MAT 290-026: "What is what in hyperbolic geometry"

Professor Michael Kapovich, MSB 2224. 


Time-space: TBD: Most likely, online via Zoom.

CRN for this class is: 70792

Credits: 1 credit for attending and 3 credits for giving a talk.

  The goal of this seminar is to be "broad and shallow", covering mostly definitions, concepts, statements 
and examples, rather than giving proofs. The seminar is a continuation of the one held in Winter quarter. Almost all talks will be given by students.

  Possible topics:

 Hyperbolic space: Upper half-space and unit ball models, angles, volume form.

Isometry group of the hyperbolic space. Classification of isometries: Hyperbolic, parabolic, elliptic. 

Definition of hyperbolic manifolds. Hyperbolic structures on the pairs of pants. Existence of hyperbolic structures on surfaces.

Riemann surfaces. Uniformization theorem.

Kleinian groups, quotient manifolds and surfaces.

Fundamental domains. Examples of Kleinian groups: Elementary, fuchsian, quasifuchsian, Appolonian packings

Beltrami differentials and quasiconformal maps in 2d.

Extremal maps between Riemann surfaces.

Teichmuller spaces. Teichmuller space of the torus.

Mapping class group and discreteness of its action on the Teichmuller space.  Moduli space.

Fenchel-Nielsen coordinates.

Compactification of the moduli space.

Dehn twists. Classification of homeomorphisms of surfaces.

Geodesic laminations. Hausdorff topology on the space of geodesic laminations. 

Measured geodesic laminations and train tracks.

Length of a measured lamination and intersection numbers.

Topology of the space of measured laminations.

Measured foliations and their relation to measured laminations. Fat train-tracks.

Harmonic maps from Riemann surfaces into trees.

Dual tree to a measured lamination. Skora's theorem: Converting trees into laminations.

Thurston's compactification of the Teichmuller space.

Earthquakes and Nielsen realization problem/theorem.

Schwarzian differential equation and quadratic differentials.

Complex-projective structures and holonomy.

Relation of complex-projective structures to quadratic differentials and measured laminations.

Ahlfors finiteness theorem.

Geometrically finite and infinite groups.

Mostow Rigidity Theorem

Pleated surfaces and how to construct them.

Ends of hyperbolic 3-manifolds, notion of tameness.

Tameness theorem. Ending laminations.

Model manifolds.

Ending Lamination Theorem.

Geometric and algebraic convergence.

Geometric structures on manifolds. Holonomy theorem.

3-dimensional geometries.

Geometrization Theorem.

McShane's Identity

Curve complex.

Hyperbolicity of curve complex.