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 12:00--12:50 in Lincoln Hall 315.
Lecturer: Izzet Coskun, icoskun@uic.edu
Office: SEO 423
Drop in hours: WF 1-1:50
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.
D. Eisenbud and J. Harris, The Practice of Algebraic Curves
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:
Homework 1, due Friday January 31
Do problems 1.1, 1.2, 1.3, 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 Friday February 14
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 Friday February 28
Do problems 16.1, 16.2 on page 131, 17.1, 17.2, 17.3, 17.4, 17.5, 17.6, 17.7 on page 145 and 18.1, 18.2 on page 152 of Otto Forster's book on Lectures on Riemann surfaces.
Homework 4, due Friday March 14
Do problems A1, A2, A3, A4, A5, A6, A7, A8, A9, A10 on pages 32-35 of ACGH.
Homework 5, due Friday April 4
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 6, due Friday April 18
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
Homework 7, due Friday May 2
Do problems