Welcome to MATH 554 Complex Manifolds I! This course will be an introduction to Riemann surfaces. Topics will include the Uniformization Theorem, the Riemann-Roch Theorem and Serre duality, the Brill-Noether Theorem and time permitting, some aspects of moduli theory.
Prerequisites: Basic graduate level courses in algebra, topology and complex analysis.
Time and Venue: MWF 10:00--10:50 in Taft 321.
Lecturer: Izzet Coskun
Office: SEO 423
Required text: The course will mostly follow standard texts on Riemann surfaces, including
Otto Forster Lectures on Riemann Surfaces
Hershel M. Farkas and Irwin Kra Riemann Surfaces.
For the later material, we will use Arbarello, Cornalba, Griffiths and Harris Geometry of Algebraic Curves I.
Grading: There will be problem sets. These problem sets will account for your entire grade.
A syllabus for the course can be found here
Homework 1, due September 11
Do problems 1.1, 1.2, 1.4, 1.5 on pages 8-9, problems 2.1, 2.2, 2.3, 2.5 on page 13 and problems 4.1, 4.4, 4.5 on pages 30-31 of Otto Forster's book on Lectures on Riemann surfaces
Homework 2, due September 27
Do problems 5.2, 5.3, 5.4, 5.5, 5.6, 5.7 on page 39, problems 9.1, 9.2, 9.3 on page 68 and problems 10.2, 10.3, 10.4 on page 81 of Otto Forster's book on Lectures on Riemann surfaces.
Homework 3, due October 18
Do problems A1, A2, A3, A4, A5, A6, A7, A8, A9, A10 on pages 32-35 of ACGH.
Homework 4, due November 8
Do problems D2, D4, D5, D6, D7 on page 40 of ACGH A1, A2, A3, on page 136 of ACGH and B1, B2, B3 on page 137 of ACGH
Homework 5, due December 2
Do problems C1, C2, C3, C4 on page 139 of ACGH and D1, D2, D3, D4, D7, D14 parts (1), (2), (3), (4) on pages 140 and 141 of ACGH