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06112


Physik Wintersemester 2007/2008
06112 Integrable Systems and Spectral Curves [S]
   
Dozent Carberry Emma
Dauer 2 SWS
Studiensemester 8
Schein Ja (Vortrag)
Inhalt The seminar will explain the role of algebraic curves have to play in the study of integrable systems. Beginning with a family of equations of the form

dA=[A,B ] (called Lax form)

one can define an algebraic curve and a flow in the Jacobian of this curve (by taking the eigenvalues and eigenlines of A, respectively). If this flow is a linear, we say that the original equation is integrable and this construction provides a linearisation of the equation! The construction can be reversed, and so we obtain a correspondence between equations in Lax form and algebraic curve data. We can use this correspondence to obtain new information about the equations, for example the genus of the algebraic curve tells us the dimension through which solutions can be perturbed. Moreover Lax equations are a basic form for integrable systems, so there is an abundance of examples.

We will use as source material Hitchin's chapter in the book "Integrable Systems: Twistors, Loop Groups and Riemann Surfaces", by Hitchin, Segal and Ward, and the article "Linearising Flows and a Cohomological Interpretation of Lax Equations" by Phillip A. Griffiths. Hitchin's article begins with an introduction to the necessary algebraic curve theory, and so we shall begin there.

Literatur "Integrable Systems: Twistors, Loop Groups and Riemann Surfaces", by Hitchin, Segal and Ward, and the article "Linearising Flows and a Cohomological Interpretation of Lax Equations" by Phillip A. Griffiths.
Weitere Informationen Das erste Treffen findet am Montag, den 10. Dezember 2007 um 14:00 Uhr im Raum 2004 (Geb. L1) statt.