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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: |
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| 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 |
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