Math 220C - Complex Analysis

Welcome to Math 220c!

Course description:

This is the third in a three-sequence graduate course on complex analysis, picking up where Math 220B left off.

Topics include: analytic continuation, sheaves of analytic functions, analytic manifolds (Riemann surfaces), harmonic/subharmonic/superharmonic functions, order of entire functions, the little/big Picard theorem.

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

Online lectures: WF 11:00-12:20.

Virtual office hours: Wednesday 12:30-1:30.

Teaching Assistant: Nandagopal Ramachandran, naramach "at" ucsd.

Virtual office Hours: Thursday, 5-7pm.

Online teaching:

We will use Zoom for both lectures and virtual office hours. 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 220 B. 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.

The problem sets are due on Fridays at 5pm on Gradescope. The grades are recorded in Canvas. Please make sure that your grades are properly recorded.

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

Qualifying Examination: Wednesday, May 27, 1-4pm.

The exam is open book (Conway)/open notes (from lectures). No outside help (including internet, collaboration, etc) is allowed. The exam will be emailed shortly before 1pm and will also be released on Gradescope. At the end of the 3 hours, you will need to upload the exam on Gradescope. There will be a 15 minute buffer period to allow everybody to upload the exams. In person follow-ups are possible to ensure A.I.

Important dates:

First class: Wednesday, April 1.
Memorial Day: Monday, May 25.
Last class: Friday, June 5.
Qualifying Exam: Wednesday, May 27, 1-4pm

Lecture Summaries

Lecture 1: Harmonic functions. Mean value property. Maximum modulus principle - PDF
Lecture 2: Poisson integral formula. Harnack inequality. Schwarz integral formula - PDF
Lecture 3: Dirichlet problem for the disc. Mean value property implies the function is harmonic. Convergence of harmonic functions - PDF
Lecture 4: Harnack's convergence theorem. Subharmonic functions. Motivation and connection with the Dirichlet problem. Definition and maximum value principles. Perron family. Perron solution. Poisson modification - PDF
Lecture 5: Properties of Poisson modifications. The Perron function is harmonic. Barriers and the Dirichlet problem - PDF
Lecture 6: Jensen formula. Poisson-Jensen formula. Connection between the growth of entire functions and the distributions of zeroes. Order of entire functions and examples - PDF
Lecture 7: More on zeroes of entire functions. Critical exponent. Genus of entire functions. Examples. Hadamard's theorem - PDF
Lecture 8: Proof of Hadamard's factorization theorem and applications - PDF
Lecture 9: Introduction to the Little and Great Picard theorems. Applications. Setting up the strategy of proof. Landau's lemma - PDF
Lecture 10: Proof of little Picard's theorem using Bloch's theorem. Constructing discs inside the image of functions. Proof of Bloch via elementary estimates. Improving the estimates via linear fractional transformations - PDF
Lecture 11: Schottky's theorem. Montel's fundamental normality test. Great Picard - PDF
Lecture 12: Analytic continuation along paths. Statement of monodromy theorem. Complete analytic functions. The Riemann surface of a complete analytic function. Sheaves. Germs. Stalks. Espace etale. Revisiting the monodromy theorem - PDF
Lecture 13: Ringed spaces. Riemann surfaces. Holomorphic functions, meromorphic functions. Examples of Riemann surfaces - PDF
Lecture 14: Identity theorem, open mapping theorem, maximum modulus for Riemann surfaces. Elliptic functions. General properties of zeros and poles of elliptic functions. Jacobi theta function. Weierstrass elliptic functions - PDF
Lecture 15: More on the Weierstrass function. Eisenstein series. The Weierstrass sigma function. Meromorphic functions on the torus in terms of the sigma function. Divisors on Riemann surfaces - PDF
Lecture 16: Divisors on curves. Sheaves attached to divisors. Weierstrass and Riemann-Roch and their formulation via sheaves. Statement of Riemann-Roch. Motivation for sheaf cohomology. Morphisms of sheaves. Exact sequences - PDF
Lecture 17: Exact sequences of sheaves, kernel, cokernel, sheafification. Examples of exact sequences. Flabby sheaves. Sheaf cohomology - PDF
Lecture 18: Sheaf cohomology. Sheaves of O_X-modules. Euler characteristic. Genus. Riemann-Roch - PDF

Homework:

Homework 1 due Friday, April 10 - PDF. Solutions PDF.
Homework 2 due Friday, April 17 - PDF. Solutions PDF.
Homework 3 due Friday, April 24 - PDF. Solutions PDF.
Homework 4 due Friday, May 1 - PDF. Solutions PDF.
Homework 5 due Friday, May 8 - PDF. Solutions PDF.
Homework 6 due Friday, May 15 - PDF. Solutions PDF.

Topics for the Qualifying Exam:

From 220A, Chapters I - IV, excluding the orientation principle III.3
From 220B, Chapters V- VIII, IX.1 but excluding VI.3, VI.4 and VII.7, VII.8.
From 220C, Chapters X - XII, excluding X.5. We gave a simplified treatment of X.4 and XII.1.

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.

The lecture on Friday, May 22 will be an improvised Qual Review.

Dragos' Office Hours: Wednesday, May 20, 12:30-1:30 PM.

Nandagopal's Extra Office Hours: Tuesday, May 26, 4-6 PM.