MIT PDE/Analysis Seminar

Abstract: Consider the triadic Cantor set equipped with the "uniform" Cantor law. It happens that its cumulative distribution function, the devil’s staircase, is Hölder regular, and its best exponent of regularity is ln(2)/ ln(3), which is exactly the Hausdorff dimension of the Cantor set. Moreover, one can show that the Fourier transform of the Cantor law decay like |ξ|^{− ln 2/ ln 3} "on average". This is no coincidence, and hint for a deeper link between Fractal Geometry and Fourier Analysis. In this talk we will detail and explore this link through the notion of Fourier Dimension. We will introduce the Fourier dimension, compute it on some easy examples, quote some natural questions that arise, and then discuss a (partial) state of the art on the topic.

Abstract: The speaker aims to report some advances in the study of irrotational oscillation of a water droplet under zero gravity. The governing physical laws, resembling the well-studied capillary water waves equation, are converted to a quasilinear dispersive para-differential system defined on the 2-sphere. Regarding this conversion, a coordinate-independent, global para-differential calculus defined on compact Lie groups and homogeneous spaces is developed as a toolbox. After discussing Cauchy theory under this novel formalism, the speaker will propose several unsolved problems concerning existence of periodic solutions, normal form reduction and generic lifespan estimates. It is pointed out that all of these problems are closely related to certain Diophantine equations.

Abstract: Although the laws of physics are quantum mechanical, the world we live in and observe is well described by classical physics. One mathematical justification for this is Egorov’s theorem which shows that the quantum and classical descriptions match as the semiclassical parameter ℏ tends to 0. However, the Ehrenfest timescale for which Egorov’s theorem holds may only be seconds or minutes for realistic systems. In this talk we consider quantum systems that are weakly coupled to a noisy external environment, as modelled by the Lindblad equation. The semiclassical limit of this Lindblad equation is a Fokker-Planck equation which includes the effects of friction and diffusion. In this talk I will show that even for very weak diffusion, the agreement between the Lindblad and Fokker-Planck equations persists for times much longer than the Ehrenfest time. This is joint work with Daniel Ranard and Jess Riedel.

Abstract: In this talk, we prove global well-posedness and scattering for the conformally invariant, radially symmetric nonlinear wave equation in the defocusing case. This result is sharp in the radial case. We spend the first half of the talk discussing the broader context of the nonlinear wave equation. We also explain why the result is sharp.

Abstract: We consider classical scalar fields in dimension 1+1 with a self-interaction potential being a symmetric double-well. Such a model admits non-trivial static solutions called kinks and antikinks. A kink cluster is a solution approaching, for large positive times, a superposition of alternating kinks and antikinks whose velocities converge to 0 and mutual distances grow to infinity. Our main result is a determination of the asymptotic behaviour of any kink cluster at the leading order.

Our results are partially inspired by the notion of "parabolic motions" in the Newtonian n-body problem. I will present this analogy and mention its limitations. I will also explain the role of kink clusters as universal profiles for formation of multi-kink configurations.

This is a joint work with Andrew Lawrie.

Abstract: In this overview talk we will explore a variational approach to the problem of Spectral Minimal Partitions (SMPs). The problem is to partition a domain or a manifold into k subdomains so that the first Dirichlet eigenvalue on each subdomain is as small as possible. We expand the problem to consider Spectral Critical Partitions (partitions where the max of the Dirichlet eigenvalues is experiencing a critical point) and show that a locally minimal bipartite partition is automatically globally minimal.

Extensions of this result to non-bipartite partitions, as well as its connections to counting nodal domains of the eigenfunctions and to a two-sided Dirichlet-to-Neumann map defined on the partition boundaries, will also be discussed.

The talk is based on several papers of Yaiza Canzani, Graham Cox, Bernard Helffer, Peter Kuchment, Jeremy Marzuola, Uzy Smilansky and Mikael Sundqvist, with and without the speaker.

Abstract: Let $E \subset \mathbb{R}^n$ be a compact set and the function $f:E \to \mathbb{R}$. Let $C_c^{1,1}(\mathbb{R}^n)$ be the space of convex, differentiable functions with Lipschitz continuous gradient. How can we tell if there exists a convex function $F \in C_c^{1,1}(\mathbb{R}^n)$ extending $f$ (satisfying $F|_E=f|_E$)? I will present recent work of mine proving that if a function satisfies a finiteness hypothesis for strongly convex functions in $C^{1,1}_c(\mathbb{R}^n)$, then there exists a strongly convex function in $C^{1,1}_c(\mathbb{R}^n)$ extending the given function. Despite obstacles to their direct application, this theorem brings techniques developed by P. Shvartsman, C. Fefferman, A. Israel, and K. Luli for smooth extension and selection to smooth, convex extension. A key component of this result is D. Azagra and C. Mudarra's theorem on extension of 1-jets of functions in $C^{1,1}_c(\mathbb{R}^n)$. We will finish with a discussion of challenges in adapting my more general 1d-theorem ($E \subset \mathbb{R}$) to higher dimensions.

Abstract: We will outline a direct approach to bounds on Weyl sums for higher degree polynomials based on ideas from dynamical systems and unitary representation theory for nilpotent Lie groups. This approach originated with Furstenberg derivation of equidistribution of fractional parts of polynomial sequences from unique ergodicity of linear toral skew shifts. In general it is a hard problem in dynamical systems (homogeneous dynamics) to prove "effective" counterparts of unique ergodicity results or more general classification results for invariant measures (Ratner theory). For nilflows (and more generally nilsequences) effective results (with power saving) were proved by Green and Tao in 2012 with no precise (and presumably not sharp) information on the exponent. Our approach to effective equidistribution (in joint work with L. Flaminion) is based on "scaling" and generalizes the "renormalization" approach to effective equidistribution (for instance, for the equidistribution of unstable manifolds of hyperbolic diffeomorphisms). From our method we derive bounds on Weyl sums comparable to the best available ones, derived by J. Bourgain, C. Demeter and L. Guth and later independently by T. Wooley from their proof of the "Vinogradov Main Conjecture".

Abstract: We consider the wave maps equation for maps from $(1+1)$-dimensional Minkowski space into a compact Lie group. The Gibbs measure of this model corresponds to a Brownian motion on the Lie group, which is a natural object from stochastic differential geometry. Our main result is the invariance of the Gibbs measure under the wave maps equation and is the first result of this kind for any geometric wave equation. The proof combines techniques from differential geometry, partial differential equations, and probability theory.

Abstract: I will talk about recent progress in the study of quantitative equidistribution of unipotent orbits in homogeneous spaces, namely, effective versions of Ratner’s equidistribution theorem. In particular, I will explain the main idea of my proof for unipotent orbits in SL(3,R)/SL(3,Z). The proof combines new ideas from harmonic analysis and incidence geometry. In particular, the quantitative behavior of unipotent orbits is closely related to a Kakeya model.

Abstract: For hyperbolic systems of conservation laws in one space dimension endowed with a single convex entropy, it is an open question if it is possible to construct solutions via convex integration. Such solutions, if they exist, would be highly non-unique and exhibit little regularity. In particular, they would not have the strong traces necessary for the nonperturbative $L^2$ stability theory of Vasseur. Whether convex integration is possible is a question about large data, and the global geometric structure of genuine nonlinearity for the underlying PDE. In this talk, I will discuss recent work which shows the impossibility, for a large class of 2x2 systems, of doing convex integration via the use of $T_4$ configurations. Our work applies to every well-known 2x2 hyperbolic system of conservation laws which verifies the Liu entropy condition. This talk is based on joint work with László Székelyhidi.