On
September 27鈥�30, 2022 at the a mini-course of lectures will be held.
Time and place: 4:00 pm, room 417, Sobolev Institute of Mathematics (Novosibirsk, 4 Acad. Koptyug avenue).
Course name:
Complex Curves, Hurwitz Formula and Their Applications
Lecturer 鈥�
Ivan A. Panin, Doctor of Physical and Mathematical Sciences, RAS corresponding member, St. Petersburg Department of the Mathematical Institute RAS.
We will talk about complex curves, holomorphic mappings between them, branched and unramified coverings. Finally, we prove the Hurwitz formula and formulate the Riemann-Roch theorem for connected compact complex curves. We immediately note that all connected compact complex curves "de facto" turn out to be complex algebraic curves (projective). But the theory is more visual and simpler precisely for complex curves (connected and compact).
Course program:
- Lecture 1. Definitions of complex curves and holomorphic mappings. Examples.
- Lecture 2. A theorem will be proved: any non-constant holomorphic mapping between connected compact complex curves is a branched covering. Corollary: the complex projective line covers only the complex projective line.
- Lecture 3. Definition of meromorphic functions on a connected compact complex curve. Examples. The Riemann-Roch question will be formulated and the theorem of the same name will be formulated.
- Lecture 4. The Riemann-Roch theorem allows you to reduce non-linear problems to linear ones. Interesting examples of the use of the Riemann-Roch theorem will be presented.
Ivan's comment:
There will be many exercises. Free knowledge of complex numbers is assumed. If we have time, then the rule of addition on an elliptic curve will be constructed and explained.
