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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" math.stanford.edu 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.).
Assignments:
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.
Schedule:
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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October 15
Midterm (topics included: Gamelin Ch. I-VIII and topics from HW1-2).
HW3.
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October 20
Uniformization of multiply-connected domains.
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October 22
Uniformization of multiply-connected
domains (continued).
HW4.
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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.
HW5.
HW3
solutions.
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November 3
Subharmonic functions: differential characterization and maximum
principle.
HW4
solutions.
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November 5
The Perron process. Harmonicity of the upper envelope construction.
HW6.
11/8: Problems 2 and 7b) have been modified.
HW5
solutions.
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November 10
Barrier functions and regularity of boundary points.
Bouligand's lemma.
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November 12
Alternative proof of the Riemann mapping theorem. Completion of the
proof of the uniformization theorem for multiply-connected domains.
HW7.
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November 17
Introduction to complex manifolds. The Riemann sphere.
HW8.
HW6
solutions.
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November 19
Green's function for domains in the plane.
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December 1
Green's function for a Riemann surface.
HW7
solutions.
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December 2, 5:15pm (note special time and day)
Green's function for a Riemann surface.
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December 3
Symmetry of Green's function and
Rado's Theorem.
Bipolar Green's function.
Uniformization theorem for Riemann surfaces.
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