First Level Degree in Mathematics
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RIEMANN SURFACES - 6 CFU
Teacher
Scheduled Period
III Year - 2 Semester | 28/02/2022 - 11/06/2022
Hours: 48 (24 exercises, 24 lesson)
Prerequisites
Algebra, geometry and analysis of the first two years. Basic knowledge on holomorphic functions of one variable.
Target skills and knowledge
The course aims to develop the fundamental concepts regarding compact Riemann surfaces (in particular, on spheres and tori), introducing the notion of genus and its interpretations (Riemann-Hurwitz formula and Riemann-Roch theorem).
Examination methods
Written examination
Assessment criteria
The exam test the acquired knowledge during the course and the capacity to apply this knowledge in particular cases. In particular the written exam will involve theory and exercises.
Course contents
Introduction to the geometry of compact Riemann surfaces. Topics:
- Definition of a Riemann surface;
- Elementary properties of holomorphic and meromorphic functions on a Riemann surface;
- Detailed study of the Riemann sphere and 1-dimensional complex tori;
- Divisors on compact Riemann surfaces, linear systems;
- Differential forms and Riemann-Roch theorem, applications;
- First notions of homology, Jacobians of Riemann surfaces, Abel-Jacobi theorem.
Planned learning activities and teaching methods
Lectures and exercise classes.
Additional notes about suggested reading
Other than the textbook, the professor's personal notes and other material will be avilable online.