Schedule

Oct 18, 14:15 - Jaume de Dios Pont (ETH Zürich)

Title: On the Hot Spots Conjecture for convex sets in high dimensions

Abstract:

A homogeneous, insulated object with a non-uniform initial temperature will eventually reach thermal equilibrium. The Hot Spots conjecture addresses which points in the object take the longest to reach this equilibrium: Where is the maximum temperature attained as time progresses?

Rauch ('74) initially conjectured that points attaining the maximum temperature would approach the boundary for larger times. Burdzy and Werner ('99) disproved the conjecture for planar domains with holes. Kawohl ('85), and later Bañuelos-Burdzy ('99), conjectured that the result should still hold for convex sets of all dimensions. This talk will explore the conjecture's validity in the high-dimensional setting.

Oct 18, 15:45 - Organizational meeting

Oct 25 - Matthias Georg Mayer (Bonn)

Title: A Theory of Structural Independence

Abstract:

We will review the usage of Bayesian networks, d-separation and causal discovery, and their limitations for making sense of structure in observed data distributions. We will highlight d-separation as the central object in classical causal discovery and present its generalization, “structural independence” as a combinatorial property of random variables on a product space. The main theorem justifies this definition by showing the equivalence to independence in all product probability distributions on the product space, generalizing soundness and completeness of d-separation.

Nov 1 - NO SEMINAR

Nov 8 - Jonas Lührmann (Texas A&M University)

Title: Asymptotic stability of the sine-Gordon kink outside symmetry

Abstract:

We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the ϕ⁴ model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes.

We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known ϕ⁴ model.

This is forthcoming joint work with Gong Chen (Georgia Tech).

Nov 15 - Chenmin Sun (Université Paris-Est Créteil)

Title: Transport properties of Gaussian measures for the energy-critical nonlinear Schrödinger equation

Abstract:

In the statistical study of Hamiltonian PDEs out of the equilibrium (lack of invariant measures), it is a natural question to understand the transport properties for canonical Gaussian measures. The first question is to understand whether Gaussian measures are quasi-invariant, i.e. Gaussian measures push-forward by the flow are equivalent to the original ones. Once we know the Gaussian measures are quasi-invariant, we would like also to understand the properties of their Radon-Nikodym densities.

In this talk, I will explain several strategies to prove quasi-invariance properties, developed in recent years. Then I will focus on the energy-critical NLS and explain the main novelties of the proof. This talk is based on a joint work with Nikolay Tzvetkov.

Nov 22 - Marc Rouveyrol (Université Paris-Saclay)

Title: Spectral estimates on non-compact manifolds

Abstract:

Spectral estimates consist in bounding the norm of frequency-localized functions by their norm on a smaller, so-called “sensor” set, up to some factor depending on the frequency threshold. They were first studied by Logvinenko and Sereda in the 1970s for flat Laplacians, in connection with the uncertainty principle for the Fourier transform. Recent investigations have focused on the manifold case, in the spirit of a 1999 paper by Jerison and Lebeau. Beyond controllability results, applications of spectral estimates include spectral geometry and the study of random Schrödinger operators.

The aim of the talk will be to give an introduction to spectral estimates and related controllability results, and present the first such results in non-compact, non-flat geometric settings. Elements of proof will be given if time allows. The tools involved draw from spectral theory, harmonic analysis, geometric analysis and control theory for the heat equation.

Nov 29 - Mateus Costa de Sousa (Basque Center for Applied Mathematics)

Title: Phase retrieval and Fourier uniqueness problems

Abstract:

A phase retrieval problem consists of recovering the phase of a function with the knowledge of physical measurements. Pauli studied this problem when the measurements are the absolute value of the function and its Fourier transform, and we are gonna discuss what can be said when one samples the absolute value of a function and its Fourier transform over discrete sets. This talk is based on joint work with João Pedro Ramos (University of Lisbon).

Dec 6 - Diogo Oliveira e Silva (Instituto Superior Técnico Lisboa)

Title: Sharp extension inequalities on finite fields

Abstract:

Sharp restriction theory and the finite field extension problem have both received much attention in the last two decades, but so far they have not intersected. In this talk, we discuss our first results on sharp restriction theory on finite fields. Even though our methods for dealing with paraboloids and cones borrow some inspiration from their euclidean counterparts, new phenomena arise which are related to the underlying arithmetic and discrete structures. The talk is based on recent joint work with Cristian González-Riquelme.

Dec 13 - Gennady Uraltsev (University of Arkansas)

Title: Probabilistic well-Posedness for the cubic NLS using directional norms and multilinear expansions

Abstract:

The nonlinear Schrödinger equation (NLS) is a case study for dispersive partial differential equation, where different frequencies travel at distinct velocities, and the linearized flow does not present a clear smoothing effect. As an infinite-dimensional Hamiltonian system, the NLS suggests the existence of invariant measures associated with its flow. Constructing such a measure requires establishing probabilistic local well-posedness of NLS in low regularity spaces.

While significant progress has been made for NLS on the torus, beginning with Bourgain's work in the 1990s, analogous results on Euclidean space remain are limited.

In this talk, we prove the probabilistic local well-posedness of the cubic NLS \[ (i\partial_{t}+\Delta)u=\pm |u|^{2}u \text{ on } [0,T)\times \mathbb{R}^{d}, \] for unit-scale Wiener randomized initial data in \(H^{S}_{x}(\mathbb{R}^{d})\). For the physically relevant case \(d=3\) we cover the range of regularities \(S>0\).

We introduce a refined decomposition of solutions: an explicit multilinear expansion of high order driven by the random initial data, and a smoother remainder term with deterministically subcritical regularity. The quantitative analysis uses directional smoothing and maximal estimates, that generalize classical multilinear Strichartz estimates.

These methods could have implications for related stochastic dispersive equations as well as to questions related to global-in-time dynamics.

Jan 10 - Fabian Höfer (University of Münster)

Title: On the growth of Sobolev norms for the periodic focusing mass-critical nonlinear Schrödinger equation under critical Gibbs dynamics

Abstract:

We investigate the growth of Sobolev norms of solutions to the focusing mass-critical nonlinear Schrödinger equation on the one-dimensional torus in the context of Gibbs measures.

We begin by introducing the focusing Gibbs measures with mass cutoffs and their renormalisability. The study of these measures was initiated by Lebowitz, Rose, and Speer (1988) and continued by Bourgain (1994). In the mass-critical case, Oh, Sosoe, and Tolomeo (2022) showed that a critical mass threshold is given by the mass of the ground state on the real line.

Next, we review a classical argument by Bourgain, which establishes almost sure square root logarithmic bounds on the growth of Sobolev norms in subcritical regimes. However, this approach breaks down at the critical mass threshold. We will discuss the challenges of this critical case.

Our approach uses the construction of a suitable Orlicz space and is based on arguments given by Oh, Sosoe, and Tolomeo. This enables us to derive logarithmic bounds on the growth of Sobolev norms in the critical regime.

This talk is based on joint work with Niko Nikov (The University of Edinburgh).

Jan 17 - Younes Zine (École Polytechnique Fédérale de Lausanne)

Title: Hyperbolic sine-Gordon model beyond the first threshold

Abstract:

In the past two decades, significant progress has been made in understanding random dispersive PDEs with polynomial nonlinearities. However, non-polynomial nonlinearities remain poorly understood. This talk will present recent advancements in this direction, focusing on the well-posedness for the two-dimensional damped wave equation with a sine nonlinearity, driven by additive space-time white noise.

I will introduce the physical Fourier restriction norm method, a novel framework that addresses the complexities of non-polynomial settings. This method leverages recent developments in the Fourier restriction theory for the cone to establish crucial deterministic estimates. Furthermore, I will discuss the proof of nonlinear smoothing for the imaginary Gaussian multiplicative chaos, which constitutes the main probabilistic component of our approach. This involves examining new Feynman diagrams, whose analysis extends beyond the classical Dyson power counting criterion. This is a joint work with Tadahiro Oh (Edinburgh, UK).

Jan 24 - James Coe (University of Edinburgh)

Title: Sharp quasi-invariance threshold for the cubic Szego equation

Abstract:

The transport properties of Gaussian measures under dispersive dynamics heavily rely on the interplay between the Gaussian structure of the data and dispersive effects, but the exact role of dispersion is not well understood. We shall consider the flow of a family of Gaussian fields under the dynamics of the cubic Szego equation; a toy model for weakly-dispersive nonlinear Hamiltonian dynamics. We see that above a critical regularity, the measures are quasi-invariant under the flow (almost-sure properties of the data are preserved), but below this regularity, quasi-invariance fails. In fact, the distribution at almost every time is singular with respect to the initial distribution.

We introduce a method to prove singularity by exhibiting an instantaneous growth of Sobolev norms of the solution, coupled with an abstract argument to show that such a growth cannot occur with positive probability at all times. We will discuss some heuristics as to when one expects quasi-invariant flow, even beyond Hamiltonian systems. In particular, we will describe how these methods can be applied to understanding invariant measures for the 2D stochastic Navier Stokes equations.

This talk is based on joint work with Leonardo Tolomeo.