### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

The seminar may occasionally take place online on the Zoom platform. In case, the Zoom details will be comunicated separately.

Please join the pdg-l mailing list or contact Gael Diebou Yomgne for further information.

## Next weeks

### February 04 - Xian Liao (KIT)

### Title:

Some mathematical results on the two dimensional inhomogeneous incompressible viscous flows

### Abstract:

In this talk we are going to present some mathematical results on the motion of the two dimensional inhomogeneous incompressible viscous flows in the presence of density/temperature variations. The PDE system is a nonlinear coupling between the density equation and the incompressible Navier-Stokes type equations, where the viscosity coefficient is density-dependent. If there is diffusion effect in the density equation, then the associated Cauchy problem is globally-in-time well-posed. As the first step toward the case with no such diffusion effect, we are going to investigate the motion of the steady flows.This talk is partially based on the joint work with Zihui He (KIT).

### February 04 - Asgar Jamneshan (Koç University)

### Title:

The structure of arbitrary Conze-Lesigne systems and an application

### Abstract:

Conze-Lesigne systems are ergodic abelian measure-preserving dynamical systems $X$ which are (abstractly) isomorphic to their second Host-Kra-Ziegler factors $Z_2(X)$. These factors (and their higher-order versions) are relevant in the area of multiple recurrence and related topics in additive combinatorics. In this talk, we present a structure theorem for Conze-Lesigne systems defined for arbitrary actions of a countable discrete abelian group $\Gamma$, describing such systems as an inverse limit of translational systems $\mathcal{G}/\Lambda$, where $\mathcal{G}$ is a locally compact nilpotent group of nilpotency class $2$, $\Lambda$ is a lattice in $\mathcal{G}$, $\mathcal{G}/\Lambda$ is equipped with Haar measure, and $\Gamma$ acts on $\mathcal{G}$ by translations. Similar structure theorems were previously known in the important special cases of finitely generated abelian groups by work of many (including Conze and Lesigne) and direct sums of finite fields by work of Bergelson, Tao, and Ziegler. We will review some of this literature by way of illustrating examples. We then use these examples to help to explain our proof in the general case. If time permits, we present an application of our structure theorem to give a qualitative proof of the inverse theorem for the Gowers uniformity norms $U^3(G)$ for arbitrary finite abelian groups $G$ via a correspondence principle.This talk is based on work jointly with Or Shalom and Terence Tao.

## Previous weeks

### January 28 - Anna Balci (Universität Bielefeld)

### Title:

Lavrentiev gap in calculus of variations: differential forms perspective, non-local problems, borderline cases and numerical analysis.

### Abstract:

Lavrentiev gap is the key phenomenon in calculus of variations to study many properties of problems with general growth. It leads to many challenges in analysis and numerics such as non-density of smooth functions and disconvergence of numerical methods. We present new density results and examples on Lavrentiev gap for general classes of differential forms using fractal contact sets. We also construct examples for the general classes of non-local problems and design the finite element scheme to study numerically so called W-minimizers for such kind of problems.This talk is based on several joint works with Lars Diening (Bielefeld), Mikhail Surnachev (Keldysh Institute of Applied Mathematics, Moscow), Johannes Storn(Bielefeld) and Christoph Ortner(UBC, Vancouver).

### January 21 - Armin Schikorra (University of Pittsburgh)

### Title:

A Harmonic Analysis perspective on $W^{s,p}$ as $s \to 1^-$

### Abstract:

We revisit the Bourgain-Brezis-Mironescu result that the Gagliardo-Norm of the fractional Sobolev space $W^{s,p}$, up to rescaling, converges to $W^{1,p}$ as $s\to 1$. We do so from the perspective of Triebel-Lizorkin spaces, by finding sharp $s$-dependencies for several embeddings between $W^{s,p}$ and $F^{s,p}_{q}$ where $q$ is either 2 or $p$. We recover known results, find a few new estimates, and discuss some open questions.Joint work with Denis Brazke, Po-Lam Yung.

### January 14 - Gianmarco Brocchi (University of Birmingham)

### Title:

Two weight estimates for the Bergman projection

### Abstract:

The space of holomorphic functions which are $L^2$-integrable on a domain are at the cross-road of Operator Theory, Harmonic Analysis and Complex analysis in several variables.An active direction of research aims to better understand the projection from $L^2$ onto this space in terms of weighted estimates. Several weighted estimates can be obtained by using a powerful technique in Harmonic Analysis called sparse domination. Sparse domination consists in controlling a non-local, non-positive operator by a sum of local, positive averages. As a consequence, plenty of quantitative weighted estimates are easily derived, and they are often optimal. In this talk we introduce the concept of sparse domination and show how this tool can be used to derive sufficient conditions for the

*two weight*inequality for the Bergman projection on the unit ball.

### December 17 - Rachel Greenfeld (UCLA)

### Title:

The periodic tiling conjecture

### Abstract:

A finite subset $F$ of a finitely generated abelian group $G$ is a translational tile of $G$ if it is possible to partition $G$ into translates of $F$. We will motivate and discuss the study of the periodic tiling conjecture, which asserts that any translational tile of $G$ admits a periodic tiling. We will present a new structure theorem for translational tilings, which implies new results regarding the periodic tiling conjecture. This is joint work with Terence Tao.

### December 10 - Christian Brennecke, Bonn Junior Fellow (Universität Bonn)

### Title:

Low-Energy Spectrum of Bose-Einstein Condensates in the Gross-Pitaevskii Regime

### Abstract:

In this talk, I will consider systems of $N$ trapped bosons in $\mathbb{R}^3$ that interact through a two-body potential whose scattering length is of order $N^{-1}$, characterizing the so-called Gross-Pitaevskii regime. At low energy, such systems form a Bose-Einstein condensate. I will review dynamical and spectral properties of such Bose-Einstein condensates, describing in particular the low-energy excitation spectrum of the system, up to errors that vanish in the limit $N\to\infty$.The talk is based on joint work with B. Schlein and S. Schraven.

### December 3 - Micheal Winkler (Universität Paderdorn)

### Title:

Gagliardo-Nirenberg-type inequalities for rapidly decreasing functions and applications in the large time analysis of nonlinear heat flows.

### Abstract:

The objective of the presentation is to discuss options of Gagliardo-Nirenberg type interpolation in situations when the considered functions are known to decay faster than algebraic in space. A result in this regard, obtained in joint work with Marek Fila, in fact establishes a quantitative connection between Lebesgue and first-order Sobolev norms of such functions, where in contrast to the classical case of algebraic decay this connection involves a non-algebraic correction. This new interpolation inequality is thereafter applied to derive temporal decay rates for solutions rapidly decreasing in space to some nonlinear diffusion equations of fast diffusion type. As these rates can be seen to be essentially optimal, this a posteriori shows optimality of the obtained interpolation results.

### November 26 - Jan Bohr (University of Cambridge)

### Title:

The transport Oka-Grauert principle for simple surfaces.

### Abstract:

In the talk we will consider matrix valued transport equations on simple Riemannian surfaces. The main result I want to explain establishes the existence of solutions with controlled Fourier modes in their velocities, so called matrix holomorphic integrating factors. This is applied to characterise the range of an associated nonlinear X-ray transform. The result can further be interpreted as non-existence result for holomorphic vector bundles on a complex surface, reminiscent of the classical Oka-Grauert principle.The talk is based on joint work with Gabriel Paternain.

### November 19 - Marco Fraccaroli (Universität Bonn)

### Title:

Outer $L^p$ spaces: K\"{o}the duality, Minkowski inequality and more.

### Abstract:

The theory of $L^p$ spaces for outer measures, or outer $L^p$ spaces, was developed by Do and Thiele to encode the proof of boundedness of certain multilinear operators in a streamlined argument. Accordingly to this purpose, the theory was developed in the direction of the real interpolation features of these spaces, such as versions of H\"{o}lder's inequality and Marcinkiewicz interpolation, while other questions remained untouched.For example, the outer $L^p$ spaces are defined by quasi-norms generalizing the classical mixed $L^p$ norms on sets with a Cartesian product structure; it is then natural to ask whether in arbitrary settings the outer $L^p$ quasi-norms are equivalent to norms and what other reasonable properties they satisfy, e.g. K\"{o}the duality and Minkowski inequality. In this talk, we will answer these questions, with a particular focus on two specific settings on the collection of dyadic intervals in $\mathbb{R}$ and the collection of dyadic Heisenberg boxes in $\mathbb{R}^2$. This will allow us to clarify the relation between outer $L^p$ spaces and tent spaces, and get a glimpse at the use of this language in the proof of boundedness of prototypical multilinear operators with invariances.

### November 12 - Phuc Cong Nguyen (Louisiana State University)

### Title:

Choquet Integrals, Capacitary Inequalities, and the Hardy-Littlewood Maximal Function

### Abstract:

We obtain the boundedness of the Hardy-Littlewood maximal function on $L^q$ type spaces defined via Choquet integrals associate to Sobolev capacities. The bounds are obtained in full range of exponents including a weak type end-point bound. We also obtain a capacitary inequality of Maz’ya type which resolves a problem proposed by D. Adams.This talk is based on joint work with Keng Hao Ooi.

### November 05 - Giuseppe Negro (University of Birmingham)

### Title:

Stability of sharp Fourier restriction to spheres.

### Abstract:

In dimension $d\in\{3, 4, 5, 6, 7\}$, we establish that the constant functions maximize the weighted $L^2(S^{d-1}) - L^4(R^d)$ Fourier extension estimate on the sphere, provided that the weight function is sufficiently regular and small, in a proper and effective sense which we will make precise. One of the main tools is an integration by parts identity, which generalizes the so-called "magic identity" of Foschi for the unweighted inequality with $d=3$ (that is, the classical Stein--Tomas estimate).Joint work with E.Carneiro and D. Oliveira e Silva.

### October 29 - Olli Saari (Universität Bonn)

### Title:

A right inverse of the divergence in time-dependent non-cylindrical domains

### Abstract:

Consider an open and connected set in the (1+n)-dimensional space time. We discuss a construction of a right inverse of divergence that respects vanishing boundary values at the lateral boundary. The domain is assumed to be Hölder regular both in time and space variables, and depending on the actual regularity, a priori estimates in Sobolev spaces of positive and negative order can be proved. In particular, we show that the construction exhibits a non-trivial amount of regularity in time variable, which is of interest from the point of view of fluid mechanics. This is based on joint work with S. Schwarzacher.

### October 22 - Dongxiao Yu (Universität Bonn)

### Title:

Asymptotic completeness for a scalar quasilinear wave equation satisfying the weak null condition

### Abstract:

I will discuss an asymptotic completeness result for a scalar quasilinear wave equation in three space dimensions. This equation satisfies the weak null condition introduced by Lindblad and Rodnianski, and it admits small data global existence which was proved by Lindblad. In this talk, I will first introduce a new notion of asymptotic profile by deriving a new reduced system for the model equation. I will then present a proof of the asymptotic completeness. In other words, given a global solution to the model equation, I will show how to find the corresponding asymptotic profile.

### October 15 - Organisational meeting

## News

Vacany: Office employee at the mathematical Institut (100%) (Application deadline: 17.03.24)

Vacancy: Office employee (Application deadline: 10.3.24)

Vacancy: Office employee (Application deadline: 3.3.24)

ERC Consolidator Grant for Dr. Evgeny Shinder

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