MAT 543: We will study some aspects of the basic theory of Riemann surfaces. Quick review: Holomorphic functions of one variable, fundamental group, covering spaces... Proper holomorphic maps of one variable. Analytic continuation: sheaves, algebraic functions, associated branched coverin gs, the monodromy theorem. Differential forms and integration. The topics covered will be a subset of the following too big set. -- Compact Riemann surfaces: calculation and finite dimensionality of Cech cohomology groups, harmonic forms, Hodge decomposition, Riemann-Roch, Serre Duality, Abel Theorem, Jacobi inversion, prescribing principal parts for functions and for differential forms. -- Open Riemann surfaces: Dirichlet problem, Weyl Lemma, Runge approximation, Mittag-Leffler, Riemann mapping, triviality of vector bundles, Riemann-Hilbert. There will be no homework and no exams. Students are expected to give in-class presentations on selected related topics/exercises. I will assist students in pr eparing these presentations.
Textbook(s): ``Riemann Surfaces" by O. Forster. This text as well as K. Siegel's ``Topics in complex function theory" (vol 1, 2, 3) are on reserve in the library.
Exams: There will be no exams.
Office Hours: By appointment: on TU and TH 10:15-11:15 and on TU 2:15-3:15. MAT Tower 5-108 (on TU am I could be in P-143).
Grade: Based on in-class presentations and in-class participation.
Schedule of in-class presentations: To be announced.
If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. All information and documentation of disability is confidential. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated.