#### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

##### April 12, 14:15 - Adolfo Arroyo-Rabasa (Bonn)

##### Title: The Interplay of Algebraic and Analytic Restrictions in PDE constraints

##### Abstract:

Fine patterns, such as oscillations and concentrations of mass, are ubiquitous in nature, from the microstructure of materials to the behaviour of turbulent fluids. These patterns are often modelled by nonlinear partial differential equations (PDEs). Originated from the work of Lions, Tartar, and many others, “Compensated compactness” is a powerful framework for understanding fine pattern formation in nonlinear PDEs, which exploits the interplay between algebraic and PDE restrictions. Algebraic restrictions in this context are constraints on the possible values of a solution to a PDE, which can strongly reduce the formation of arbitrary fine patterns. This means that, even if a PDE does not have an explicit solution, we can still learn a lot about its behaviour by studying the interplay of these restrictions. While oscillatory behaviour is somewhat well understood, much less is known about the formation of mass concentrations and the shape of their generated singularities. I will give a general overview on how the import of Fourier Analysis tools has ignited substantial advances in compensated compactness theory and particularly mass concentrations. I will then discuss a superposition principle for concentrations that could lead to substantial progress at the intersection of various subfields of analysis.

##### April 12, 15:45 - Organizational meeting

##### April 19 - Leonidas Daskalakis (Rutgers)

##### Title: Roth's theorem and the Hardy--Littlewood majorant problem for thin subsets of Primes

##### Abstract:

We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions. We also prove that the Hardy--Littlewood majorant property holds for these sets. Notably, our considerations recover the results for the Piatetski--Shapiro primes for exponents close to 1, which are primes of the form $\lfloor n^c\rfloor$ for a fixed c>1.

##### April 26 - Valentina Ciccone (Bonn)

##### Title: Endpoint estimates for higher-order Marcinkiewicz multipliers

##### Abstract:

Marcinkiewicz multipliers on the real line are bounded functions of uniformly bounded variation on each Littlewood-Paley dyadic interval. The corresponding multiplier operators are well known to be bounded on $L^p(\mathbb{R})$ for all $1< p < \infty $. Optimal weak-type endpoint estimates for these operators have been studied by Tao and Wright who proved that they map locally $L\log^{1/2}L$ to weak $L^1$. In this talk, we consider higher-order Marcinkiewicz multipliers, that is multipliers of uniformly bounded variation on each interval arising from a higher-order lacunary partition of the real line. We discuss optimal weak-type endpoint estimates for the corresponding multiplier operators. These are established as a consequence of a more general endpoint result for a higher-order variant of a class of multipliers introduced by Coifman, Rubio de Francia, and Semmes and further studied by Tao and Wright. The seminar is based on joint work with Odysseas Bakas, Ioannis Parissis, and Marco Vitturi.

##### May 3 - Mouhamadou Sy (AIMS - Senegal)

##### Title: Probabilistic global well posedness for the incompressible 3D Euler system

##### Abstract:

The Euler system describes the evolution of an ideal fluid (where the viscosity is neglected). Its mathematical study is central in theoretical fluid dynamics and, in some regimes, remains an outstanding open problem. The Cauchy theory associated with the system is locally well known for smooth enough data. However, the persistence of such local solutions up to arbitrary time is far from being completely understood. This question was shown to be equivalent to essential boundedness of the vorticity by Beale-Kato-Majda. Due to the structure of the vorticity equations in 2 dimensions, one can show several conservation properties leading to global well posedness in this context. However, none of such information on the dynamics is available in three dimensions. The only control at our disposal is the preservation of the kinetic energy (given by the L^2-norm of the velocity); which is too 'supercritical' to match the regularity of the known local solutions. Hence the essential obstruction to the deterministic theory. By employing a probabilistic method, we make use of the kinetic energy and show a global well posedness result on invariant subsets of Sobolev spaces and establish long-time dynamic properties of the solutions. The argument relies on a combination of the Gibbs measures approach and the fluctuation-dissipation method. A major difficulty here, compared to earlier works on e.g. energy supercritical NLS, is the lack of a second coercive conservation law. This is a joint work with Juraj Földes.

##### May 10 - Margherita Disertori (Bonn)

##### Title: Functional integral representations for discrete random operators

##### Abstract:

Transport properties in disordered materials can be inferred from the spectral behavior of certain random operators, the most famous example being the random lattice Schrödinger operator (Anderson model). In certain cases, the problem can be reformulated as a complex valued integral. The main technical issue is to control the integral in the thermodynamic limit, when the number of variables diverges. This is achieved by a combination of analytic and algebraic tools, among which complex saddle analysis, cluster expansions around gaussian measures and Brascamp-Lieb inequality. I will give an overview of the problem and tools.

##### May 17 - NO SEMINAR

##### May 24 - Pfingstferien

##### May 31 - Lillian Pierce (Bonn)

##### Title: Counterexamples disproving pointwise convergence for solutions of dispersive PDE’s

##### Abstract:

In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space that implies pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose counterexample construction for the Schrödinger maximal operator proved a necessary condition on the regularity, and Du and Zhang, who proved a sufficient condition. In this talk, we will briefly sketch how Bourgain’s counterexamples can be constructed from first principles. Then we will describe a new flexible number-theoretic method for constructing counterexamples, which proves a necessary condition for a broad class of dispersive PDE to be bounded from H^s to local L^1.

##### June 7 - Lars Becker (Bonn)

##### Title: A degree one Carleson operator along the paraboloid

##### Abstract:

Carleson proved in 1966 that the Fourier series of any square integrable function converges pointwise to the function, by establishing boundedness of the maximally modulated Hilbert transform from L^2 into weak L^2. This talk is about a generalization of his result, where the Hilbert transform is replaced by a singular integral operator along a paraboloid. I will review some previous extensions of Carleson's theorem, and then discuss the two main ingredients needed to deduce our result: sparse bounds for singular integrals along the paraboloid, and a square function argument relying on the geometry of the paraboloid.

##### June 14 - Eric Sawyer (McMaster University)

##### Title: A probabilistic analogue of the Fourier extension conjecture and an implication for the Kakeya conjecture

##### Abstract:

We prove that the Fourier extension conjecture holds when averaged over all smooth Alpert multipliers of +/-. We also show that if this averaged extension conjecture holds for all conjugations of the smooth Alpert multipliers by a unimodular function, then the Kakeya conjecture holds. To prove the probabilistic analogue of the extension conjecture, we use frames for Lp consisting of smooth compactly supported Alpert wavelets having a large number of vanishing moments, along with standard estimates on oscillatory integrals and probabilistic interpolation of L2 and L4 estimates, as part of a two weight testing strategy using pigeonholing via the uncertainty principle to define various subforms. It is crucial to use probability in our method to obtain L4 estimates with the correct decay when dealing with resonant subforms. See arXiv:2311.03145v7.

##### June 21, 14:00 - Rajula Srivastava (Bonn)

##### Title: Counting Rational Points near Manifolds

##### Abstract:

How many rational points with denominator of a given size lie within a certain distance from a compact, "non-degenerate" manifold? This talk is about some recent progress towards answering this question, based on joint work with Damaris Schindler and Niclas Technau. We shall discuss a new way of examining the problem: under a combined lens of harmonic analysis (oscillatory integrals) and homogeneous dynamics (quantitative non-divergence estimates). We shall also talk about applications to problems concerning Hausdorff dimension and measure refinements for the set of well-approximable points on non-degenerate manifolds.

##### June 21, 15:00 - Lorenzo Pompili (Bonn)

##### Title: On the stability of the line soliton of the KP-II equation on $\mathbb R^2$

##### Abstract:

Wave fronts in the ocean are inherently nonlinear. Two questions arise naturally: (1) Are these wave fronts stable? (2) What happens when two such wave fronts intersect transversally? We study these questions in the setting of the KP-II equation, a 2D generalization of KdV. Wave fronts for this PDE are called "line-solitons". We prove the stability of line-solitons in a codimension-1 manifold of initial perturbations in a weighted space. The main tool is a "soliton-addition-map" which allows to decouple the evolution of the pure soliton from the rest of the solution: this tool is related to the inverse scattering transform (a.k.a. nonlinear Fourier transform) of KP-II. We finally discuss how our method can be applied to the study of interactions between several line-solitons.

##### June 28 - Alexander Volberg (Bonn)

##### Title: 3/2>1: a counterexample to $A_2$ conjecture for matrix weights

##### Abstract:

How to estimate the norm of the Hilbert transform (or other singular operators) in space $L^2(w)$ by characteristics of weight $w$? This classical question appeared from probability (namely, from the regularity theory of stationary stochastic processes). Initial answer was given by Hunt—Muckenhoupt—Wheeden and Helson—Szegö in the 1970’s. But the sharp estimates are only from 2000’s due to Petermichl, Volberg, Hytönen. The answer (obtained first for Ahlfors—Beurling transform, then for the Hilbert transform, and then for all Calder\’on—Zygmund operators) is that the bound is linear in $A_2$ Hunt—Muckenhoupt—Wheeden norm of weight $w$. This was a solution of the so-called $A_2$ conjecture. But what if the stationary process is a vector one (as in Wiener and Masani question from 1957)? Then the weight $W$ is a matrix weight. It was a long standing problem to prove that the same linear estimate holds. Recently we, Domelevo—Petermichl—Treil—Volberg, showed that this is not the case, $A_2$ conjecture fails, and the right sharp estimate is $[W]_{A_2}^{3/2}$.

##### July 5 - Michel Alexis (Bonn)*

*Joint seminar with PDE and Inverse Problems group.

##### Title: How to represent a function in a quantum computer?

##### Abstract:

Quantum Signal Processing (QSP) is a process by which one represents a signal $f: [0,1] \to [-1,1]$ as the imaginary part of the upper left entry of a product of $SU(2)$ matrices parametrized by the input variable $x \in [0,1]$ and some ``phase factors'' $\{\psi_k\}_{k \geq 0}$ depending on $f$. QSP was well-understood for polynomial signals $f$, but not for arbitrary signals $f :[0,1] \to (-1,1)$. Our recent work addresses more general classes of signals by using the $L^2$ theory of nonlinear Fourier analysis. Namely, after a change of variables, QSP is actually the $SU(2)$ model of the nonlinear Fourier transform, and the phase factors $\{\psi_k\}_k$ correspond to the nonlinear Fourier coefficients. Then, by exploiting a nonlinear Plancherel identity and a contraction mapping, we will show that QSP can be extended to all signals $f$ bounded in absolute value by $\frac{1}{\sqrt{2}}$. This is joint work with Gevorg Mnatsakanyan and Christoph Thiele.

##### July 12 - Zexing Li (University of Cambridge)

##### Title: On asymptotic stability for self-similar blowup of mass supercritical NLS

##### Abstract:

For slightly mass supercritical semilinear Schrodinger equations, self-similar blowup has been proven to exist and generate stable blowup dynamics, but a detailed asymptotic structure was missing. We will discuss two results leading to the asymptotic stability. Firstly we prove a finite codimensional version by introducing Strichartz estimate for the linearized matrix operator; and secondly, in a forthcoming work, we count all the unstable directions of the matrix operator and then prove the asymptotic stability without losing codimensions. New techniques are introduced to determine the spectrum for such non-self-adjoint and non-relatively-bounded perturbed operator in high dimensions, which might be useful in other context as well.

##### July 19 - TBA

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