Mathematics

Program of the course:

• Integrable systems:

1. Integrability via Lax pair
2. The Korteweg de Vries equation as an example of an integrable system
3. Cauchy problem for the Korteweg de Vries equation with rapidly decreasing initial data via inverse scattering
4. The Cauchy problem for the Korteweg de Vries equation with periodic initial data

• Riemann Surface:

1. Definition and basic examples, genus
2. Differentials on a Surface, Jacobi Variety and Abel’s theorem
3. Riemann-Roch theorem
4. Theta-functions and Riemann vanishing theorem
5. Baker -Akhiezer function and periodic solutions of integrable equations