The lecture "Fourier restriction, and its applications to nonlinear dispersive equations" will be held at the University of Bonn in Summer 2026.
Time: 13th April 2026 - 20th July 2026; always on Monday, 4:15 pm - 5:45 pm. (No lecture on 25th May due to Pentecoste Holiday.)
Location: Seminar room 1.007
Office hours: Last regular office hours in Summer 2026 will be on Monday, 20th July, 10 am - noon (note the unusual time). Later appointments are possible by email.
Oral exams (30 minutes) will take place on 27th and 28th July, 2026 (first examination period) and on 29th and 30th September, 2026 (second examination period). Please send an email to make an appointment two weeks before the examination period.
Here are the lecture notes, which will be updated continuously: Download
Topics of the lectures:
1. Basic properties of the Fourier transform, motivation: Fourier restriction and nonlinear dispersive equations, non-stationary phase estimates
2. Stationary phase: Van der Corput lemma, Morse lemma, stationary phase in higher dimensions
3. Wave packet decompositions: Proof of dispersive properties for Schrödinger and wave equations.
4. Non-endpoint Strichartz estimates for SEQ: TT^* argument, Hardy-Littlewood-Sobolev inequality. Application to 1d cubic NLS.
5. Connection between Fourier restriction and dispersive equations. Strichartz for wave equations via Littlewood-Paley theory.
More on local well-posedness: role of H^s-spaces, scaling, Persistence of regularity.
6. conservation laws for well-posed solutions, critical well-posedness for 2d SEQ, Endpoint Strichartz estimates.
7. Bilinear Strichartz estimates, Bilinear Fourier extension estimates for elliptic hypersurfaces due to Tao
8. Bilinear Fourier extension estimates for the paraboloid, II
9. Bilinear Fourier restriction for the cone, Bilinear-to-linear reduction
10. Generalizations of bilinear restriction estimates, square function estimates, Kakeya estimates
11. Bochner-Riesz estimates, L^p-estimates for the wave equation
12. Local smoothing for the wave equation
13. Periodic Strichartz estimates and remarks on decoupling
14. Decoupling II (*this is not compulsory material for the exam)
Here is the teaching evaluation: Link