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: 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: I will present a construction of a singular measure on three dimensional sphere $S^3$ whose non-zero components in spherical harmonic decomposition are concentrated around lines $y=ax$ and $x=ay$. This could be seen as a generalization of Aleksandrov's singular pluriharmonic measure or an inverse to the Brummelhuis condition of absolute continuity. The main ingredient is a construction of bounded spherical harmonics with small, localized Fourier spectra.

Abstract: Qualitatively, actions of unipotent groups on homogeneous spaces are very well understood after breakthrough works of Ratner from the 90's. The past two decades have seen dramatic progress in providing rates in these results. In this talk, we will be interested in the distributional properties of periodic orbits of semisimple groups. Here, the desire is to show that such a periodic orbit 'approximates' a periodic orbit of a larger group with a rate that is polynomial in the volume.

In joint work with Einsiedler, Lindenstrauss, and Mohammadi, we prove such a result for compact (congruence) quotients with a rate this is explicit in all aspects (including the volume of the ambient space). This relies heavily on a variety of tools including uniformity of the spectral gap for periodic orbits in congruence quotients and polynomial properties of unipotent flows. We will discuss these results with an emphasis on the dynamical methods and mention applications in number theory briefly.

Abstract: Suppose $f$ is an $m$-smooth function on the unit ball that is small (for instance, vanishes) on an epsilon-net $E$ for a sufficiently small epsilon. Then the maximum of $f$ is controlled by the $L^1$-norm of its $m-th$ derivative and its uniform norm on $E$. This estimate is dimensionless. The proof is not long and uses only undergraduate analysis.

The talk is based on an ongoing joint work with Aleksei Kulikov and Fedor Nazarov on multi-dimensional Fourier uniqueness.

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.

Abstract: We will talk about a variant of the well-known local smooth- ing conjecture with times restricted to a fractal set. The frac- tal conjecture involves a new dimensional spectrum called the Legendre-Assouad function, which also comes up in the char- acterization of $L^{p}$-improving properties of spherical maximal functions. Joint work with David Beltran, Alex Rutar and An- dreas Seeger.

Abstract:We study integer-valued polynomials that grow at most exponentially on the integers, a question originally raised by Dimitrov. A result of Elkies and Speyer shows a sharp threshold: below a certain growth rate only finitely many such polynomials exist, while above it there are infinitely many. In joint work with Alon Nishry, we obtained quantitative asymptotic estimates for how many such polynomials of large degree exist at or above this threshold. Our approach uses convex-geometric volume estimates together with potential theory and orthogonal polynomials. We also extend the framework to allow different growth bounds on the positive and negative integers, where the threshold is again determined by the logarithmic capacity of an explicit planar set.

Abstract: The first part of this talk deals with the numerical approximation to nonlinear dispersive equations, such as the prototypical nonlinear Schrödinger or Korteweg-de Vries equations. We introduce integration techniques allowing for the construction of schemes which perform well both in smooth and non-smooth settings. Higher order extensions will be presented, following techniques based on decorated trees series inspired by singular stochastic PDEs via the theory of regularity structures.

In the second part, we introduce a new approach for designing and analyzing schemes for some nonlinear and nonlocal integrable PDEs, including the Benjamin-Ono equation. This work is based upon recent theoretical breakthroughs on explicit formulas for nonlinear integrable equations. It opens the way for studying the asymptotic behavior of the solutions, including their soliton resolution and small dispersion limit.

Abstract: We study the dynamics of perturbations around inhomogeneous stationary states of the Vlasov-HMF (Hamiltonian Mean-Field) model. These stationary solutions are built with compact support and satisfy a linearized stability criterion (Penrose criterion). We show a scattering behavior to a modified state over long (but finite) time depending on the size of the perturbation. This implies a Landau damping effect with an algebraic rate. The key ingredients are based on the analysis of echoes in the dynamics generated by the action-angle variables of the inhomogenous stationary state. This is a joint work with Torryanand Seetohul and Frédéric Rousset.