TA1022 Riemann Surfaces (Riemannytor)

Why Riemann Surfaces:

This course is an introduction to Riemann surfaces with an combinatorial and geometric viewpoint as the subtitle of the book followed by the course says

In 1851 Riemann studies (using Dirichlet Principle) inverses of conformal functions. Riemann shows that inverses of complex analytical functions are (analytical) functions defined on surfaces (to be constructed): the so called Riemann surfaces. Which functions are analytical, meromorphic, determines the conformal structure of the surface.

Notice that we can think of having different geometries (distances, given by isometric differential structures) on a topological surface X and the corresponding Riemann surface is the class of conformal geometries.

In fact we are not interested in analytical complex functions (they are too few) but on functions on the Riemann Sphere (which is the complex projective line, bringing infinity to Earth), so we consider in general meromorphic functions.

Now a Riemann surface is a topological surface with a class of meromorphic functions.

Contents:

The course will study these topics:

• Holomorphic (analytical) maps between Riemann surfaces are branched coverings. In fact one can see a Riemann surface as a branched covering of the Riemann Sphere (earlier work of Schwarz, Hurwitz, Weiertrass, Clebsch, Klein, and many more).



• The group of meromorphic functions on a Riemann surface is (functorially) the group of of fractional functions of a projective (the infinity brought to Earth) smooth curve. So compact Riemann surfaces are (smooth) projective complex curves (earlier work of Schottky, Wiman, Torelli, Fricke, and many more).



• Finally any R. S. is the quotient of either the Riemann sphere,the complex plane or the complex disc by a discrete subgroup of the corresponding group of motions (Poincaré, Koebe)

Teaching and Organization:

The total hours of the course are 180. The contact hours of the course consist of ten lectures. Jones-Singerman's and Miranda's books contain extensive lists of relevant exercises. Try to solve a good account of them!!

Relevant materials for the course are to be found in this webpage under Extra Materials.

Examination

The examination of the course is by Hand-in Exercises that cover the contents of the course.

• To get Pass (3) the course Master students must have at least 16, out of 25, correctly solved exercises. To get a 5 master students must have at least 22 correctly solved exercises.
  
• Doctoral students must have at least 22 correctly solved exercises to get Pass.
  Doctoral students shall give a 50- minutes seminar on a topic related to the course.

Credit points

6 hp for TA1022, 8 hp for 6FMAI21.

Examiner

Milagros Izquierdo

Schedule

Timeedit

Subject area

Mathematics

Page responsible: milagros.izquierdo@liu.se
Last updated: 2024-10-02