#### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

##### April 16 - Organisational meeting

##### April 23 - Andreia Chapouto, University of Edinburgh

##### Title:

Invariance of the Gibbs measures for the periodic generalized KdV equations

##### Abstract:

In this talk, we consider the periodic generalized Korteweg-de Vries equations (gKdV). In particular, we study gKdV with the Gibbs measure initial data. The main difficulty lies in constructing local-in-time dynamics in the support of the measure. Since gKdV is analytically ill-posed in the L2-based Sobolev support, we instead prove deterministic local well-posedness in some Fourier-Lebesgue spaces containing the support of the Gibbs measure. New key ingredients are bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Once we construct local-in-time dynamics, we apply Bourgain's invariant measure argument to prove almost sure global well-posedness of the defocusing gKdV and invariance of the Gibbs measure. Our result completes the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations. This talk is based on joint work with Nobu Kishimoto (RIMS, University of Kyoto).

##### April 30 - Itamar Oliveira, Cornell University

##### Title:

The Fourier Extension problem through a time-frequency perspective

##### Abstract:

An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $ L^{2+\frac{2}{d}}([0,1]^{d}) $ to $ L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1}) $ for every $ \varepsilon>0 $. It has been fully solved only for $ d=1 $ and there are many partial results in higher dimensions regarding the range of $ (p,q) $ for which $ L^{p}([0,1]^{d}) $ is mapped to $ L^{q}(\mathbb{R}^{d+1}) $. In this talk, we will take an alternative route to this problem: one can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for**tensor functions**, meaning for all $ g $ of the form $ g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d})$. Time permitting, we will also address multilinear versions of the statement above and get similar results, in which we will need only one of the many functions involved in each problem to be of such kind to obtain the desired conjectured bounds. This is joint work with Camil Muscalu.

##### May 7 - No speaker

##### May 14 - Andriy Bondarenko, NTNU

##### Title:

Fourier interpolation with zeros of zeta and L-functions

##### Abstract:

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other L-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on strengthening of Knopp's abundance principle for Dirichlet series with functional equations and a careful analysis of the associated Dirichlet series kernel, with coefficients arising from certain modular integrals for the theta group.

##### May 21 - Alix Deleporte, Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay

##### Title:

Szegő kernels and Toeplitz operators

##### Abstract:

The Berezin-Toeplitz quantization allows to associate, to functions on Kähler (complex+symplectic) manifolds, self-adjoint operators on Hilbert spaces, depending on a semiclassical parameter. When the manifold is $ \mathbb{R}^{2n}= \mathbb{C}^n $, we retrieve (up to the FBI or Bargmann transform) the standard classes of pseudodifferential operators. Toeplitz operators also contain quantum spin systems as an important class of examples (when the manifold is the sphere $S^2$ or its powers) which describe the interaction between a solid and a magnetic field.In this talk, I will introduce Toeplitz operators and their principal geometric ingredient, the Szegő kernel, with main motivation a concrete example of quantum spin system with exotic behaviour. I will then describe the semiclassical techniques that are developed and used in the study of eigenfunction localisation in Berezin-Toeplitz quantization.

##### May 28 - Pfingstferien

##### June 4 - Dorothee Frey, KIT

##### Title:

Wave equations with low regularity coefficients

##### Abstract:

In this talk we discuss fixed-time $ L^p $ estimates and Strichartz estimates for wave equations with low regularity coefficients. It was shown by Smith and Tataru that wave equations with $ C^{1,1} $ coefficients satisfy the same Strichartz estimates as the unperturbed wave equation on $ \mathbb{R}^n $, and that for less regular coefficients a loss of derivatives in the data occurs. We improve these results for Lipschitz coefficients with additional structural assumptions. We show that no loss of derivatives occurs at the level of fixed-time $ L^p $ estimates, and that existing Strichartz estimates can be improved. The permitted class in particular excludes singular focussing effects. We also discuss perturbation results, and related results on Strichartz estimates for Schrödinger equations.

##### June 11 - Chenmin Sun, Laboratoire de Mathématiques AGM, CY Cergy-Paris Université

##### Title:

On the probabilistic well-posedness for the fractional cubic NLS

##### Abstract:

Gibbs measure is an important object in the study of macroscopic behavior of solutions of dispersive equations. Motivated by testing the dispersive effect for the construction of invariant measures, we consider the fractional cubic NLS with weak dispersion. For some ranges of dispersion, we constructed global weak solutions and strong solutions (resulting the flow property) via different methods. The construction of strong solutions is based on a "good" probabilistic local well-posedness result, which is difficult as the dispersion becomes very weak. Our resolution relies on a very recent refined resolution ansatz introduced by Deng-Nahmond-Yue. To overcome the difficulties caused by the weak dispersion in our problem, we benefit from some "physical-space" properties of the random averaging operators. This talk is based on collaborations with Nikolay Tzvetkov (CY Cergy-Paris Université).

##### June 18 - João P.G. Ramos, ETH Zütrich

##### Title:

Stability for geometric and functional inequalities

##### Abstract: **Abstract.**

##### June 25 - Stefan Buschenhenke, Christian-Albrechts-Universität Kiel

##### Title:

Factorisation and near-extremisers in restriction theory

##### Abstract:

We give an alternative argument to the application of the so-called Maurey-Nikishin-Pisier factorisation in Fourier restriction theory. Based on an induction-on-scales argument, our comparably simple method applies to any compact quadratic surface, in particular compact parts of the paraboloid and the hyperbolic paraboloid. This is achieved by constructing near extremisers with big "mass", which itself might be of interest.

##### July 2 - Maria Ntekoume, Rice University

##### Title:

Symplectic non-squeezing for the KdV flow on the line

##### Abstract:

In this talk we prove that the KdV flow on the line cannot squeeze a ball in $\dot H^{-\frac 1 2}(\mathbb R)$ into a cylinder of lesser radius. This is a PDE analogue of Gromov’s famous symplectic non-squeezing theorem for an infinite dimensional PDE in infinite volume.

##### July 9 - Moritz Egert, Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay

##### Title:

Dirichlet problem for elliptic systems with block structure

##### Abstract:

I’ll consider a very simple elliptic PDE in the upper half-space: divergence form, transversally independent coefficients and no mixed transversal-tangential derivatives. For L2-data the Dirichlet problem can be solved via a semigroup. For other data classes X (Lebesgue, Hardy, Sobolev, Besov,…) the question, whether the corresponding Dirichlet problem is well-posed, is inseparably tied to the question, whether there is a compatible semigroup on X.On a "semigroup space" the infinitesimal generator has most properties that one can dream of and these can be used to prove well-posedness. However, there are genuinely more "well-posedness spaces" than "semigroup spaces". For example, up to boundary dimension n=4 there is a well-posed BMO-Dirichlet problem, whose unique solution has no reason to keep its tangential regularity in the interior of the domain.

I’ll give an introduction to the general theme and discuss some new results, all based on a recent monograph jointly written with Pascal Auscher.

##### July 9 - Alexander Volberg, Michigan State University

##### Title:

Why do we not have embedding theorems in the bi-disc?

##### Abstract:

Embedding theorems for the classical Hardy space on bidisc are not known when I am writing this abstract.On the other hand, embedding theorems for the Dirichlet space of analytic functions on bidisc were recently found. And they were also found for the Dirichlet space of analytic functions on tri-disc. But not on four-disc…

We will explain the obstacles that preclude us to have those answers.

This talk is based on joint works with N. Arcozzi, I. Holmes, P. Mozolyako, G. Psaromiligkos and P. Zorin-Kranich.

##### July 16 - Alexei Poltoratski, University of Wisconsin-Madison

##### Title:

Pointwise convergence of scattering data

##### Abstract:

Тhe scattering transform, appearing in the study of differential operators, can be viewed as an analog of the Fourier transform in non-linear settings. This connection brings up numerous questions on finding non-linear analogs of classical results of Fourier analysis. One of the fundamental results of linear analysis is a theorem by L. Carleson on pointwise convergence of the Fourier series. In this talk I will discuss convergence for the scattering data of a real Dirac system on the half-line and present an analog of Carleson's theorem for the non-linear Fourier transform.

##### July 23 - Dominique Maldague, MIT

##### Title:

A new approach to small cap decoupling for the parabola

##### Abstract:

I will discuss forthcoming work in collaboration with Yuqiu Fu and Larry Guth. Strichartz estimates for solutions to the periodic Schrodinger equation are a direct corollary of the (l^2,L^p) decoupling theorem of Bourgain and Demeter. In the setting for small cap decoupling (see the paper of Demeter, Guth, Wang), we continue to measure the L^p norm of solutions to the periodic Schrodinger equation (as well as more general functions) but over a spatial scale which does not see the full periodicity of the solutions. By further developing the approach used by Guth, Maldague, and Wang to re-prove decoupling for the parabola, we obtain sharp level set estimates for the size of the solutions on these smaller spatial domains. The level set estimates refine and recover the results of Demeter, Guth, Wang for the parabola, and lead to new (l^q,L^p) small cap decoupling inequalities.## News

Jessica Fintzen wins Cole Prize

Dr. Regula Krapf receives university teaching award

Prof. Catharina Stroppel joined the North Rhine-Westphalia Academy for Sciences and Arts

Prof. Daniel Huybrechts receives the Compositio Prize for the periode 2017-2019

Prof. Catharina Stroppel receives Gottfried Wilhelm Leibniz Prize 2023

Grants for Mathematics students from Ukraine

Prof. Jessica Fintzen is awarded a Whitehead Prize of the London Mathematical Society

Prof. Peter Scholze elected as Foreign Member of the Royal Society