#### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

##### July 15 - Alexander Volberg (Michigan State University)

##### Title:

Title: An estimate of Sidon constant for complex polynomials with unimodular coefficients##### Abstract:

In this paper we are concerned with the Bohnenblust--Hille type inequalities for certain polynomials of bounded degree but of very large number of variables. As the polynomials will be defined on groups, one can think about the problem as the estimate of Sidon constants. In most cases the sharp constants are unknown. We estimate the universal constant concerning the Sidon type estimates of degree $d$ polynomials of $n$ variables $z_1, \dots, z_n$ with unimodular coefficients. For polynomials that have constant absolute value of coefficients, this allows us to improve the estimate obtained by Bayart, Pellegrino and Seoane-Sepulveda in their work on the Bohr radius of the n-dimensional polydisc.

## Journey back in time

##### July 08 - Theresa C Anderson, (Carnegie Mellon University)

##### Title:

Two meetings of analysis and number theory##### Abstract:

In many recent works, analysis and number theory go beyond working side by side and team up in an interconnected back and forth interplay to become a powerful force. Here I describe two distinct meetings of the pair, which result in sharp counts for equilateral triangles in Euclidean space and statistics for how often a random polynomial has Galois group not isomorphic to the full symmetric group.

##### July 01 - Amru Hussein, (TU Kaiserslautern)

##### Title:

The hydrostatic approximation for the primitive equations with full and horizontal viscosity##### Abstract:

The primitive equations for the ocean and atmosphere are considered to be a fundamental model for geophysical flows. They are obtained formally as the hydrostatic limit from the anisotropic Navier-Stokes equations. Here, this is justified by rigorous convergence results in maximal $L_t^p$-$L_x^q$-regularity spaces to the primitive equations with full viscosity and the primitive equations with only horizontal viscosity. Similar results have been obtained by Li and Titi for the $L^2$-setting using global energy estimates. In contrast, the approach presented here relies only on local quadratic norm estimates, and this strategy is based on the usual construction of global-in-time solutions for small data.

##### June 24 - Martin Spitz, (Universität Bielefeld)

##### Title:

Almost sure scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data##### Abstract:

The local and global wellposedness theory of nonlinear dispersive equations with randomized data has attracted a lot of interest over the last years. In particular in the scaling-supercritical regime, where a deterministic wellposedness theory fails, randomization has become an important tool to study the generic behaviour of solutions. In this talk we study the energy-critical NLS on $\mathbb{R}^4$ with supercritical initial data. We present a randomization based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and a unit-scale decomposition in physical space. We then discuss the resulting (almost surely) improved space-time estimates for solutions of the linear Schrödinger equation with randomized data and how these estimates yield almost sure scattering for the energy-critical cubic NLS.

##### June 17 - Georgios Sakellaris, (Aristotle University of Thessaloniki)

##### Title:

Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces##### Abstract:

We will discuss local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation $-div(A\nabla u+bu)+c\nabla u+du=-div f+g$, assuming that $A$ is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions $b,f\in L^{n,1}$, $d,g\in L^{\frac{n}{2},1}$ and $c\in L^{n,q}$ for $q\leq\infty$, we will see that, with the surprising exception of the reverse Moser estimate, estimates with ``good" constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on $b,d$ or $c,d$, we will discuss a maximum principle and Moser's estimate for subsolutions with ``good" constants. We will also see that the reverse Moser estimate for nonnegative supersolutions with ``good" constants always holds, under no smallness assumptions when $q<\infty$, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, in the setting of Lorentz spaces, we will show that our assumptions are the sharp ones to guarantee these estimates.

##### May 27 - Rafael Granero Belinchón, (University of Cantabria)

##### Title:

On the vanishing depth for incompressible free boundary flows##### Abstract:

In this talk we will present some new results regarding the dynamics of incompressible free boundary flows. We will show a number of results studying the case where the free surface is near the impervious bottom. In particular, we will show that a smooth capillary-gravity water waves cannot touch the bottom while being smooth. The results in this talk are in collaboration with Zhiyuan Geng (BCAM)

##### May 20 - Sy Mouhamadou, (Imperial College London)

##### Title:

Constructing global solutions for energy supercritical PDEs##### Abstract:

In this talk, we will discuss invariant measures techniques to establish probabilistic global well-posedness for PDEs. We will go over the limitations that the Gibbs measures and the so-called fluctuation-dissipation measures encounter in the context of energy-supercritical PDEs. Then, we will present a new approach combining the two aforementioned methods and apply it to the energy supercritical Schrödinger equations. We will point out other applications as well.

##### May 13 - Olli Saari (University of Bonn)

##### Title:

Phase space projections##### Abstract:

A partition into tiles of the area covered by a convex tree in the Walsh phase plane gives an orthonormal basis for a subspace of L2. There exists a related projection operator, which has been an important tool for dyadic models of the bilinear Hilbert transform. Extending such an approach to the Fourier model is strictly speaking not possible, but satisfactory substitutes can be constructed. This approach was pursued by Muscalu, Tao and Thiele (2002) for proving uniform bounds for multilinear singular integrals with modulation symmetry in dimension one. I discuss a multidimensional variant of the problem. This is based on joint work with Marco Fraccaroli and Christoph Thiele.

##### May 06 - Joseph Adams, (Heinrich Heine University Düsseldorf)

##### Title:

Local well-posedness theory of the Novikov-Veselov equation##### Abstract:

The Novikov-Veselov equation is another generalization of the celebrated KdV equation. It is an interesting object of study in low-regularity well-posedness theory. In this talk, these merits are discussed, and recently published results concerning its local well-posedness both in the periodic and nonperiodic setting are presented. The arguments are mainly based on a symmetrization of the equation and, in the periodic case, a new bilinear estimate derived from number theory. Possible further paths for investigation and open questions are also discussed.(based on the joint work: https://arxiv.org/abs/2111.04575)

##### April 29 - Organizational meeting

##### April 22 - Xiaoyutao Luo, Duke University

##### Title:

Sharp nonuniqueness for the Navier-Stokes equations##### Abstract:

Weak solutions of the incompressible Navier-Stokes equations are unique in the so-called Ladyzhenskaya-Prodi-Serrin regime. A scaling analysis suggests that classical uniqueness results are sharp, but previous nonuniqueness constructions of convex integration are far below the critical threshold. In this talk, I will discuss sharp nonuniqueness results on two end-points of the Ladyzhenskaya-Prodi-Serrin regime. Joint work with Alexey Cheskidov.## 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