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, 3/8, will be given by John Anderson online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

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.

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, 2/8, will be given by Kihyun Kim online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

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.

The APDE seminar on Monday, 11/30, will be given by Khang Manh Huynh online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Construction of the Hodge-Neumann heat kernel, local Bernstein estimates, and Onsager’s conjecture in fluid dynamics.

Abstract: Most recently, in arXiv:1907.05360 [math.AP], we introduced the theory of heatable currents and proved Onsager’s conjecture on Riemannian manifolds with boundary, where the weak solution has $B_{3,1}^{\frac{1}{3}}$ spatial regularity. In this sequel, by applying techniques from geometric microlocal analysis to construct the Hodge-Neumann heat kernel, we obtain off-diagonal decay and local Bernstein estimates, and then use them to extend the result to the Besov space $\widehat{B}_{3,V}^{\frac{1}{3}}$, which generalizes both the space $\widehat{B}_{3,c(\mathbb{N})}^{1/3}$ from arXiv:1310.7947 [math.AP] and the space $\underline{B}_{3,\text{VMO}}^{1/3}$ from arXiv:1902.07120 [math.AP] — the best known function space where Onsager’s conjecture holds on flat backgrounds.

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, 11/16, will be given by Shuang Miao online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: The stability of blow up solutions to critical Wave Maps beyond equivariant setting

Abstract: In 2006, Krieger, Schlag and Tataru (KST) constructed a family of type II blow up solutions to the 2+1 dimensional wave map equation with unit sphere as its target. This construction provides the first example of blow up solutions to the energy-critical Wave Maps. A key feature of this family is that it exhibits a continuum of blow up rates. However, from the way it was constructed, the stability of this family was not clear and it was believed to be non-generic. In this talk I will present our recent work on proving the stability and rigidity of the KST family, beyond the equivariant setting. This is based on joint works with Joachim Krieger and Wilhelm Schlag.

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.