65-412 Seminar on Rieman Surfaces and Integrable Systems (in English)

Instructors: Prof. Dr. Jörg Alexander Teschner

Event type: Seminar

Displayed in timetable as: M-Sem

Hours per week: 2

Language of instruction: English

Min. | Max. participants: 5 | 14

Comments/contents:
Goal of this seminar, and of the accompanying lectures 65-411 is to introduce into important aspects of the theory of classically integrable systems, many of which involve beautiful parts of the theory of Riemann surfaces. The main goal of the seminars will be to understand the main ideas of Hitchin's spectral transform, a geometric construction exhibiting the exact integrability of large classes of the known integrable models. If time permits and if there is interest we may also discuss the relation between exact integrability, Special Geometry and supersymmetry.

1. Background from Poisson-geometry, in particular the

Liouville-Arnold theorem and the notion of

algebraically completely integrable system

2. Relevant aspects from the theory of Riemann surfaces

- line bundles, sheaves and vector bundles

- branched covers and the direct image

3. Matrix polynomials and Lax pairs

(4. Integrability, Special Geometry and supersymmetry)

While it is certainly highly recommended to follow both the seminar accompanying lectures 65-411 "Riemann Surfaces und Integrable Systems", it should in principle also be possible to participate only in the seminar.

Literature:
Literature:

1. V. I. Arnold. Mathematical Methods of Classical Mechanics.

Springer-Verlag, New York, 1978.

2. N. J. Hitchin, G. Segal, N. Woodhouse, and R.S. Ward.

Twistors, Loop Groups and Riemann Surfaces.

Oxford University Press, Oxford, 1997.

3. Hitchin, Nigel.

Stable bundles and integrable systems.

Duke Math. J. 54 (1987), no. 1, 91 - 114.

4. D. Freed.

Special Kahler Manifolds.

Commun. Math. Phys. 203 (1999), 31 - 52.