Class videos.
First class met Tuesday, Aug 29. This meeting was not recorded, and we just discussed the basic plan of the course. Actual course content started on Thursday, Aug 31. The first six lectures will give an overview of the class, using slides I created for a 2-lecture short coruse in Barcelona, April 2021. We then cover the unifomization theorem (following Don Marshall's proof) and Belyi's theorem (following the text of Girondo and Gonzalez-Diez). I will then do a rapid review of quasiconformal mappings in the plane, and then a more careful discusssion of the QC folding theorem. We will end the course with some applications of QC folding: wandering domains, equilateral triangulations of Riemann surfaces, constucting transcendental Julia sets of small dimension, and building meromorphic functions with (nearly) specified singular dynamics.
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Lecture 01: Thur Aug 31, Course Overview, Part I. Harmonic measure,
conformally balanced trees.
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Lecture 02: Tue Sept 5, Course Overview, Part II. True trees, Belyi functions, equilateral triangulations.
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Lecture 03: Thur Sept 7, Course Overview, Part III. True trees are dense, non-compact surfaces.
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Lecture 04: Tue Sept 12, Course Overview, Part IV. Infinite trees
and the folding theorem.
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Lecture 05: Thur Sept 14, Course Overview, Part V. Quasiconformal folding
and the folding theorem.
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Lecture 06: Tue Sept 19, Course Overview, Part VI.
The infinite 3-regular tree; some applications of QC folding to dynamics
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- No class on Thur Sept 21.
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Lecture 06: Tue Sept 26, Uniformization Theorem, Part I. Statement,
sketch proof following D. Marshall.
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Lecture 06: Thur Sept 28, Uniformization Theorem, Part II.
Prelimnary Lemmas
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Lecture 07: Tue Oct 3, Uniformization Theorem, Part III. Dipole
Green's Functions, finish proof.
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Lecture 08: Thur Oct 5, Riemann surfaces, diverence/convergence type,
fundamental domains, Y-pieces.
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- No class on Tue Oct 10, SBU Fall break.
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Lecture 09: Thur Oct 12, Meromorphic functions, equilateral triangulations,
algebraic curves.
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Lecture 10: Tue Oct 17, Belyi's theorem: algebraic implies Belyi function exists
algebraic curves.
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Lecture 11: Thur Oct 19, Finsh Belyi's theorem: Belyi function exists implies algebraic
algebraic curves.
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Lecture 12: Tue Oct 24, Modulus, Definitions of QC maps
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Lecture 13: Thur Oct 26, Equicontinuity of K-QC maps
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Lecture 14: Tue Oct 21, 1-QC maps are conformal
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- No class on Thur Nov 2.
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Lecture 15: Tue Nov 7, Analytic properties of QC maps
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Lecture 16: Thur Nov 9, Complete proof of MRMT, Gehring's reverse Holder lemma.
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Lecture 17: Tue Nov 14, Start proof of QC folding theorem
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Lecture 18: Thur Nov 16, Finish proof of QC folding theorem
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Lecture 19: Tue Nov 21, Applications of QC folding: rapid growth, spirals, area conjecture,
wandering domains
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- No class on Thur Nov 23 (Thanksgivng break).
- No class on Tue Nov 28.
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Lecture 20 Thur Nov 30, Guest lecture by Lasse Rempe on equilateral triangulations
of Riemann surfaces.
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Lecture 21 Tue Dec 5, Guest lecture by Kirill Lazebnik on prescribing the postsingular set
of a rational map.
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Lecture 22 Thur Dec 7 (last class), Guest lecture by Kirill Lazebnik on
new version of Runge's approximation theorem via QC folding.
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