Course Code : | 2600WETDGA |

Study domain: | Mathematics |

Academic year: | 2019-2020 |

Semester: | 2nd semester |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | English |

Exam period: | exam in the 2nd semester |

Lecturer(s) | Sonja Hohloch Tom Mestdag |

At the start of this course the student should have acquired the following competences:

an active knowledge of

general notion of the basic concepts of

specific prerequisites for this course

- competences corresponding the final attainment level of secondary school

an active knowledge of

- English

a passive knowledge ofIf everybody in the audience speaks Dutch the subject may/will be taught in Dutch. The lecture notes and/or literature are/stay nevertheless in English.

- English

- general knowledge of the use of a PC and the Internet

general notion of the basic concepts of

Information may be posted on the webpages of the professors and/or in blackboard, so the student should be able to access this information...

specific prerequisites for this course

Necessary "pre-knowledge":

- Standard theorems of ODE theory and basic notions of dynamical systems (flow). One possible source are the (Dutch) lecture notes on "ODEs and dynamical systems" by Sonja Hohloch (---> her webpage) or any standard text book on ODEs and dynamical systems.
- Standard notions of differential geometry. One possible source are the (Dutch) lecture notes by Tom Mestdag (---> webpage) or any standard text book on differential geometry.

Useful/helpful knowledge, but not mandatory: The master classes

- Integrable Hamiltonian systems
- Functional analysis
- Algebraic topology

- Broader knowledge in advanced analysis and geometry and their interactions.

Tom Mestdag's part of the course starts with

(1) Elementary calculus of variations, and its applications.

(2) Lie groups, actions of Lie groups on differentiable manifolds.

(3) Fibre bundles (vector bundles, principal fibre bundles, etc), connections on fibre bundles.

Then the course usually focuses on

(a) Vector fields with a symmetry group, reduction, reconstruction.

(b) Applications in the context of Lagrangian systems, of symplectic geometry and of Poisson geometry.

Sonja Hohloch's part of the course studies properties of integrable Hamiltonian PDEs with the Korteweg-de Vries (KdV) equation as leading example, i.e., several items of Tom Mestdag's part reappear in an infinite dimensional setting. More precisely, we study (at least) the following topics concerning the KdV equation:

(1) Motivation from physics via soliton waves and intuition how to get from the wave equstion to KdV.

(2) Hamiltonian and integrable properties of KdV: the KdV hierarchy and its motivation.

(3) Existence and uniqueness questions of solutions of KdV.

(4) Symmetries of KdV.

(5) Scattering and inverse scattering of KdV.

The course has an international dimension.

Class contact teachingLectures Practice sessions

ExaminationOral without written preparation Oral with written preparation Closed book Open book

The lecturers will provide either course notes and/or relevant literature.

Sonja Hohloch will post literature or references for her half of the course on her WEBPAGE, NOT ON BLACKBOARD.

S. Kobayashi & K. Nomizu, Foundations of Differential Geometry, Wiley- Interscience (1996)

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer-Verlag, 2008.

Ablowitz & Clarkson: "Solitons, Nonlinear Evolution, Equations and Inverse Scattering", Cambridge University Press 1992

Tom Mestdag: tom.mestdag AT uantwerpen.be

Sonja Hohloch: sonja.hohloch AT uantwerpen.be