]>Title: Blow-up dynamics for the self-dual Chern-Simons-Schrödinger equation

Abstract: We consider the blow-up dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) under equivariance symmetry. (CSS) is $L^2$-critical, has the pseudoconformal symmetry, and admits a soliton $Q$ for each equivariance index $m \geq 0$. An application of the pseudoconformal transformation to $Q$ yields an explicit finite-time blow-up solution $S(t)$ which contracts at the pseudoconformal rate $|t|$. In the high equivariance case $m \geq 1$, the pseudoconformal blow-up for smooth finite energy solutions in fact occurs in a codimension one sense, but also exhibits an instability mechanism. In the radial case $m=0$, however, $S(t)$ is no longer a finite energy blow-up solution. Interestingly enough, there are smooth finite energy blow-up solutions whose blow-up rates differ from the pseudoconformal rate by a power of logarithm. We will explore these interesting blow-up dynamics (with more focus on the latter) via modulation analysis. This talk is based on my joint works with Soonsik Kwon and Sung-Jin Oh.

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