# Graduate Seminar, S4B1: Nonlinear Fourier Analysis, WS 2015/16

Organizers: Prof. Dr. Christoph Thiele, Dr. Diogo Oliveira e SilvaWebpage: www.math.uni-bonn.de/ag/ana/WiSe1516/S4B1_WS_1516.html

## Introduction

The 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.## Structure

We 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.

## Assignments

1. (Oct 27)*[Diogo Oliveira e Silva]*

__The nonlinear Fourier transform on l^0, l^1 and l^p__2. (Nov 3)

*[Diogo Oliveira e Silva]*

__The nonlinear Fourier transform on l^2 (half-line)__3. (Nov 10)

*[Marco Fraccaroli]*

__The nonlinear Fourier transform on l^2 (full line, bounded a)__4. (Nov 17)

*[Angus Griffith]*

__The nonlinear Fourier transform on l^2 (full line, unbounded a)__5. (Nov 24)

*[Damian Dabrowski]*

__The Riemann-Hilbert problem for rational functions__6. (Dec 1)

*[Joris Roos]*

__Maximal functions associated to filtrations__7. (Dec 8)

*[Wenhui Shi]*

__Absolutely continuous spectrum for one-dimensional SchrÃ¶dinger operators with slowly decaying potentials: some optimal results__8. (Dec 15)

*[Gianmarco Brocchi]*

__Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations__9. (Jan 12)

*[Polona Durcik]*

__Uniform constants in Hausdorff-Young inequalities for the Cantor group model of the scattering transform__10. (Jan 19)

*[Michal Warchalski]*

__Nonlinear phase unwinding of functions__11. (Jan 26)

*[Gennady Uraltsev]*

__A Carleson theorem for a Cantor group model of the scattering transform__12. (Feb 2)

*[Christian Zillinger]*

__On the absolutely continuous spectrum of one-dimensional SchrÃ¶dinger operators with square summable potentials__**Letzte Änderung:**28.10.2015

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