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, the little/big Picard theorem, Riemann surfaces, holomorphic sheaves, and others.

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

Online lectures: MWF 3:00-3:50

Virtual office hours: W 4:00 - 5:30

Teaching Assistant: Shubham Sinha, shs074 "at" ucsd

Virtual office Hours: Thursday 9:00AM - 11:00AM

Online teaching:

We will use Zoom for both lectures and virtual office hours. Regular attendance is expected (barring unforeseen circumstances and special arrangements). You can access the Zoom Room for this course from Canvas by cliking on "Zoom LTI PRO" at the bottom of the menu on the left. The Zoom Meeting ID is available via Canvas.

The lectures will be given synchronously. Class sessions will be recorded and later made available on Canvas. Sharing the recordings or the links to the recordings with anyone is prohibited.

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. 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 on 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 on Gradescope. A possible method is described here.

Important dates:

First class: Monday, March 29
University Holiday: Monday, May 31
Last class: Friday, June 4
Qualifying Exam: Tuesday, May 18, 5-8pm

Lecture Summaries

Lecture 1: Harmonic functions. Harmonic conjugates and simply connected domains. The mean value property. The maximum and minimum principle for harmonic functions - 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 unit disc. Mean value property implies the function is harmonic. Convergence of harmonic functions - PDF
Lecture 6: Harnack's convergence theorem. Subharmonic functions. Poisson modification (bumping) of subharmonic functions - PDF
Lecture 7: Properties of the Poisson modification. The Perron family. The Perron function is harmonic - PDF
Lecture 8: Barriers and the Dirichlet problem. Jensen's formula - PDF and Supplement
Lecture 9: Poisson-Jensen formula. Connection between the growth of entire functions and the distributions of zeroes. Order of entire functions and examples - PDF
Lecture 10: Introduction to Hadamard's factorization. Rank. Critical exponent. Genus. Examples - PDF
Lecture 11: Applications of Hadamard to weak forms of Picard's theorems. Bounding the order in terms of the genus - PDF
Lecture 12: Proof of Hadamard's Theorem - PDF
Lecture 13: Introduction to the Little and Great Picard theorems. Applications. The strategy of the proof. Landau's lemma and its consequences - PDF
Lecture 14: Proof of Little Picard from Bloch's Theorem. A discussion of Bloch's constant. A sharpened open mapping theorem. Strategy for the proof of Bloch - PDF
Lecture 15: Proof of Bloch's Theorem. Remarks on the original proof of Little Picard - PDF
Lecture 16: Recap of the proof of Little Picard and onto Great Picard. The family of functions omitting two values. Schottky's Theorem - PDF
Lecture 17: The family of functions omitting two values is normal in the extended sense. The proof of Great Picard - PDF
Lecture 18: Presheaves. Sheaves. Examples. Germs. Stalks - PDF
Lecture 19: Ringed spaces. Morphisms. Riemann surfaces. Holomorphic functions - PDF
Lecture 20: More on the definiton of Riemann surfaces. Some examples of Riemann surfaces. Holomorphic and meromorphic functions. Order of zeroes and poles. Divisors - PDF
Lecture 21: More examples of Riemann surfaces. Basic results: identity theorem, open mapping theorem, maximum modulus. More on divisors - PDF
Lecture 22: Sheaves attached to divisors. Three questions about divisors on Riemann surfaces. Motivation for sheaf cohomology - PDF
Lecture 23: Morphisms of sheaves. Exact sequences of sheaves. Three important examples - PDF
Lecture 24: Kernels. Cokernels. Sheafification. Flabby sheaves - PDF
Lecture 25: The canonical flabby resolution. Sheaf cohomology - PDF
Lecture 26: Sheaves of modules. Locally free sheaves. Coherent sheaves. Holomorphic Euler characteristics. The Riemann-Roch theorem - PDF

Homework:

Homework 1 due Friday, April 2 - PDF. Solutions PDF.
Homework 2 due Friday, April 9 - PDF. Solutions PDF.
Homework 3 due Friday, April 16 - PDF. Solutions PDF.
Homework 4 due Friday, April 23 - PDF. Solutions PDF.
Homework 5 due Friday, April 30 - PDF. Solutions PDF.
Homework 6 due Friday, May 7 - PDF. Solutions PDF.

Topics for the Qualifying Exam. Please see the course websites for the Fall and Winter quarters.

Math 220A covered roughly Conway, Chapters I - IV
Math 220B covered roughly Conway, Chapters V- VIII, IX.1 but excluding VI.3, VI.4 and VII.7, VII.8.
Math 220C roughly covers Conway, Chapters X - XII, excluding most of X.4, and excluding all of X.5, XII.3, XII.4.

Lecture Notes

Past Exams and F20 Exam.

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 Session: Friday, May 14.

Office hours (Dragos): May 12, 4-5pm, May 14, 1-2pm, May 17, 3-4pm (instead of the lecture that day).

Office hours (Shubham): May 11, 12-2pm, May 15, 4-6pm, May 18, 12-2pm.

There will be no lecture/office hours on Wednesday, May 19.