PS = Problem Session
Yael Algom-Kfir - The geometry of outer space
Syllabus: Outer Space, folding, the Lipchitz metric, asymmetry, train track maps, dilitations, dynamics and growth, the compactification of Outer Space, isometries, the free factor complex.
Lecture 1. Definition of Outer Space, simplicial structure, folding, topology, Out(F_n) action, the Lipschitz metric.
Lecture 2. Some examples of asymmetry of the metric, quasi-symmetry in the thick part, train track maps and axes of irreducible automorphisms, asymmetry of dilitations, other types of isometries - parabolic and elliptic.
Lecture 3. The compactification of Outer Space, topology, examples: leaf spaces, direct limit of an axis, a tree with non-trivial edge stabilizer. North-South dynamics of a fully-irreducible outer automorphism. (time permitting - the metric completion of Outer Space+ the isometry group of Outer Space).
Lecture 4. The free factor complex and free splitting complex, maps between them and Outer Space, images of fold paths, sub-factor projections, the search for a distance formula.
Tara Brendle - Description of Teichmüller space in terms of hyperbolic geometry
Syllabus: Metrics on Teichmuller space. Action of the mapping class group. Classification of mapping classes. Connections with combinatorial complexes.
Lecture 1. Brief definitions of Teich(S) and Mod(S), including statement of the classification of mapping class group with examples, nuts and bolts of Teich(S) including: length functions, Teich(pants), Teich(torus), Fenchel-Nielsen coordinates.
- Farb-Margalit (Chapter 10)
- Schwartz (Chapter 21)
Lecture 2. 9g - 9 theorem, measured foliations, Teichmuller mappings.
- Farb-Margalit (Chapters 10, 11)
- Fathi-Laudenbach-Poenaru (all of it)
Lecture 3. Grotzsch's problem (Teichmuller existence and uniqueness in the special case of rectangles), proper discontinuity of action of Mod(S) on Teich(S). If time permits, return to Nielsen-Thurston classification of mapping classes in more detail, canonical form.
- Farb-Margalit (Chapters 11, 12, 13)
- Birman-Lubotzky-McCarthy's Duke paper
Lecture 4. Sketch of Thurston's proof of classification and Thurston's compactification of Teich(S), connections with curve complex etc.
- Farb-Margalit (Chapters 13, 15)
- Ivanov's curve complex paper.
Nathan Dunfield - Methods for computation of geometric structures and invariants
Syllabus: Exploration of low-dimensional manifolds. Application of computational tools.
Lecture 1. The Geometrization Theorem and its connection to the homeomorphism problem for 3-manifolds. Contrast with higher dimensions. Teaser demo of SnapPy.
- Bruno Martelli, Introduction to Geometric Topology, https://arxiv.org/abs/1610.02592
- Greg Kuperberg, Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization, https://arxiv.org/abs/1508.06720
Lecture 2. Review of hyperbolic geometry in dimension 3 focusing on the upper-halfspace model. Example of a topological ideal triangulation (Borromean rings). Ideal geodesic tetrahedra in H^3. Gluing equations and perhaps cusp equations if time permits.
- Jeffery Weeks, Computation of Hyperbolic Structures in Knot Theory, https://arxiv.org/abs/math/0309407
Lecture 3. Finish discussion of Thurston’s gluing equations. Brief discussion of canonical triangulations and their usefulness regarding the homeomorphism problem. Extended demonstration of SnapPy, including large random knots/links.
- Jeffery Weeks, Convex hulls and isometries of cusped hyperbolic 3-manifolds. Topology Appl. 52 (1993), no. 2, 127–149.
- Culler, Dunfield, Goerner, and Weeks, SnapPy, a computer program for studying the geometry and topology of 3-manifolds, http://snappy.computop.org
Lecture 4. Proving hyperbolic structures exist via numerical methods. Interval arithmetic and effective versions of the inverse function theorem. Topological applications, specifically solving the word problem.
- Hoffman et. al. Verified computations for hyperbolic 3-manifolds, https://arxiv.org/abs/1310.3410
- Dunfield, Hoffman, and Licata, Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling, https://arxiv.org/abs/1407.7827
Erwan Lanneau - Teichmüller dynamics
Syllabus: Flat surfaces and billiards. The torus and Veech examples. Spaces of flat surfaces. The SL(2, R) action on strata and the method of renormalisation. Applications to counting problems and to windtree models.
Lecture 1. Introduction to translation surfaces and their modulispaces, periods coordinates. I will use MCG, Teichmuller space, moduli space, ergodic theory (definition of invariant measure, ergodicity). The SL(2,R) action and its applications.
Lecture 2. Applications to billiards and Veech surfaces. Several examples of SL(2,R) action. I will give a proof of Masur's criterion. I will then focus on applications to Veech surfaces. I will use hyperbolic geometry (PSL(2,R), unit tangent bundle of H^2). I will also use Dehn twists and affine action on translation surfaces.
Lecture 3. I will focus on the wind-tree model (billiard in a non compact case). Definition (cocycle, recurrence, Theorem of Schmidt). Briefly explain non ergodicity (Fraczek-Ulcigrai). Then I will give a detailed proof of Avila-Hubert's theorem about recurrence. If time permits I will explain the diffusion with rate 2/3.
Lecture 4. I will explain the notion of R-linear manifolds in periods coordinates. Then I will state Eskin-Mirzakhani-Mohammadi theorem. The rest of the lecture will be devoted to some application of this theorem to various problems (classification of Veech surfaces, characterisation of complete periodic/parabolic surfaces). If time permits I will explain several advances by Eskin-Filip-Wright.
Julien Marché - Geometric structures viewed in terms of representations
Syllabus: Real and complex and hyperbolic geometry. Geometries of 3-manifolds. Deformation spaces of structures.
Lecture 1. Character varieties of fundamental groups in SL_2(C). Definition, tangent space, examples. Some rough ideas about Culler-Shalen theory as a motivation.
- Shalen, Representations of 3-manifolds groups.
Lecture 2. The case of surfaces. Symplectic structure, the cases of SU_2 and SL_2(R), the Euler class.
- Goldman, Representations of fundamental groups of surfaces.
Lecture 3. Dynamics in the SU_2 case. Some motivations for studying the SU_2 character variety (moduli space of holomorphic vector bundles, TQFT).
Proof of ergodicity (Goldman's theorem).
- Goldman, Ergodicity of mapping class group actions on SU_2-character varieties.
Lecture 4. Dynamics in the SL_2(R) case. Teichmüller component, low genus examples, relation with the Bowditch conjecture.
- Goldman, The mapping class group action on real SL_2-characters of a punctured torus
- Marche-Wolff, The modular action on real SL_2 characters in genus 2.
Problem sets and references
- Webpage for lectures, problem sets, and references.
- Lecture notes 1, Problem set 1
- Lecture notes 2, Problem set 2
- Lecture notes 3, Problem set 3
- Lecture notes 4, Problem set 4
References (suggested by Stergios Antonakoudis)
- Casson and Bleiler's book: Automorphisms of surfaces after Nielsen and Thurston
- Farb and Margalit's book: A primer on mapping class groups
- Fathi, Laudenbach, and Poénaru's book: Thurston's work on surfaces
- McMullen's course notes: Riemann surfaces, dynamics, and geometry
- Hubbard's books: Teichmüller theory and applications to geometry, topology, and dynamics
- Buser's book: Geometry and spectra of Riemann surfaces
- Nag's book: Complex analytic theory of Teichmüller spaces
Lehto and Virtanen's book: Quasiconformal mappings in the plane
- Ahlfors' books: Lectures on quasi-conformal mapping, Conformal invariants
- Marden and Strebel's paper: A characterization of Teichmüller differentials
- Thurston's paper: Zippers and univalent functions