# Graduate seminar on Advanced topics in PDE

## Organizers

• Prof. Dr. Herbert Koch
• Prof. Dr. Christoph Thiele
• Marco Fraccaroli
• ## 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 Marco Fraccaroli (mfraccar at math.uni-bonn.de) for further information.

##### Title:
Invariance of the Gibbs measures for the periodic generalized KdV equations

##### Abstract:
In this talk, we consider the periodic generalized Korteweg-de Vries equations (gKdV). In particular, we study gKdV with the Gibbs measure initial data. The main difficulty lies in constructing local-in-time dynamics in the support of the measure. Since gKdV is analytically ill-posed in the L2-based Sobolev support, we instead prove deterministic local well-posedness in some Fourier-Lebesgue spaces containing the support of the Gibbs measure. New key ingredients are bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Once we construct local-in-time dynamics, we apply Bourgain's invariant measure argument to prove almost sure global well-posedness of the defocusing gKdV and invariance of the Gibbs measure. Our result completes the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations. This talk is based on joint work with Nobu Kishimoto (RIMS, University of Kyoto).

##### Title:
The Fourier Extension problem through a time-frequency perspective

##### Abstract:
An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $L^{2+\frac{2}{d}}([0,1]^{d})$ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1})$ for every $\varepsilon>0$. It has been fully solved only for $d=1$ and there are many partial results in higher dimensions regarding the range of $(p,q)$ for which $L^{p}([0,1]^{d})$ is mapped to $L^{q}(\mathbb{R}^{d+1})$. In this talk, we will take an alternative route to this problem: one can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions , meaning for all $g$ of the form $g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d})$. Time permitting, we will also address multilinear versions of the statement above and get similar results, in which we will need only one of the many functions involved in each problem to be of such kind to obtain the desired conjectured bounds. This is joint work with Camil Muscalu.

##### Title:
Fourier interpolation with zeros of zeta and L-functions

##### Abstract:
We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other L-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on strengthening of Knopp's abundance principle for Dirichlet series with functional equations and a careful analysis of the associated Dirichlet series kernel, with coefficients arising from certain modular integrals for the theta group.

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