Graduate Seminar, S4B1 - Nonlinear Fourier Transform, WS 2012

Organizers: Prof. Dr. C. Thiele, Dr. D. Oliveira e Silva
Webpage: www.math.uni-bonn.de/people/thiele/teaching/2012NLFT/

Introduction

The nonlinear Fourier transform and Fourier series is a basic model 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 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 seminar, it is also an excellent addition for students interested in related fields such as mathematical physics or integrable nonlinear equations.

Structure:

We meet once a week during the winter semester , Mo 14ct-16 In the first session we will make assignments from the list of chapters below. Students having been unable to attend the session on Monday Oct 8 have still time to enroll until October 15 please contact me if you are interested. Participants will then present their assignments throughout the semester. Students interested may request an examination for credit. Students unable to come to the first meeting should contact thiele at math.uni-bonn.de.

Script :

We use some preprint notes by Tao, T., Tsai found here: NLFT Preprint

Assignments :

The following are a tentative list of assigments. Further topics will be provided as needed, participants are welcome to suggest further related topics.
Group A:

Nonlinear Fourier series, finite case and l1 case
pages 3-11
[presenter: Diogo]
Nonlinear Fourier series, l2 on half line
pages 16-26
[presenter: Christian]
NLFS in l2, full line, forward map
pages 27-36
[presenter: Joris]
NLFS in l2, full line, existence of inverse
pages 36-44
[presenter: Stefan]
NLFS inverse of rational data
pages 44-55
[presenter: Polona]