Short course on

Compact Riemann surfaces and the KdV equation

by Ian McIntosh (University of York)

Waseda University

September 2020

real time Zoom lectures*

 

 

This series of lectures will be an introduction to compact Riemann surfaces, with a view towards applications to integrable systems such as the KdV equation.The author is a renowned specialist in this area and an excellent speaker. The only prerequisites are first course in complex analysis and calculus of several variables.

Each lecture will include ample time for discussion and questions. Graduate students and researchers in mathematics, physics, and engineering are welcome (see below for registration information).

LECTURE NOTES ARE AVAILABLE ! (updated 25 September 2020)

EXERCISES !


Schedule:

Week 1

Monday September 14, 17:30-19:00

Tuesday September 15, 17:30-19:00

Wednesday September 16, 17:30-19:00

Thursday September 17, 17:30-19:00

Week 2

Monday September 21, 17:30-19:00

Tuesday September 22, 17:30-19:00

Wednesday September 23, 17:30-19:00

Thursday September 24, 17:30-19:00


Topics

1. Riemann surfaces as complex 1-manifolds.
2. Holomorphic and meromorphic functions.
3. Vector fields and differentials.
4. Integration and Poincare duality.
6. The residue theorem and period integrals.
7. Divisors, the Jacobi variety and the Abel map.
8. The Riemann theta-function.
9. Baker-Akhiezer functions and solutions of the KdV equation.

Some references

Basic Riemann surface theory:
- Introduction to Riemann surfaces, G. A. Springer (Wiley)
- Algebraic curves and Riemann surfaces, R. Miranda (AMS Graduate Studies in Math., Vol. 5),
- Lectures on Riemann surfaces, O. Forster (Springer Graduate Texts in Math., Vol 81)
Theta-functions and KdV:
- Theta-functions and nonlinear equations, B. Dubrovin, (with an appendix by I. M. Krichever). Uspekhi Math Nauk 36 (1981), no. 2 (218), 11-80. (English translation: Russian Math. Surveys 36 (1981), no. 2, 11-92 (1982).)


* Registered students will receive Zoom log in information via Waseda Moodle. Visitors/guests are welcome to attend. In order to receive Zoom log in information, please send an e-mail to Martin Guest (martin at waseda.jp) stating your name, university affiliation, and position/student status.


The course is an activity of the

Mathematics and Physics Unit "Multiscale Analysis, Modelling and Simulation" Top Global University Project, Waseda University

Students may register to obtain credit for this course (MATX72ZL Advanced Study of Nonlinear Mechanics).


These lectures are also supported by the Institute for Mathematical Science, Waseda University