Arbeitsgruppe Analysis und Partielle Differentialgleichungen

Graduate seminar on Advanced topics in PDE

Organizers

  • Prof. Dr. Herbert Koch
  • Prof. Dr. Christoph Thiele
  • Dr. Leonardo Tolomeo
  • Schedule

    This seminar takes place regularly on Fridays, at 14.00 (c.t.). Because of the current regulations regarding the Corona pandemic, the seminar will take place online on the Zoom platform. Please join the pdg-l mailing list or contact Dr. Tolomeo (tolomeo at math.uni-bonn.de) for further information.

    November 13 - Mihajlo Cekic

    Title:
    The Ruelle Zeta Function for nearly hyperbolic 3-manifolds

    Abstract:
    The Ruelle Zeta Function (RZF) is defined in analogy to the Riemann Zeta Function, where primes correspond to primitive closed orbits of an Anosov flow X. The RZF extends meromorphically to the whole complex plane and carries rich information about the flow. Using microlocal methods, Dyatlov-Zworski recently showed that the order of vanishing at zero n(X) of the RZF equals the minus Euler characteristic, if X is the geodesic vector field of a negatively curved surface. In this talk, I will explain an exciting novel result showing the instability of n(X) close to hyperbolic 3-manifolds, starkly contrasting the case of surfaces. The proof is based on studying the pushforward of a certain pairing between resonant states (“eigenstates of X”), regularisation arguments and wavefront set calculus. Joint work with Dyatlov, Küster and Paternain.

    November 20 - Gael Diebou

    Title:
    On the parabolic Hardy-Hénon equation in Marcinkiewicz spaces

    Abstract:
    The (elliptic) Hardy-Hénon equation was proposed as a model for rotating stellar systems. Its parabolic analogue has recently attracted more attention. In this talk, recent results pertaining to the global wellposedness theory of the Cauchy problem for the parabolic Hardy-Hénon equation will be discussed. I will emphasize on the role played by the choice of the initial data class in the analysis of the problem and present the main results which essentially deal with existence/non-existence (of global solutions), their long-time asymptotic behavior as well as self-similarity properties. Further interesting related results will be briefly mentioned if time allows.

    November 27 - Tristan Robert

    Title:
    Invariant measures for the nonlinear wave equations in 2d

    Abstract:
    In this talk we will discuss some recent results regarding the construction and invariance of Gibbs measures under the flow of nonlinear wave equations in dimension two. The case of polynomial nonlinearities being well-understood, we will discuss two examples of non-polynomial nonlinearities. These are joint works with T. Oh, P. Sosoe and Y. Wang.

    December 4 - Alex Amenta

    Title:
    Recent developments in Banach-valued time-frequency analysis

    Abstract:
    Over the last year Gennady Uraltsev and I have managed to prove some interesting results involving time-frequency analysis for functions taking values in abstract Banach spaces. In this talk I will give an overview of our results, the underlying methods (Banach-valued outer Lebesgue spaces and modulation-invariant Carleson embeddings), and some interesting problems that we (or you) may tackle in the future. In particular I will discuss
    - the bilinear Hilbert transform (on functions valued in intermediate UMD spaces),
    - bounds for variational Carleson operators, i.e. variational estimates for partial Fourier integrals (of functions valued in intermediate UMD spaces),
    - multilinear modulation-invariant Fourier multipliers with operator-valued symbols (once more, on functions valued in intermediate UMD spaces).

    December 11 - Bjoern Bringmann

    Title:
    Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity

    Abstract:
    In this talk, we discuss the construction and invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree-nonlinearity. We start with a brief review of finite-dimensional Hamiltonian ODEs, which serves as a stepping stone towards the main topic of this talk. After introducing the wave equation with a Hartree-nonlinearity, we briefly discuss the construction of the Gibbs measure, which is based on earlier work of Barashkov and Gubinelli. We also discuss the mutual singularity of the Gibbs measure and the so-called Gaussian free field. In the main part of this talk, we study the dynamics of the nonlinear wave equation with Gibbsian initial data. Our argument combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. At this point in time, this is the only proof of invariance of any singular Gibbs measure under a dispersive equation.

    December 18 - Justin Forlano

    Title:
    Normal form approach to the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces

    Abstract:
    In recent years, the normal form approach has provided an alternative method to establishing the well-posedness of solutions to nonlinear dispersive PDEs, as compared to using heavy machinery from harmonic analysis. In this talk, I will describe how to apply the normal form approach to study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the real-line and prove local well-posedness in almost critical Fourier-amalgam spaces. This involves using an infinite iteration of normal form reductions (namely, integration by parts in time) to derive the normal form equation, which behaves better than NLS for rough functions. This is joint work with Tadahiro Oh (U. Edinburgh).

    January 15 - Ruoci Sun

    Title:
    Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds

    Abstract:
    This presentation, which is based on the work Sun [2], is dedicated to describing the complete integrability of the Benjamin-Ono (BO) equation on the line when restricted to every N-soliton manifold, denoted by $ U_N$. We construct (generalized) action-angle coordinates which establish a real analytic symplectomorphism from $ U_N$ onto some open convex subset of $ R^{2N}$ and allow to solve the equation by quadrature for any such initial datum. As a consequence, $ U_N$ is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gérard-Kappeler [1]. The global well-posedness of the BO equation on $U_N$ is given by a polynomial characterization and a spectral characterization of the manifold $U_N$. Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the very shift semigroup. The construction of action-angle coordinates for each $U_N$ constitutes a first step towards the soliton resolution conjecture of the BO equation on the line.

    Bibliography:
    [1] Gérard, P., Kappeler, T. On the integrability of the Benjamin-Ono equation on the torus, arXiv:1905.01849, to appear in Commun. Pure Appl. Math., https://doi.org/10.1002/cpa.21896 , 2020.
    [2] Sun, R. Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds, Version Dec 15th.

    January 22 - Leonard Cadilhac

    Title:
    Weak (1,1) type inequalities for noncommutative singular integrals

    Abstract:
    Noncommutative Lp-spaces are originally a product of operator theory but have been discovered to be a rich object from the harmonic analytic point of view. For example, in the last decade, operators derived from singular integrals have been introduced and found some remarkable applications in that context. After presenting these various notions, I will discuss some related ongoing work with J. Parcet and J. Conde-Alonso in which we investigate weak (1,1) type inequalities for these operators. This work is meant to complement known results of the theory that usually focus on the better understood BMO approach.

    January 29 - Emiel Lorist

    Title:
    Singular stochastic integral operators

    Abstract:
    Singular integral operators play a prominent role in harmonic analysis. By replacing the integration with respect to some measure by integration with respect to Brownian motion, one obtains a stochastic singular integral operators of the form $S_K G(t) :=\int_{0}^\infty K(t,s) G(s) d W(s),$ which appear naturally in questions related to SPDE. Here G is an adapted process, W is a Brownian motion and K is allowed be singular. In this talk I will introduce Calderón--Zygmund theory for such singular stochastic integrals with operator-valued kernel K. I will first discuss $L^p$-extrapolation under a Hörmander condition on the kernel. Afterwards I will treat sparse domination and sharp weighted bounds under a Dini condition on the kernel, leading to a stochastic analog of the solution to the $A_2$-conjecture.

    February 5 - Jonathan Hickman

    Title:
    Fourier restriction estimates via real algebraic geometry

    Abstract:
    In this talk I will discuss the classical Fourier restriction conjecture in harmonic analysis. This longstanding problem investigates basic mapping properties of the Fourier transform and directly relates Fourier analysis to geometric concepts such as curvature. Underpinning the conjecture are deep questions in (continuum) incidence geometry. I will describe some recent joint work with Josh Zahl which obtains new partial results on the restriction conjecture. To do this, we use tools from real algebraic geometry to study the underlying incidence problems.

    February 12 - Marina Iliopoulou

    Title:
    A discrete Kakeya-type inequality

    Abstract:
    The Kakeya conjectures of harmonic analysis claim that congruent tubes that point in different directions rarely meet. In this talk we discuss the resolution of an analogous problem in a discrete setting (where the tubes are replaced by lines), and provide some structural information on quasi-extremal configurations. This is joint work with A. Carbery.

    Aktuelles


    18./19. Oktober: Symposium of the Global Math Network

    Plücker Lectures Kevin Buzzard & Johan Commelin 3-8 November

    Prof. Ana Caraiani erhält einen New Horizon Preis in Mathematik 2023

    17. Oktober 2022: Vorstellung der Hirzebruch-Sammlung und Auftaktveranstaltung des Otto Toeplitz-Gedächtnisstiftungsfonds

    Pius XI-Medaille an Prof. Peter Scholze verliehen

    Prof. Dr. Ana Caraiani neuer Hausdorff Chair am Mathematischen Institut

    Prof. Valentin Blomer und Prof. Georg Oberdieck erhalten ERC grants

    Prof. Peter Scholze erhält Lehrpreis der Universität

    Ausschreibung: W3-Professur Reine Mathematik

    Bonner Mathematik im Shanghai Ranking unter den Top 15 weltweit

    Prof. Carl-Friedrich Bödigheimer hat Daidalos-Münze der Studienstiftung erhalten

    Prof. Valentin Blomer erhält Lehrpreis der Mathematisch-Naturwissenschaftlichen Fakultät 2021

    Stipendien für Mathematikstudierende aus der Ukraine

    Angriff auf die Ukraine

    Corona: Maßnahmen im Mathematik-Zentrum

    Prof. Jessica Fintzen erhält einen Whitehead Prize der London Mathematical Society

    Prof. Georg Oberdieck erhält Dubrovin Medaille 2022

    Prof. Peter Scholze zum Foreign Member der Royal Society ernannt

    Prof. Dr. Jessica Fintzen neu am Mathematischen Institut

    Otto Toeplitz-Gedächtnisstiftungsfonds eingerichtet

    Prof. Catharina Stroppel ist plenary speaker beim ICM 2022 in Sankt Petersburg

    Bonner Mathematik belegt bei Shanghai Ranking den 1. Platz in Deutschland und weltweit den 13. Platz

    Prof. Georg Oberdieck erhält Heinz Maier-Leibnitz-Preise 2020

    Prof. Daniel Huybrechts erhält gemeinsam mit Debarre, Macri und Voisin ERC Synergy Grant