#### RG Analysis and Partial Differential Equations

## V5B2 - Selected Topics in Analysis and PDE - Dispersive PDEs: deterministic and probabilistic perspectives

#### Summer Semester 2021

- Dr. Leonardo Tolomeo
- Instructor

### Lectures

- Tue 14-16 on zoom

Every Tuesday there will be a lecture, taking place on Zoom. The handwritten notes taken during the lectures will be made available on this website. Recordings of the lectures are available, please contact Dr. Tolomeo via email if you are interested. For convenience, the recording of lecture 1 (and only lecture 1) is available on this page.

- Notes of lecture 1, Recording of Lecture 1,
- Notes of lecture 2,
- Notes of lecture 3,
- Notes of lecture 4,
- Notes of lecture 5,
- Notes of lecture 6,
- Notes of lecture 7,
- Notes of lecture 8,
- Notes of lecture 9,
- Notes of lecture 10,
- Notes of lecture 11,
- Notes of lecture 12,
- Notes of lecture 13.

The information about the zoom session can be found on Basis.

### Topics

This course aims at providing the basis for the study of dispersive equations, both in the deterministic setting, and in the probabilistic one. Our goal is to show how probabilistic effects affect the behaviour of these equations, greatly improving the results available. We will cover1. Strichartz estimates for Schrödinger and wave equations.

2. Local well posedness theory Schrödinger and wave equations in subcritical Sobolev spaces H^s.

3. Global well posedness theory for Schrödinger and wave equations in H^1.

4. Ill posedness in supercritical Sobolev spaces.

5. Local well posedness in

*supercritical*Sobolev spaces for random initial data.

6. Global well posedness for random initial data.

If time allows, we will also discuss some features of the associated stochastic PDEs.

### Prerequisites

Analysis: basic real and complex analysis, basic knowledge of Fourier analysis.Probability: measure theoretical approach to probability, Gaussian random variables, independence.

### Literature

- L. Grafakos, Classical and modern Fourier analysis.
- T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis.
- M. Gubinelli, T. Souganidis, N. Tzvetkov, Singular Random Dynamics, Chapter 4.

## News

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

Hirzebruch lecture by Rahul Pandharipande on Monday, May 15, 2023

Lucas Mann recevices the Hausdorff memorial award 2021/22

Bachelor Prize 2021/22 of the BMG awarded

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