RG Analysis and Partial Differential Equations

Graduate seminar on Advanced topics in PDE

Organizers

Schedule

This seminar takes place regularly on Fridays, at 14:15. The seminar will take place in person in SemR 0.011. Please join the pdg-l mailing list for further information.

April 17 - Pauwel Van Den Eeckhaut (Bonn)

Title: Local rigidity of self-joinings and factors of pro-nilsystems

Abstract:

Pro-nilsystems play a central role in ergodic structure theory: they are precisely the structured systems governing multiple ergodic averages. In this talk I will discuss a fundamental property of this class of dynamical systems, namely its stability under taking factors. More precisely, I will explain a direct proof that any factor of an ergodic $k$-step pro-nilsystem is again a $k$-step pro-nilsystem. The main new ingredient is a local rigidity result for ergodic self-joinings of nilsystems: any ergodic self-joining sufficiently close to the diagonal joining is in fact the graph joining of an automorphism. The key geometric input here is a ‘no small subnilmanifolds’ lemma. I will then sketch how this rigidity result can be used to prove the factor-closure of pro-nilsystems.

April 24 - Alexander Volberg (MSU and Bonn)

Title: The Boolean surface area of polynomial threshold functions

Abstract: HCM event calendar

Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman–Linial conjecture.

In this work we exhibit a new geometric sense in which PTFs are tightly constrained, by studying them through the lens of the Boolean surface area (or Talagrand boundary):

$\textbf{BSA} [f] = \textbf{E} |∇ f | = \textbf{E} \sqrt{Sens_f (x)},$

which is a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-d PTF $f$ has subpolynomial Boolean surface area:

$\textbf{BSA} [f] ≤ exp(C(d) \sqrt{log n}).$

This is a superpolynomial improvement over the previous bound of $n^{1/4} (log n)^{C(d)}$ that follows from Kane’s landmark bounds on average sensitivity of PTFs [?].

Degree-$d$ PTFs thus satisfy a stronger form of geometric regularity than was previously visible from influence bounds alone. As an application, we obtain improved noise sensitivity estimates in the case of small noise parameter.

May 1 - No Seminar

May 8 - Davi Castro-Silva (University of Cambridge)

Title: An algorithmic Polynomial Freiman-Ruzsa theorem

Abstract: HCM event calendar

In a major advance in additive combinatorics, Gowers, Green, Manners, and Tao resolved the long-standing Polynomial Freiman-Ruzsa (PFR) conjecture, which characterizes approximate subgroups with only polynomial loss in parameters. This result bridges combinatorial and algebraic notions of structure, and has wide-ranging implications across combinatorics and theoretical computer science.

In this talk, I will introduce the context of the PFR theorem and describe recent joint work with Jop Briët, Srinivasan Arunachalam, Arkopal Dutt, and Tom Gur, in which we develop efficient algorithms for several equivalent formulations of this theorem. A key feature of our work is the development of new bridges connecting additive combinatorics with symplectic geometry and quantum computation.

Based on the preprint arXiv:2604.04547.

May 15 - Stefanos Lappas (Bonn)

Title: Sharp bilinear estimates for singular integral operators and their maximal counterparts with kernels in weighted spaces

Abstract: HCM event calendar

We discuss the boundedness properties of bilinear singular integral operators (including their maximal versions) associated with rough homogeneous kernels on $R$. In particular, we focus on the $L^{p_1} (R) \times L^{p_2} (R) \to L^p (R)$ bounds in the optimal quasi-Banach range of exponents $1 < p_1 , p_2 < \infty$ and $1/2 < p < \infty$, when the angular component $\Omega$ of the kernel belongs to weighted $L^q$-spaces on the unit sphere $S^1$ and has vanishing integral. This talk is based on two joint works with Petr Honzík, Lenka Slavíková and Bae Jun Park.

May 22 - Elias Schuster (Bonn)

Title: Polynomial maps between abelian groups

Abstract: HCM event calendar

Using discrete derivatives, one can define a notion of polynomials between arbitrary groups. Such polynomials arise naturally in inverse Gowers theory through a fundamental (and still only partially established) dichotomy: a bounded function f: G \to C either behaves pseudorandomly, or it correlates with a polynomial phase. This principle is crucial in establishing the existence of arithmetic patterns in subsets A \subset G.

Despite their importance, polynomial maps are only partially understood yet. To remedy this, it is valuable to develop algebraic characterizations of such functions. In this talk, we describe the construction of a universal group \Pol_k^{ab}(G) which classifies all unital polynomials of degree at most k from G into an abelian group, building on work of Jamneshan and Thom. We then present classification results of \Pol_k^{ab}(G) for a large class of abelian groups G.

May 29 - No Seminar

June 5 - Bonn-Cologne Analysis & PDE Workshop 2026

Location: Lipschitz Lecture Hall

Schedule and Abstract: HCM event calendar

Please also use the link to webpage of this event

Our group member Robert Schippa will give a talk in this event.

June 12 - Jesse Reimann (TU Delft)

Title: Schur multipliers of divided differences and their boundedness

Abstract: HCM event calendar

In this talk I will introduce multilinear Schur multipliers and their connection with multilinear Fourier multipliers and Calderon-Zygmund operators. Moreover, I will discuss recent work with Martijn Caspers, in which we found sharp constants for the multilinear Schur multipliers of divided differences.

June 19 - Double Session

Talk 1 - 14:15 - Sebastián Muñoz-Thon (Paris)

Title: Decay of correlations on Abelian covers of isometric extensions of Anosov flows

Abstract: HCM event calendar

I will report on a recent work with M. Cekić and T. Lefeuvre. We prove a complete asymptotic expansion of the correlation function in inverse powers of the time variable, for flows which arise as Abelian extensions, that is, extensions to Z^d-covers, of certain partially hyperbolic flows. This includes the frame flow of an Abelian cover of a negatively curved closed Riemannian manifold (M, g), if the frame flow on (M, g) is ergodic. As a special case, our theorem also applies to Abelian extensions of Anosov flows. The proof uses Fourier series in Z^d (Floquet theory), (microlocal) anisotropic Sobolev spaces tailored to the dynamics, as well as the semiclassical Borel-Weil calculus on principal bundles.

Talk 2 - 15:15 - Ying Wang (Basque Center for Applied Mathematics)

Title: Intertwining operators beyond the stark effect

Abstract: HCM event calendar

The main mathematical manifestation of the Stark effect in quantum mechanics is the shift and the formation of clusters of eigenvalues when a spherical Hamiltonian is perturbed by lower order terms. Understanding this mechanism turned out to be fundamental in the description of the large-time asymptotics of the associated Schrödinger groups and can be responsible for the lack of dispersion [1, 2, 3]. Recently, Miao, Su, and Zheng introduced in [4] a family of spectrally projected intertwining operators, reminiscent of the Kato’s wave operators, in the case of constant perturbations on the sphere (inverse-square potential), and also proved their boundedness in $L^p$. Our aim is to establish a general framework in which some suitable intertwining operators can be defined also for non constant spherical perturbations in space dimensions 2 and higher, furthermore we investigate the mapping properties between $L^p$-spaces of these operators. In 2D, we prove a complete result, for the Schrödinger Hamiltonian with a (fixed) magnetic potential an electric potential, both scaling critical. In higher dimensions, apart from recovering the example of inverse-square potential, we can conjecture a complete result in presence of some symmetries (zonal potentials), and open some interesting spectral problems concerning the asymptotics of eigenfunctions. This is a jointed work with Luca Fanelli, Xiaoyan Su, Junyong Zhang and Jiqiang Zheng.

[1] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1966.

[2] C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55 (1987), 329--347.

[3] B. G. Korenev, Bessel Functions and Their Applications, An International Series of Monographs in Mathematics, Taylor and Francis, 11 New Fetter Lane, London EC4P 4EE, 2002.

[4] G. N. Watson, A Treatise on the Theory of Bessel Functions. Second Edition, Cambridge University Press, 1944.

June 26 - Double Session

Talk 1 - 14:15 - Joseph Adams (Heinrich-Heine-Universität Düsseldorf)

Title: On the well-posedness theory for higher-order (d)NLS hierarchy equations

Abstract: HCM event calendar

Complete integrability of dispersive PDEs has become a central property in the well-posedness and stability analysis of PDEs that possess this property, usually yielding an infinite number of conserved quantities and often explicit families of solutions available. Classical examples of such equations are the cubic nonlinear Schrödinger (NLS) equation and its close relative the derivative nonlinear Schrödinger (dNLS) equation.

With both of these equations there are associated, higher-order, PDEs that are derived from their conservation laws - so called integrable hierarchies. In this talk we review the recent developments in the well-posedness theory of equations in these hierarchies, while not explicitly relying on their complete integrability. We will be covering said equation's derivation, (multilinear refinements of) Strichartz estimates leading to low-regularity well-posedness, as well as complementary ill-posedness results establishing optimality. We conclude with a discussion of some open questions related to the presented material.