Complex Analysis, Geometry, and Topology (course 215A)

Stanford University

Department of Mathematics

Autumn 2009

Time: Tuesdays, Thursdays at 2:15pm.
Room: Building 380, room 381-T.

Teacher: Y.A. Rubinstein. Office hours: Tuesday 3:30-5:30pm, Thursday 12-1pm in 382-F.

CA: J. Perea. Email: jperea "at" Office hours: Tuesday, Thursday 4-6pm in 381-D.

Course plan:
This is the first course (of three) in the 215 sequence "Complex Analysis, Geometry, and Topology." It is a first-year graduate level course on complex analysis. The course will be divided roughly into three parts. The first part will be a quick review of some essential facts from a basic undergraduate complex analysis course. In the second part we will concentrate on conformal mappings and give a proof of the Riemann Mapping Theorem. The third part will include a collection of topics, largely depending on time constraints, among which we hope to touch upon the following: the Dirichlet problem for Laplace's equation, univalent functions and Loewner evolution, Riemann surfaces and the Uniformization Theorem.

Main reference:
T.W. Gamelin, Complex Analysis.

Additional reference:
L. Ahlfors, Complex Analysis (3rd Ed.).

Eight homeworks on a weekly basis. Homework from the previous week due on the next Thursday in-class or by 3:30pm in the CA's mailbox. Homework solutions by the CA will be posted on this webpage. There will be one in-class midterm (October 15 in class) and one take-home final (handed out on Tuesday, December 1 in class, due Wednesday, December 9, noon, by email to me or in the CA's mailbox). Grade: the best seven homeworks will each contribute ten percent as will the midterm, the final will contribute twenty percent.


  • September 22
    Overview/syllabus/references. Review of undergraduate complex analysis I: complex numbers and the Fundamental Theorem of Algebra, analytic functions, Cauchy-Riemann equations, conformal maps, harmonic functions and their basic properties, Cauchy's theorem and Green's theorem, Cauchy's integral formula, Cauchy estimates, Liouville's theorem.

  • September 24
    Review of undergraduate complex analysis II: Proof of the Fundamental Theorem of Algebra, Morera's Theorem, Goursat's Theorem, reformulation of Green's Theorem and dbar notation, Pomepeiu's Formula, power series and radii of convergence, analytic functions and power series, analyticity at infinity.

    HW1. 10/1: further correction posted to Prob. 2.

  • September 29
    Review of undergraduate complex analysis III: zeros of analytic functions, Laurent series, isolated singularities: Riemann's theorem on removable singularities, characterization of poles, Casorati-Weierstrass' Theorem.

  • October 1
    Review of undergraduate complex analysis IV: Characterization of meromorphic functions on the Riemann sphere (Chow's theorem for the Riemann sphere), periodic functions and Fourier series, Residue and Fractional Residue Theorems, residue calculus, Argument Principle, Rouche's Theorem.

    HW2. 10/7: Prob. 7 has been shortened and Prob. 8 has been modified.

  • October 6
    Hurwitz's Theorem, winding numbers, simply-connected domains and their various characterizations, formulation of the Riemann Mapping Theorem and a proof of a weak version of it.

    HW1 solutions.

  • October 8
    Strategy of proof of the Riemann Mapping Theorem, the Schwarz Lemma, automorphisms of the unit disc, Montel's thesis theorem and the Arzela-Ascoli theorem.

    HW2 solutions. 10/14: some typos corrected in Prob. 10.

  • October 14, 4:30pm, in: Herrin T 185 (note special time and place)
    Pick's version of the Schwarz Lemma, automorphisms of the disc and some hyperbolic geometry on the unit disc, conclusion of the proof of the Riemann Mapping Theorem.

  • October 15
    Midterm (topics included: Gamelin Ch. I-VIII and topics from HW1-2).


  • October 20
    Uniformization of multiply-connected domains.

  • October 22
    Uniformization of multiply-connected domains (continued).


  • October 27
    Properties of harmonic functions and introduction to the Dirichlet problem for the Laplace equation on domains in the plane. The Dirichlet problem on the unit disk and Poisson's kernel.


    HW3 solutions.

  • November 3
    Subharmonic functions: differential characterization and maximum principle.

    HW4 solutions.

  • November 5
    The Perron process. Harmonicity of the upper envelope construction.

    HW6. 11/8: Problems 2 and 7b) have been modified.

    HW5 solutions.

  • November 10
    Barrier functions and regularity of boundary points. Bouligand's lemma.

  • November 12
    Alternative proof of the Riemann mapping theorem. Completion of the proof of the uniformization theorem for multiply-connected domains.


  • November 17
    Introduction to complex manifolds. The Riemann sphere.


    HW6 solutions.

  • November 19
    Green's function for domains in the plane.

  • December 1
    Green's function for a Riemann surface.

    HW7 solutions.

  • December 2, 5:15pm (note special time and day)
    Green's function for a Riemann surface.

  • December 3
    Symmetry of Green's function and Rado's Theorem. Bipolar Green's function. Uniformization theorem for Riemann surfaces.