#### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

##### 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.