MIT PDE/Analysis Seminar

Abstract:In weakly nonlinear dispersive wave systems, long-time dynamics are typically governed by time resonances, where wave phases evolve coherently due to exact frequency matching. Recent advances in measuring the spatio-temporal spectrum, however, reveal prominent excitations beyond those predicted by time resonances. In this talk, I will present an alternative mechanism: space resonances. These occur when wave packets share the same group velocity and remain co-located, producing long-lived interactions. I will illustrate these ideas in the context of surface gravity waves, where triadic interactions occur without exact three-wave time resonances.

This talk is based on collaborations with Fabio Pusateri, Jalal Shatah, Yulin Pan, Miguel Onorato, Tristan Backmaster and Yongji Wang.

Abstract: In this talk we will review past and recent results pertaining to the emerging field of integrable space-time nonlocal nonlinear evolution equations. In particular, we will discuss blow-up in finite time of soliton solutions as well as the physical derivations of many integrable nonlocal systems.

Abstract: We will talk about recent work establishing a quantitative nonlinear scattering theory for asymptotically de Sitter solutions of the Einstein vacuum equations in $(n+1)$ dimensions with $n\geq4$ even, which are determined by small scattering data at future infinity and past infinity. We will also explain why the case of even spatial dimension $n$ poses significant challenges compared to its odd counterpart and was left open by the previous works in the literature.

Abstract: There is a hierarchy of regularity for continuous $\mathbb{Z}^n$-periodic functions in $\mathbb{R}^n$, $C^0\supset C^1\supset \cdots \supset C^\infty \supset$ analytic $\supset$ trigonometric polynomial, and the decay of the Fourier coefficients pre- cisely reflects this regularity. In particular, the support supp(f̂) is finite if and only if $f$ is a trigonometric polynomial. Periodic hypersurfaces in $\mathbb{R}^n$ exhibit a similar regularity hierarchy, but there is no analogous Fourier description.

In this talk, I will present a joint work with Mario Kummer in which we provide a sufficient Fourier-criterion for a $C^{1+\epsilon}$ peri- odic hypersurface $\Sigma\subset\mathbb{R}^n$ to be the zero set of a trigonomet- ric polynomial of the form $p(e^{2\pi i x_{1}},\ldots,e^{2\pi i x_{n}})$ with $p$ Lee–Yang polynomial.

The criterion can be stated using a recent notion introduced by Yves Meyer: a periodic and positive Radon measure $m$ on $\mathbb{R}^n$ is a lighthouse measure if $\mathrm{supp}(m)$ has zero Lebesgue measure and $\mathrm{supp}(\widehat{m})$ is contained in a proper double cone.

Our proof relies on the classification of one-dimensional Fourier quasicrystals. No field specific background is assumed. This work is based on collaborations with Alex Cohen, Pavel Kurasov, and Cynthia Vinzant.