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  • King's College London, Fall 2023-24: Manifolds

  • London Taught Course Center, Fall 2022-2023: Foliations and 3-Dimensional Manifolds

  • University of Oxford, Michaelmas Term, 2020–2021: Intercollegiate Tutorial for Topology and Groups

  • University of Oxford, St Peter's College, 2020–2021: Tutorial for Prelim Analysis

  • University of Oxford, Michaelmas Term, 2019–2020: Intercollegiate Tutorial for Topology and Groups

  • University of Oxford, Trinity Term, 2018–2019: Mapping Class Groups of Surfaces

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Description: Let S be a closed orientable surface of positive genus. Nielsen and Thurston proved that any homeomorphism of S can be isotoped to be in one of three `standard geometric forms', called Nielsen-Thurston classification. The applications of this geometric perspective cannot be overemphasised.

During the course, examples of three standard forms and methods for constructing them will be discussed, and we briefly experiment with the software Flipper to visualise examples. We give a proof of the classification based on the hyperbolic geometry of surfaces, following Casson-Bleiler's book, as the core part of the course. The importance of delving into the proof is to obtain insight into the structure of homeomorphisms, which can be useful in other contexts.

The main references are 

  1. Automorphisms of surfaces after Nielsen and Thurston, Casson and Bleiler

  2. Fuchsian groups, Katok

  3. A primer on mapping class groups, Farb and Margalit

  4. Thurston's work on surfaces, Fathi and Laudenbach and Poenaru

The class is held Tuesdays and Thursdays at11 am. This page will be updated, as we proceed during the term.

  • Sessions 1 and 2: Definition of the mapping class group, and examples of mapping classes. Computing the mapping class group of the torus and classifying elements into periodic, reducible and Anosov. Poincare-Hopf formula to justify introducing singular foliations for higher genus surfaces. Statement of Thurston-Nielsen classification and examples of pseudo-Anosov maps. Experimenting with Flipper software to visualise the three geometric forms in action. Exercises and Solutions

  • Sessions 3 and 4: Review of Hyperbolic geometry in dimension two covering the topics: upper half-plane and unit disk models, classification of isometries, Hyperbolic structures on closed orientable surfaces, Gauss-Bonnet theorem, Lower bound for the hyperbolic area of a complete hyperbolic surface with geodesic boundary. Exercises 

  • Session 5: Geodesic representatives of essential closed curves, and essential closed 1-submanifolds. Alexander trick. Bigon criterion. A characterisation of periodic maps in terms of their actions on simple closed curves. 

  • Session 6: Geodesic laminations on closed hyperbolic surfaces and their examples. The directions of geodesics in a geodesic lamination vary continuously. Geodesic laminations are nowhere dense and can be expressed as a union of geodesics in just one way. A transverse arc to a minimal geodesic lamination intersects it in a Cantor set.

  • Session 7: Introducing the space of geodesic laminations on a closed hyperbolic surface, equipped with the Hausdorff metric, and showing that this space is compact. Introducing isolated leaves in geodesic laminations.

  • Session 8: The lift of a homeomorphism of a closed hyperbolic manifold extends continuously to the boundary. A concrete description of the unit tangent bundle for a closed hyperbolic surface. The action of a mapping class on the space of geodesic laminations (respectively the unit tangent bundle of the surface).

  • Princeton University 2016–2017: Instructor for Calculus I

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