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 either online on the Zoom platform or in person in SR 0.011. Please join the pdg-l mailing list for further information.

Oct 13 - Organizational meeting

Oct 20 - Jianghao Zhang (Bonn)

Title: A Survey on Superorthogonality

Abstract:

This talk is about the superorthogonality phenomenon in harmonic analysis. We will mainly focus on the newest type of superorthogonality introduced by Gressman-Pierce-Roos-Yung. Some applications will also be briefly discussed.

Oct 27 - Kornélia Héra (Bonn)

Title: Hausdorff dimension of Besicovitch sets of Cantor graphs

Abstract:

It is well known that planar Besicovitch sets- sets containing a unit line segment in every direction- have Hausdorff dimension 2. In a joint work with Iqra Altaf and Marianna Csörnyei we consider Besicovitch sets of Cantor graphs in the plane- sets containing a rotated (and translated) copy of a fixed Cantor graph in every direction, and prove lower bounds for their Hausdorff dimension.

Nov 3 - Michel Alexis (Bonn)

Title: Some counterexamples in two-weight norm inequalities for Calderón-Zygmund operators

Abstract:

I will discuss some bad behavior that two-weight norm inequalities for Calderón-Zygmund operators on $L^p$ exhibit that are not present in the well-known (Muckenhoupt) one-weight theory for Singular Integrals. Namely, we will see via some counterexamples that two-weight norms lack a characterization in terms of the weights alone, are unstable under biLipschitz change of variables, and when p not 2, are only known to be characterized by some technical vector-valued conditions.

Nov 10 - Alexandros Eskenazis (IMJ-PRG, Sorbonne University)

Title: Some recent advances in discrete harmonic analysis

Abstract:

Boolean analysis has evolved into a multifaceted field of mathematics, blending techniques and intuition from analysis, probability and combinatorics. In this talk, we shall survey a line of recent developments in the field that has been motivated by problems in functional analysis and discrete geometry. Time permitting, selected applications in theoretical computer science will also be discussed.

Nov 17 - Yung-Chang (Martin) Hsu (Purdue)

Title: Triangular Hilbert Transform along Parabola: An alternate proof with basic Van der Corput lemma

Abstract:

In this talk, we go over a few recent developments on multilinear singular integrals to motivate the study of Triangular Hilbert Transform along parabola. We then browse through the original proof and comment on a few technicalities. Lastly, we will see a sketch of an alternate proof that circumvents those technicalities with a few remarks on the comparison and applications.

Nov 24 - Ruoyuan Liu (Edinburgh)

Location: Lipschitz Saal (joint session with RG Functional Analysis, IAM)

Title: Local well-posedness of a quadratic nonlinear Schroedinger equation on the two-dimensional torus

Abstract:

In this talk, I will present results on local well-posedness of the nonlinear Schroedinger equation (NLS) with the quadratic nonlinearity |u|^2, posed on the two-dimensional torus, from both deterministic and probabilistic points of view. For the deterministic well-posedness, Bourgain (1993) proved local well-posedness of the quadratic NLS in H^s for any s > 0. In this talk, I will go over local well-posedness in L^2, thus resolving an open problem of 30 years since Bourgain (1993). In terms of ill-posedness in negative Sobolev spaces, this result is sharp. As a corollary, a multilinear version of the conjectural L^3 -Strichartz estimate on the two-dimensional torus is obtained. For the probabilistic well-posedness, I will talk about almost sure local well-posedness of the quadratic NLS with random initial data distributed according to a fractional derivative of the Gaussian free field. I will also mention a probabilistic ill-posedness result when the random initial data becomes very rough. The first part of the talk is based on a joint work with Tadahiro Oh (The University of Edinburgh).

Dec 1 - Leonard Busch (University of Amsterdam)

Title: An Inverse Problem with Partial Neumann Data and $L^{n/2}$ Potentials

Abstract:

Motivated by electrical impedance tomography we consider a partial data inverse problem with unbounded potentials. Rather than rely on functional analytic arguments, we construct an explicit Green's function with which we construct complex geometric optics (CGO) solutions and show unique determinability of potentials in $L^{n/2}$ for the Schroedinger equation with partial Neumann data.