#### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

##### 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.

##### Dec 8- Alex Rutar (University of St. Andrews)

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##### Dec 15 - Edward McDonald (Penn State)

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##### Dec 22 -

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##### Jan 12- Thomas Alazard (CNRS and ENS, Paris-Saclay)

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##### Jan 19 - Enno Lenzmann (University of Basel)

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##### Jan 26, 14.15 - Mikel Florez Amatriain (BCAM)

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##### Jan 26, 15.15 - Tamás Keleti (ELTE)

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##### Feb 2 -

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## News

ERC Consolidator Grant for Dr. Evgeny Shinder

Jessica Fintzen wins Cole Prize

Vacancy: Office employee (Application deadline: 3.12.23)

Dr. Regula Krapf receives university teaching award

Prof. Catharina Stroppel joined the North Rhine-Westphalia Academy for Sciences and Arts

Prof. Daniel Huybrechts receives the Compositio Prize for the periode 2017-2019

Prof. Catharina Stroppel receives Gottfried Wilhelm Leibniz Prize 2023

Rajula Srivastava receives Association for Women in Mathematics Dissertation Prize

Prof. Ana Caraiani wins a New Horizons in Mathematics Prizes 2023

Pius XI Medal awarded to Professor Peter Scholze

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

Grants for Mathematics students from Ukraine

Prof. Jessica Fintzen is awarded a Whitehead Prize of the London Mathematical Society

Prof. Peter Scholze elected as Foreign Member of the Royal Society

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

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