Wintersemester 2019/20: Integrable Systems

Prof. Dr. Katrin Wendland
Dr. Mara Ungureanu

Wann und wo: Di 14 - 16, SR 125, Ernst-Zermelo-Str. 1

Vorbesprechung: Dienstag, 23.07.2019, 13:00, SR318, Ernst-Zermelo-Str. 1
Um teilzunehmen, kommen Sie bitte in die Vorbesprechung des Seminares; eine Teilnehmerliste wird nicht vorab ausliegen.

Integrability is a feature of certain physical models which simplifies calculations, as it allows one to compute quantities not just approximately and numerically, but exactly and analytically. It can be understood as the absence of chaotic motion, or more precisely, as a hidden enhancement of symmetries which substantially constrain the motion. The first examples of systems that could be solved exactly appeared in classical mechanics: planetary motion, spinning tops, or harmonic oscillators. A common property of these systems is that they can be solved by computing the integral of a known function (hence the name "integrable systems"). In the 19th century, Liouville provided a theoretical framework to characterise such systems, but the real revolution in the field took place in the 20th century, when truly general mathematical structures emerged. More recently, the extension of these results to quantum mechanics, classical and quantum field theories, statistical mechanics, and string theory led to many important results and is still a very active field of research.

In this seminar we investigate the mathematical structures that underlie integrable systems. We focus on the interplay between the group theoretical aspects embodied by the Lax pairs of operators and the geometric ones represented by certain Riemann surfaces called spectral curves. Using examples from classical mechanics, we see how the problem of solving equations of motion transforms into a problem in group theory, how dynamical variables can be expressed in terms of theta functions associated to the spectral curve, and develop a dictionary between the two approaches. On the way, we shall learn some useful tools from symplectic geometry, Riemann surfaces, and Lie and Poisson algebras (no previous knowledge required!).

Nützliche Vorkenntnisse: Funktionentheorie, Differentialgeometrie I

Die Links führen auf Webseiten, von denen aus dem Universitätsnetz die jeweiligen Referenzen zugänglich sind. Falls kein Link gesetzt ist, finden Sie die Referenz in der Bibliothek des Mathematischen Institutes Freiburg.

Das vorläufige Vortragsprogramm finden Sie hier.
Die Vorträge können auf Deutsch oder auf Englisch präsentiert werden.

Tutorium: Dr. Mara Ungureanu