Instructor: M. Kapovich

Special TA: Y. Lai

Well, a Riemann surface is a certain kind of Hausdorff space. You know what a

Hausdorff space is, don’t you? Its also compact, ok. I guess it is also a manifold.

Surely you know what a manifold is. Now let me tell you one non-trivial theorem,

the Riemann–Roch Theorem.

— Gian-Carlo Rota’s recollection of Lefschetz lecturing in the 1940’s, quoted in

A Beautiful Mind by Sylvia Nasar.

Course description:

The main objectives of the course are:

1. Uniformization theorem for Riemann surfaces.

2. Riemann-Roch theorem for Riemann surfaces.

3. Ahlfors-Schwartz lemma: Curvature, distance and conformality.

4. Introduction to Teichmuller spaces and moduli of Riemann surfaces.

I will try to make the treatment of the Uniformization theorem and Riemann-Roch as elementary as possible, avoiding harmonic analysis and PDEs.

All the tools that we will need are: the Riemann mapping theorem, reflection principle, convergence for normal families of holomorphic maps, residues of meromorphic functions and intergation of differential forms along curves.

Prerequisites: Algebraic Topology (215AB), Differential Topology (239), Differential Geometry (240AB), Complex Analysis (185AB and 205).

Sources:

1. H. Farkas and I. Kra "Riemann surfaces", Springer Verlag.

2. R. Narasimhan "Compact Riemann surfaces", Birkhauser.

3. O. Lehto anf K. Virtanen "Quasiconformal mappings in the plane", Springer Verlag.

4. R. Miranda "Algebraic curves and Riemann surfaces", Publ. AMS, GSM, Vol. 5.

5. Various handouts.

1. Uniformization theorem for Riemann surfaces.

2. Riemann-Roch theorem for Riemann surfaces.

3. Ahlfors-Schwartz lemma: Curvature, distance and conformality.

4. Introduction to Teichmuller spaces and moduli of Riemann surfaces.

I will try to make the treatment of the Uniformization theorem and Riemann-Roch as elementary as possible, avoiding harmonic analysis and PDEs.

All the tools that we will need are: the Riemann mapping theorem, reflection principle, convergence for normal families of holomorphic maps, residues of meromorphic functions and intergation of differential forms along curves.

Prerequisites: Algebraic Topology (215AB), Differential Topology (239), Differential Geometry (240AB), Complex Analysis (185AB and 205).

Sources:

1. H. Farkas and I. Kra "Riemann surfaces", Springer Verlag.

2. R. Narasimhan "Compact Riemann surfaces", Birkhauser.

3. O. Lehto anf K. Virtanen "Quasiconformal mappings in the plane", Springer Verlag.

4. R. Miranda "Algebraic curves and Riemann surfaces", Publ. AMS, GSM, Vol. 5.

5. Various handouts.