Math 220C - Complex
Welcome to Math 220c!
This is the third in a three-sequence graduate course on complex
analysis, picking up where Math 220B left off.
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"
Online lectures: WF 11:00-12:20.
Virtual office hours: Wednesday 12:30-1:30.
Teaching Assistant: Nandagopal Ramachandran, naramach "at"
- Virtual office Hours: Thursday, 5-7pm.
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
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.
- Math 220 B. However, this is a
graduate level course, so at times, we may use notions from related
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.
The final grade is based on homework.
The problem sets are
mandatory and are a very important part of the
The problem sets are due on Fridays at 5pm on Gradescope.
grades are recorded in Canvas.
Please make sure that your grades
are properly recorded.
There are several ways to upload your homework on
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
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.
- First class: Wednesday, April 1.
- Memorial Day: Monday, May 25.
- Last class: Friday, June 5.
- Qualifying Exam: Wednesday, May 27, 1-4pm
- 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
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 -
- 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 -
- Lecture 7: More on zeroes of entire functions. Critical
exponent. Genus of entire functions. Examples. Hadamard's theorem -
- 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 -
- 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
linear fractional transformations - PDF
- Lecture 11: Schottky's theorem. Montel's fundamental
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 -
- Lecture 13: Ringed spaces. Riemann surfaces. Holomorphic
functions, meromorphic functions. Examples of Riemann surfaces -
- 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
Weierstrass and Riemann-Roch and their formulation via sheaves. Statement
of Riemann-Roch. Motivation for sheaf cohomology. Morphisms of sheaves. Exact sequences -
- Lecture 17: Exact sequences of sheaves, kernel, cokernel,
sheafification. Examples of exact sequences. Flabby sheaves. Sheaf
- Lecture 18: Sheaf cohomology. Sheaves of O_X-modules.
Euler characteristic. Genus. Riemann-Roch -
Homework 1 due Friday, April 10 - PDF.
Homework 2 due Friday, April 17 - PDF.
Homework 3 due Friday, April 24 - PDF.
Homework 4 due Friday, May 1 - PDF.
Homework 5 due Friday, May 8 - PDF.
Homework 6 due Friday, May 15 - PDF.
Topics for the Qualifying Exam:
- From 220A,
I - IV, excluding the orientation principle III.3
Chapters V- VIII, IX.1 but excluding VI.3, VI.4 and VII.7, VII.8.
220C, Chapters X - XII, excluding X.5. We gave a simplified treatment of
X.4 and XII.1.
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.