Class videos.
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Lecture 01: Mon. Jan 22, Course Introduction, Chapter 1, complex arithmetic and stereographic projection .
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Lecture 02: Wed. Jan 24, Start Chapter 2, fundamental theorem of algebra, power series, isolated zeros.
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Lecture 03: Mon. Jan 29, Finish Chapter 2, Start Chapter 3, manipulating power series,
complex differentiability, maximum prinicples.
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Lecture 04: Wed. Jan 31, Chapter 3, open mapping, local behavior, Liouville's theorem,
Schwarz's lemma.
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Lecture 05: Mon Feb 5, Finish Chapter 3, Start Chapter 4,
hypebolic metric, some examples, integrals along curves, analytic =
holomoprhic.
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Lecture 06: Wed Feb 7, More Chapter 4,
Morera's theorem, Runge's theorem.
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Lecture 07: Mon Feb 12, Finish Chapter 4,
Start Chapter 5,
uniform limits of analytic functions,
applications of Runge's theorem, homologous curves.
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Lecture 08: Wed Feb 14, Continue Chapter 5,
winding numbers, removable sets.
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Lecture 09: Mon Feb 19, Finish Chapter 5, Laurent series, argument principle, Rouche's theorem.
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Lecture 10: Mon Feb 26, Start Chapter 7, Harmonic functions, Cauchy Riemann equations..
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Lecture 11: Wed Feb 28, Continue Chapter 7, Harmonic conjugates,
Lindelof's maximum principle, Harnack's inequality.
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Lecture 12: Mon Mar 4, Finish Chapter 7, Start Chapter 10, Harnack's principle,
normal families, spherical derivative, Marty's theorem.
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Lecture 13: Wed Mar 6, Continue Chapter 10, Hurwitz theorem, Riemann mapping theorem, Schwarz reflection theorem
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Spring break Mon Mar 11, Wed Mar 13.
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Lecture 14: Mon Mar 18, Finish Chapter 10, Zalcman's lemma, Montel's theorem, introduction
to Julia sets and Fatou sets.
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Midterm Wed Mar 25.
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Lecture 15: Mon Mar 25, Start Chapter 12, Jordan curve theorem
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Lecture 16: Wed Mar 27, Finish Chapter 12, Start Chapter13, Caratheodory-Torhorst theorem,
Dirichlet problem, Perron Families
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No class Mon Apr 1.
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Lecture 17: Wed April 3, Finish Chapter 13, local barriers, Riemann mapping theorem.
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Lecture 18: Mon April 8, Start Chapter 14, Analytic continuation,
the monodromy theorem, Riemann surfaces.
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Lecture 19: Wed April 10, Finish Chapter 14,
Riemann surfaces, universal covers, deck transformations.
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There was a technical problem, and it seems the audio was not
recorded for the last 20 minutes of the April 10 lecture (no Zoom transcript either).
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Lecture 20: Mon April 15, Start Chapter 15, Skip Section 1, Green's function:
definitions, properties, existence.
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Lecture 21: Wed April 17, Finish Chapter 15, proof of the
uniformization theorem.
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Lecture 22: Mon April 22, Survey Chapter 9, residue calculus, compute integrals.
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Lecture 23: Wed April 24, Survey Chapter 11, MIttag-Leffler theorem, Blaschke products.
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Lecture 24: Mon April 25, Lecture on true trees; uses slides from lectures in
Barcelona, April 2021.
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Lecture 25: Wed May 1, Last class. Continues lecture on Belyi surfaces
and equilateral triangulations; uses slides from lectures in Barcelona, April 2021.
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