Mitarbeiter
Forschung
Lehre
WS 2010/2011
SS 2010
WS 2009/2010
SS 2009
WS 2008/2009
SS 2008
WS 2007/2008
SS 2007
WS 2006/2007
 Lineare Algebra I Integr. Systems Graphs & Surfaces Oberseminar Kolloquium
Drittmittel
Startseite

## Topics on integrable systems

 Titel: Topics on integrable systems Dozent(in): Dr. Igor Mencattini Termin: Mo. 10:00-11:30 (L1 1007) Gebäude/Raum: L1 1007 (Mathematik) Ansprechpartner: Dr. Igor Mencattini

## Inhalt der Lehrveranstaltung:

The goal of these lectures is to give an introduction to the theory of classical integrable systems.
Even if integrable systems are highly non-generic dynamical systems, they seem to pervade the landscape of modern mathematics. In fact, the notion of integrability plays a fundamental role in the modern understanding of Huyghen's principle for hyperbolic PDE, in the theory of special functions (e.g. orthogonal polynomials and their generalizations), in algebraic geometry (e.g. the Kontsevich-Okounkov-Padharipande proof of Witten's conjecture about intersection theory on the moduli space of algebraic curves) and, finally, in number theory (e.g. the geometric approach to Langland's program).
It is easier to give examples of integrable systems than to give a precise definition of integrability. Nevertheless, we can say that an integrable system is a hamiltonian system having the maximum possible number of (global) first integrals. For dynamical systems having a finite number of degrees of freedom, the property of being integrable has a nice geometrical interpretation: the phase-space is foliated by affine manifolds, where the motion of the system takes place. In some cases such affine manifolds are tori, where the hamiltonian flow linearizes.
When we move to the case of infinite numbers of degrees of freedom, we lose the geometric intuition. In this case we can only say that the equations are to some degree soluble, that such solutions can be found explicitly and that there exist general methods to find them.
Because the subject uses the methods of several different areas of mathematics (e.g. algebraic geometry, representation theory of Lie algebras and Lie groups) instead to dive in the deep of the theoretical background, I will focus the lectures on examples and I will discuss in detail the integrability of some remarkable dynamical systems. The lectures are meant to be elementary. A general knowledge of the rudiments of differential and algebraic geometry as well as of Lie groups and Lie algebras is welcome but not necessary.
A tentative plan of the lectures is the following:
1) Introduction to symplectic and Poisson geometry
2) Integrability {it \$acute{a}\$ la} Liouville, Arnold's theorem and action-angle variables
3) Synopsis of algebraic and analytical methods (\$r\$-matrices, AKS scheme, spectral curves)
4) Toda systems and spinning tops
5) Hamiltonian reduction, projections' method
6) Rational and trigonometric Calogero-Moser's systems
7) KdV's equation and its interpretation as hamiltonian system
8) Remarks on the inverse scattering method, action-angle variables, multi-solitons and rational solutions for KdV's equation
9) KdV's equation as an Euler's equation
10) Calogero-Moser's system and rational solutions of the KdV's equation

Time permitting, the following topics could be discussed:
10) Integrable hierarchies and Sato-Segal-Wilson's grassmannian
11) Bispectral problem.

## Vorkenntnis für die Lehrveranstaltung:

Linear Algebra I,II, Analysis I,II, some knowledge of manifolds

## Literatur zur Lehrveranstaltung:

1) Arnold, V.I. Mathematical methods of classical mechanics. Graduate Texts in Mathematics,60. Springer, Berlin (1984)
2) Audin, M. Spinning Tops. Cambridge studies in advanced mathematics, 51. Cambridge University Press (1996)
3) Reyman, A.G and Semenov-Tian-Shanski Group theoretical methods in the theory of finite dimensional integrable systems in Dynamical Systems VII, Encyclopedia of Mathematical Sciences, 16. Springer, (1994)
4) Guest, M. A. Harmonic maps, Loop Groups, and Integrable Systems. Student Text, 38. London Mathematical Society, (1997)
5) Ablowitz, M.J and Segur, H. Solitons and the Inverse Scattering Transform. SIAM, Studies in Applied Mathematics (1981)
6) Babelon, O. and Bernard, D. and Talon, M. Introduction to Classical Integrable Systems. Cambridge Monographs on Mathematical Physics, Cambridge University Press (2003).

## Stundenplan

 Mo Di Mi Do Fr Vorlesung10:00 - 11:30, L1 1007 Übung10:00 - 11:30, L1 1010

## weitere Informationen zu der Lehrveranstaltung:

 empfohlenes Studiensemester der Lehrveranstaltung: ab dem 6. Semester Fachrichtung Lehrveranstaltung: Mathematik, Physik Nummer der Lehrveranstaltung: 06056 Dauer der Lehrveranstaltung: 2 SWS Typ der Lehrveranstaltung: WV - Wahlvorlesung Leistungspunkte: 6 LP für Vortrag Bereich: Geometrie Lehrveranstaltungspflicht: Wahl Begleitende Lehrveranstaltung(en): 06057 Semester: WS 2006/07

 [Institut für Mathematik]   [Universität Augsburg]   [Math.-Nat. Fakultät]   [Impressum]