On Tuesday, April 16th, at 2pm in room 630, starts a series of four lectures on the topic “Solutions in Mathematical Physics and Riemann Surfaces“. The lecturer will be Prof. Susumu Tanabe from the Moscow Institute of Physics and Technology (MIPT).
The purpose of this lecture series is to explore the algebraic-geometric nature of solutions to certain nonlinear differential equations in mathematical physics, specifically those that arise in the theory of shallow water. After introducing the theory of Riemann surfaces, we will consider theta functions with quasi-periodicity relative to shifts on a lattice. Using these theta functions, we will express specific solutions to equations such as the sine-Gordon, Korteweg-De Vries, and Kadomtsev-Petviashvili (KP).
Content of the lectures:
1. Definition of a Riemann surface. The branching point and its multiplicity. The genus of a Riemann surface.
2. Meromorphic functions and differential forms (differentials) on a Riemann surface.
3. Periods of differential forms (differentials) and the canonical basis of homological cycles.
4. Jacobi manifold and Jacobi’s inversion problem.
5. Periods of holomorphic differentials. Elliptic functions.
6. Basic properties of theta functions. Riemann’s theorem on the zeros of theta functions.
7. Baker-Akhiezer function.
8. Application of the Baker-Akhiezer function to the solution of the sine-Gordon, Korteweg-de Vries and Kadomtsev-Petviashvili equations.
The course is designed for students in their third year and above, as well as undergraduates and graduates who are familiar with complex analysis. The course consists of four lectures of 1.5 hours each.
- 2024-04-16 14:00:00
- 2024-04-16 15:30:00