Complex Analysis
Fall 2010

Time and place: Tue 17:30 -- 18:45 West Hall 4, Wed 17:15 -- 18:30 Research I, 120
Instructor: Pavel Tumarkin
e-mail: p dot tumarkin at jacobs-university dot de
Office: Research I, 130; Phone: 200-3517
Office hours: Wed 16:00 -- 17:00 and by appointment

Teaching Assistant: Alexey Petukhov
e-mail: a dot petukhov at jacobs-university dot de
Office: Research I, 129a
Office hours: by appointment

Textbooks: R. Miranda, Algebraic curves and Riemann surfaces
H.Farkas, I.Kra, Riemann surfaces

Preliminary course content (subject to change): Real and complex manifolds, complex structure; various ways to introduce Riemann surfaces; biholomorphic equivalence of Riemann surfaces; automorphisms of Riemann surfaces, Hurwitz theorem; meromorphic functions, differentials, Riemann-Roch theorem and its applications; uniformization, hyperbolic geometry; introduction to moduli spaces, stable curves, compactifications

Topics covered:
- Real and complex manifolds, basic definitions
- Affine and projective curves
- Maps between Riemann surfaces, local structure, holomorphic and meromorphic functions
- Riemann surfaces of algebraic functions
- Holomorphic maps and ramified coverings
- Group actions on Riemann surfaces, Galois coverings
- Topological and holomorphic uniformization, classification of Riemann surfaces
- Riemann-Hurwitz formula, classification of Riemann surfaces with signature
- Hurwitz theorem and hyperbolic geometry
- Hadamard theorem, hyperbolic uniformization, moduli spaces of hyperbolic and complex structures
- Around Riemann-Hurwitz: genus of projective curve, hyperelliptic surfaces
- Singular curves: resolving singularities
- Meromorphic 1-forms on Riemann surfaces, dimension of the space of holomorphic 1-forms
- Divisors, linear systems, maps to projective spaces, Riemann-Roch theorem
- Around Riemann-Roch: meromorphic functions, hyperelliptic and non-hyperelliptic curves, moduli space

Midterms: Midterm (take-home), due Monday, 01.11.10

Final exam: Final exam, Saturday, December 4, 10:00 -- 14:00, Lecture Hall Res. II.

Homeworks: There will be occasional homework assignments.

Grading policy (subject to change):

Grades will be computed according to the following rule (subject to minor changes):

Cutoff score: 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% 45%
Grade: 1.0 1.33 1.66 2.0 2.33 2.66 3.0 3.33 3.66 4.0 4.33