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 Riem. Geometrie Sem. math. Phys. Sem. alg. curves Seminar int. syst. Oberseminar gen. cplx. geom. Kolloquium
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Seminar: Integrable Systems and Spectral Curves

 Dozent(in): Visiting Professor Dr. Emma Carberry Termin: 07.01.2008 - 04.02.2008 montags im Raum 3008 (L1) Gebäude/Raum: Ansprechpartner: Visiting Professor Dr. Emma Carberry

Inhalt der Lehrveranstaltung:

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.

The first meeting will be on Monday 10th December, 14:00, in room 2004.
The seminar will consist of one lecture per week plus an additional hour in which we will discuss. The additional hour will be scheduled at the first meeting at a time convenient for the participants.

Literatur zur Lehrveranstaltung:

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.

weitere Informationen zu der Lehrveranstaltung:

 empfohlenes Studiensemester der Lehrveranstaltung: ab dem 8. Semester Fachrichtung Lehrveranstaltung: Dipl.-Mathematik Nummer der Lehrveranstaltung: 0 Dauer der Lehrveranstaltung: keine Angabe Typ der Lehrveranstaltung: S - Seminar Semester: WS 2007/08

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