Math 220C - Complex Analysis

Course description: This is the third in a three-sequence graduate course on complex analysis, picking up where Math 220B left off. Topics to be presented include: Mittag-Leffler's theorem, Schwarz reflection principle, analytic continuation, sheaves of analytic functions, analytic manifolds (Riemann surfaces), harmonic/subharmonic/superharmonic functions, order of entire functions, the big Picard theorem. If time permits, we may say more about Riemann surfaces in the direction of the Riemann-Roch theorem.

Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera]. Office hours: Thu 11-12 or by appointment.

Lectures: MWF 9-10, in APM 7421.

Textbook: Required: J.B. Conway, Functions of One Complex Variable, Second Edition (same text as for Math 220B). From any UCSD computer, the book is available as a legal free download via this link.

Prerequisites: Math 220B or equivalent (with instructor's permission).

Homework: Weekly assignments, due on Fridays.

Final exam: None. Instead, the qualifying exam will be held Wednesday, May 31, 9am-12pm.

Grading: 100% homework.

Announcements:

• First lecture: Monday, April 3.
• Holidays this term: Monday, May 29.
• No office hours on May 4 or May 11; email me for assistance instead.
• Due to the qualifying exam, course evaluations are due earlier than usual (Sunday, May 14).

Assignments:

• HW 1 (due Friday, April 14): pdf.
• HW 2 (due Friday, April 21): pdf.
• HW 3 (due Friday, April 28): pdf.
• HW 4 (due Friday, May 5): pdf.
• HW 5 (due Friday, May 12): pdf.
• HW 6 (due Friday, May 19): pdf.

Topics:

• Apr 3: Schwarz reflection principle (IX.1).
• Apr 5: germs of analytic functions, analytic continuation along a path (IX.2).
• Apr 7: more on analytic continuations (IX.3.1-IX.3.5).
• Apr 10: the monodromy theorem (IX.3.6-3.9, plus a few points from IX.4).
• Apr 12: the sheaf of germs of analytic functions (IX.5.1-5.9).
• Apr 14: the Riemann surface of a complete analytic function, analytic manifolds (IX.5.10--6.2).
• Apr 17: more on analytic manifolds; Maximum Modulus Theorem, Liouville's Theorem (IX.6.3--6.13).
• Apr 19: Open Mapping Theorem (IX.6.14-6.18).
• Apr 21: complete analytic functions of local inverses (IX.6.19-21).
• Apr 24: covering spaces (IX.7).
• Apr 26: harmonic functions (X.1).
• Apr 28: construction of harmonic functions on a disc from their restriction to the boundary (X.2).
• May 1: Harnack's theorem (X.2.16); subharmonic and superharmonic functions (X.3).
• May 3: the Dirichlet problem (up to X.4.2).
• May 5: more on the Dirichlet problem (X.4.3-4.9; lecture by Alina Bucur).
• May 8: more on the Dirichlet problem, Green's functions (X.4.9-X.5.2; lecture by Alina Bucur).
• May 10: Weierstrass factorization, order and genus of an entire function (XI.1-XI.2; lecture by Dragos Oprea).
• May 12: Jensen and Poisson-Jensen formulas, Hadamard's factorization theorem (XI.1, XI.3; lecture by Dragos Oprea).
• May 15: Hadamard's factorization theorem and applications (XI.3).
• May 17: Bloch's theorem (XII.1).
• May 19: little Picard theorem (XII.2).
• May 22: Schottky's theorem, big Picard theorem (XII.3-4).
• May 24: qual review.
• May 26: qual review.
• May 29: NO LECTURE (Memorial Day).
• May 31: QUALIFYING EXAM (9am-12pm, APM 6402).
• Jun 2: prime number theorem (lecture notes: pdf).