Graduate Seminar, S4B1: Nonlinear Fourier Analysis, WS 2015/16Organizers: Prof. Dr. Christoph Thiele, Dr. Diogo Oliveira e Silva
IntroductionThe nonlinear Fourier transform and the nonlinear Fourier series are basic models for a type of analysis done in many parts of mathematics. Related key words are: scattering transform, orthogonal polynomials, Schur's algorithm, Riemann-Hilbert problems, integrable systems, operator theory. This seminar will present an introduction to the theory that runs parallel to the usual basic development of linear Fourier analysis, with particular emphasis on the intricacies in the nonlinear setting that occur on the space of square summable sequences. For interested students, the seminar may lead to a Master thesis assignment. Due to the foundational nature of the subject, this seminar is also an excellent addition for students interested in related fields such as mathematical physics or integrable nonlinear equations.
StructureWe will meet once a week starting on the second week of the semester on Tuesdays 16:00(s.t.)-18:00 in SR0.003. A preliminary organizational meeting took place on Thursday, October 15, 14:00(s.t.)-14:30. This occasion was used to make assignments from this list of topics and to decide the final schedule for our subsequent meetings, see below. Participants will present their assignments throughout the semester. Interested students may request an examination for credit.
Assignments1. (Oct 27) The nonlinear Fourier transform on l^0, l^1 and l^p [Diogo Oliveira e Silva]
2. (Nov 3) The nonlinear Fourier transform on l^2 (half-line) [Diogo Oliveira e Silva]
3. (Nov 10) The nonlinear Fourier transform on l^2 (full line, bounded a) [Marco Fraccaroli]
4. (Nov 17) The nonlinear Fourier transform on l^2 (full line, unbounded a) [Angus Griffith]
5. (Nov 24) The Riemann-Hilbert problem for rational functions [Damian Dabrowski]
6. (Dec 1) Maximal functions associated to filtrations [Joris Roos]
7. (Dec 8) Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: some optimal results [Wenhui Shi]
8. (Dec 15) Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations [Gianmarco Brocchi]
9. (Jan 12) Uniform constants in Hausdorff-Young inequalities for the Cantor group model of the scattering transform [Polona Durcik]
10. (Jan 19) Nonlinear phase unwinding of functions [Michal Warchalski]
11. (Jan 26) A Carleson theorem for a Cantor group model of the scattering transform [Gennady Uraltsev]
12. (Feb 2) On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials [Christian Zillinger]
Letzte Änderung: 28.10.2015