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.

]]>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.

]]>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.

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.

]]>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.

]]>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.

Title: Finite time blow up for focusing supercritical NLS and compressible fluids

Abstract: I will present recent results in collaboration with Frank Merle, Pierre Raphaël and Igor Rodnianski concerning finite time blow up for the focusing supercritical NLS equation, for compressible Euler equations, and for compressible Navier-Stokes equations.

]]>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.

Title: Global stability for nonlinear wave equations with multi-localized initial data.

Abstract: The classical global existence theory for nonlinear wave equations requires initial data to be small and localized around a point. In this work, we initiate the study of the global stability of nonlinear wave equations with non localized data.

In particular, we extend the classical theory to data localized around several points. This is achieved by generalizing the vector field method to the multi-localized case.

The core of our argument lies in a close inspection of the geometry of two interacting waves emanating from different localized sources. We show trilinear estimates to control such interaction, by means of a physical space method. This is joint work with John Anderson (Princeton University).