This is a first semester course for the M2 of the Master Program in Mathematical Physics at the IMB.
The course will be an introduction to Riemann surfaces and to some of their relations with integrable systems. We will start by reviewing some known fact complex analysis and considering some motivation for introducing Riemann surfaces. Then we will develop some of the classical theory of Riemann surfaces. If time permits we will consider some application to the theory of integrable systems, for example the construction of solutions to the KP hierarchy via the Baker-Akhiezer functions.
Topics will include: definition of Riemann surface and basic examples, plane algebraic curves, hyperelliptic curves, holomorphic coverings, fundamental group, Riemann-Hurwitz theorem, homology groups, abelian differentials, meromorphic functions, divisors, Abel theorem, Riemann-Roch theorem.
The course material will be posted on the Teams group of the course.
Main references: