Math 220C - Complex Analysis

Welcome to Math 220c!

Course description:

This is the third in a three-sequence graduate course on complex analysis.

Topics include: harmonic functions, entire functions and Hadamard factorization, Riemann surfaces, holomorphic sheaves.

Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu

Lectures: MWF 1:00-1:50 in APM B-402A

Office hours: W 2:00 - 3:30 in APM 6-101

Teaching Assistant: Nandagopal Ramachandran, naramach "at" ucsd

Office Hours: TH 4:30 - 6:30 in APM 5-748

Lectures:

The lectures will be given in person. It may be necessary to make adjustments during the term, depending on how the current pandemic situation progresses. Regular attendance is expected (barring unforeseen circumstances and special arrangements).

Textbook: Functions of One Complex Variable, by J. B. Conway.

Additional Reading: Complex Analysis, by Lars Ahlfors.

Prerequisites:

Math 220B. However, this is a graduate level course, so at times, we may use notions from related fields, including topology and real analysis. I am happy to discuss prerequisites on an individual basis. If you are unsure, please don't hesitate to contact me.

Grading:

The final grade is based on homework and regular attendance. The Qualifying Exam score is not part of the final grade.

The problem sets are mandatory and are a very important part of the course. We will drop the lowest problem set.

The problem sets are due on Fridays in Gradescope. In order not to interfere with lecture time, there will be a grace period on gradescope.

There are several ways to upload your homework in Gradescope. A possible method is described here.

Important dates:

First class: Monday, March 28
University Holiday: Monday, May 30
Last class: Friday, June 3
Qualifying Exam: May 17, 1-4 PM

Lecture Summaries

Lecture 1: Introduction. Harmonic functions. Harmonic conjugates and simply connected domains. Mean value property - PDF
Lecture 2: More on the maximum principle. Introduction to Poisson's integral formula and Dirichlet problem - PDF
Lecture 3: The Poisson kernel. Poisson's formula. Harnack's inequality - PDF
Lecture 4: Schwarz integral formula. The Dirichlet problem for the unit disc - PDF
Lecture 5: More on the Dirichlet problem for the disc. Mean value property and harmonic functions - PDF
Lecture 6: Convergence of harmonic functions. Harnack's convergence theorem. Subharmonic functions - PDF
Lecture 7: Poisson modification (bumping) of subharmonic functions. Properties of the Poisson modification. The Perron family - PDF
Lecture 8: The Perron function is harmonic - PDF
Lecture 9: Barriers and characterization of Dirichlet regions - PDF
Lecture 10: Jensen's formula. Poisson-Jensen - PDF
Lecture 11: Applications of Jensen. Connection between the growth of entire functions and the distributions of zeroes. Order of entire functions. Rank. Critical exponent - PDF
Lecture 12: Rank. Critical exponent. Genus. Examples. Introduction to Hadamard's factorization. Applications of Hadamard to weak forms of Picard's theorems - PDF
Lecture 13: Bounding the order in terms of the genus. Bounding the rank in terms of the order - PDF
Lectures 14 - 15: Proof of Hadamard's theorem - PDF
Lecture 16: Presheaves. Sheaves. Germs and stalks. Ringed spaces. Morphisms of ringed spaces. Riemann surfaces. Holomoprhic functions on Riemann surfaces. Holomorphic maps between Riemann surfaces - PDF
Lecture 17: More on coordinate charts, holomorphic functions, holomorphic maps. Meromorphic functions. Divisors. Principal divisors - PDF
Lecture 18: Compatible charts. Examples of Riemann surfaces: Riemann sphere, curves, projective curves, tori - PDF
Lecture 19: Basic results on Riemann surfaces: identity theorem, open mapping theorem, maximum modulus. More on divisors - PDF
Lecture 20: Sheaves attached to divisors. Three questions about divisors on Riemann surfaces. Motivation for sheaf cohomology - PDF
Lecture 21: Exact sequences of sheaves. Three important examples - PDF
Lecture 22: Kernels and cokernels of morphisms of sheaves. Sheafification. Flabby sheaves - PDF
Lecture 23: Flabby sheaves. The canonical flabby resolution. Sheaf cohomology - PDF
Lecture 24: Cohomology of flabby sheaves. Long exact sequence in cohomology. Sheaves of modules. Locally free sheaves. Coherent sheaves - PDF
Lecture 25: Coherent sheaves. Holomorphic Euler characteristics. The Riemann-Roch theorem - PDF

Homework:

Homework 1 due Friday, April 1 - PDF. Solutions PDF.
Homework 2 due Friday, April 8 - PDF. Solutions PDF.
Homework 3 due Friday, April 15 - PDF. Solutions PDF.
Homework 4 due Friday, April 22 - PDF. Solutions PDF.
Homework 5 due Friday, April 29 - PDF. Solutions PDF.

Past Exams

A review sheet for the Qualifying Exam, written by former graduate students. Please use at your own risk, and be advised that some of the topics may be different than what was covered this year.

Review: Friday, May 13, 1-2PM

There will be no lecture on Monday, May 9, Monday, May 16 and Wednesday, May 18.

Dragos will hold in-person office hours for the Qualifying Exam: May 9, 1-2PM, May 11, 2-3:30PM, May 16, 1-2PM

Nandagopal will hold in-person office hours for the Qualifying Exam: May 11, 4-6PM, May 13, 4-6PM, May 16, 3-5PM. (These times will replace his regular office hours on Thursday this week).