27th International Internet Seminar

Welcome to the 27th Internet Seminar on Evolution Equations

Poster download as pdf here (opens in new tab) .

This year’s Internet Seminar is on Harmonic Analysis Techniques for Elliptic Operators. Starting in October 2023, your organisers and virtual lecturers

Moritz Egert (opens in new tab)

Robert Haller (opens in new tab)

Sylvie Monniaux (opens in new tab)

Patrick Tolksdorf (opens in new tab)

welcome you on a journey into cutting-edge techniques in harmonic analysis and their applications to differential operators with rough coefficients.


Registration starts in early September.

In order to register for the internet seminar please write an e-mail to with the following information: your name, your institution with country and whether you are a student or a local coordinator.

You will then get further information and access to the discussion platform.

Programme of the course

The study of the Laplacian on ℝn through the Fourier transform lies at the center of classical harmonic analysis. It is Plancherel's theorem that intimately links the space L2(ℝn) with the theory of weak derivatives and a symbolic calculus for the Laplacian. Examples are
the Littlewood-Paley (in)equality \[ \| u \|_2^2 = C \int_0^{\infty} \| f(- t \Delta) u \|_2^2 \, \frac{\mathrm{d} t}{t},\]
the Riesz transform "estimates" \[\| (- \Delta)^{\frac{1}{2}} u \|_2 = \| \nabla u \|_2, \]
or the fact that the resolvent (λ – Δ)-1 is given by a nice kernel that yields bounds in Lp (ℝn) for p ≠ 2.
Over the last decades, the quest to generalize these properties to elliptic operators L in divergence form with bounded measurable coefficients has triggered the development of new techniques that led to a surge of spectacular results in elliptic and parabolic PDE-theory. We will give an introductory course that covers the cornerstones of this "L-adapted Fourier analysis". The generalization of the Riesz transform estimates is the famous Kato square root problem whose solution will also be presented in the lectures. A variety of very recent results relying on these techniques will be covered in the project phase.

History of the Internet Seminar

The Internet Seminar is an international academic event dedicated to modern analysis. It was started in 1997 by the functional analysis group of Tübingen (lead by Rainer Nagel) and, since then, has been held every year, seeing the participation of more and more universities from all around the world. The aim of the course is to introduce master students, Ph.D students and post-docs to subjects related to functional analysis and evolution equations.

(For past iterations see here (opens in new tab).)

Phase 1: The Lectures (October 2023 – February 2024)

A weekly lecture will be provided on the discussion platform (opens in new tab) as lecture notes and a video recording. These lectures will be self-contained, and references for additional reading will be provided. The weekly lecture will be accompanied by exercises, and the participants are supposed to solve these problems.

Phase 2: The Projects (April – June 2024)

The participants will form small international groups to work on diverse projects which supplement the theory of Phase 1 and provide some applications. The list of projects and further details concerning the application process will be published in February 2024.

Phase 3: The Workshop (17.06. – 21.06.2024)

The final workshop takes place at the CIRM (opens in new tab) in Luminy (Marseille, France). There the project teams of Phase 2 will present their projects and additional lectures will be delivered by leading experts in the field. The official announcement of the workshop can be found here (opens in new tab) .


We expect the participants to have a basic knowledge in functional analysis, bounded operators, foundations of Hilbert spaces and some familiarity with the Fourier transform and functions in one complex variable.