2Course start Friday Jan 17, 2025, 10.15-12.00, Room 3418 KTH Mathematics. We will meet roughly once per week, either on Mondays or Fridays.
Next lecture Friday May 9, 10-12 in 3418.
Lecturer: Fredrik Viklund, frejo //at// kth.se.
Examination will include an oral exam and essays.
Essay topics (send an email if/when you decide on a topic or want to discuss one): Your own suggestion, definition of Weil-Petersson Teichmuller space T_0(1), and the Weil-Petersson metric on T_0(1), more details for the Fenchel-Nielsen coordinates, Teichmuller space of a surface with boundary, Thurston compactification of Teichmuller space, Kleinian groups, length spectra of hyperbolic surfaces, quadratic differentials, the Weil-Petersson metric, conformal welding, conformal removability, holomorphic motion, ... in principle, many topics are enough for several essays, let me know in this case. Some instructions. For the essay, write it with your fellow students as intended reader. It is probably hard to say anything meaningful in less than 5 pages. Take care to spell out all the basic definitions you need; the goal is to understand the topic and explain it to your fellow students, not to impress. Try if you can to rewrite proofs in your own ``words'' and I encourage you to carefully discuss/work out simple examples if possible. Of course, you should include appropriate references.
* Calderon-Zygmund (Schmaltzer).
Thurson's earthquake theorem (Eriksson)
* Conformal welding: (Andina)
* Teichmüller curves from the algebraic point of view (Mason)
* Riemann-Roch (Kiehn, Kapatsori)
Thurston compactification (Rosenblad)
Translation surfaces (Rajasekar)
* Fenchel-Nielsen (Guo)
Holomorphic motion (Yuksel)
* Quadratic differentials (Lindström)
Beurling-Ahlfors, Douady-Earle (Avelin)
Literature:
Imayoshi, Taniguchi: An introduction to Teichmuller spaces (available as e-book from KTHB): https://link.springer.com/book/10.1007/978-4-431-68174-8
Lehto: Univalent functions and Teichmuller spaces (available as e-book from KTHB): https://link.springer.com/book/10.1007/978-1-4613-8652-0
Farb, Margalit: A primer to the Mapping Class Group. (Search for the book online.)
Lecture 1: Introduction and overview of the course. Riemann surfaces, complex structures. The Teichmuller space of tori in two different ways, Part 1. We constructed the torus as a quotient space.
Lecture 2. Moduli space of tori, fundamental group, marked tori, Teichmuller space T_1, markings by diffeomorphism. The Teichmuller space T(R), R torus. Definition of T_g and T(R), R genus g closed surface.
Lecture 3. Proof of theorem that T_1 = T(R), R tori. Interpretation of T(R) as deformation of the complex structure on R (for closed genus g surface), Beltrami coefficient of diffeomorphism. Recap of facts about Riemann surfaces: uniformization theorem, covering surfaces, universal covering surface. Every RS has a universal cover, which is conformally equivalent to one of \hat{C}, C, H. The group of covering transformations. (Imayoshi Chp 1, 2.)
Lecture 4. Fundamental group and covering transformation group isomorphism, properties of the covering transformations. Fuchsian model of a surface, Möbius transformations, hyperbolic surfaces and PSL(2,R), classification of elements in PSL(2,R), Fuchsian groups. (Imayoshi Chp 2.)
Lecture 5. Basic properties of Fuchsian groups: equivalent definitions (discrete subgroup of PSL(2,R), proper discontinuous action, no convergent sequence in Aut(H)), ambiguity in Fuchsian model, Theorem (Thm 2.22): Elements in Fuchsian group for closed genus g \ge 2 surface are hyperbolic (except identity), Fundamental domains, Fricke coordinates. (Imayoshi Chp 2.)
Lecture 6: Proof of injectivity of Fricke coordinates (Thm 2.25). Hyperbolic geometry (Imayoshi Chp 3.) Hyperbolic metric, geodesics, hyperbolic metric on Riemann surfaces, every free homotopy class has a unique geodesic representative.
Lecture 7: proof of the theorem that elements in Fuchsian group for closed genus g \ge 2 surface are hyperbolic (except identity). Fenchel-Nielsen coordinates: Teichmuller space as a space of hyperbolic metrics (see Farb-Margalit), pair of pants, construction of marked right-angled hyperbolic hexagons with any prescribed 3 alternating sides, cutting into pairs of pants, length and twist coordinates.
Lecture 8: Quasiconformal maps. Motivations. Diffeomorphic QC maps: ellipse field, dilatation, Beltrami coefficient and some properties (composition rules). A description of T(R) in terms of Beltrami coeffiecients. General definitions: ACL property and L^2_loc property. The geometric definition. Beltrami equation.
Lecture 9. An overview of the proof of MRMT using singular integral operators: Cauchy transform, Beurling transform, the Calderon-Zygmund estimate. Translation of the Beltrami equation into an integral equation and solution by Neumann series. Quasisymmetric functions and the Beurling-Ahlfors extension. Universal Teichmuller space: two models.
Lecture 10. More on Universal Teichmuller space. Quasicircles and conformal welding, solution via Beltrami equation. Teichmuller metric on T(1). Teichmuller space of a general Riemann surface. Relation to Fuchsian group.
Lecture 11. Teichmuller space of general surface, group isomorphism induced by a QC map, relation to homotopy. Teichmuller metric. Guest lecture by Steffen Rohde on Circle Packings.
Lecture 12 (May 2). Symmetry of Beltrami coefficients under Fuchsian group action. Proof that Teichmuller metric is indeed a metric, completeness of Teichmuller metric. Quadratic differentials, Part 1.
Lecture 13. (May 9). More on quadratic differential, Schwarzian derivatives, Bers embedding.