Math 9201 – Seminar in Topology – Spring 2018
Simple curves and the volumes of moduli spaces: an exploration of the thesis of Maryam Mirzakhani
Office hours:
MWF 2:00–3:00pm and by appointment
Information:
Term: Spring 2018 Time: MWF 1:10–2:00pm Location: 1117 Stevenson Center
Links:
View the Course Syllabus for the course description and course policies.
Announcements:
- There will be NO CLASS during the week of April 9–13.
- In the week of April 16–20 our postdoc Caglar Uyanik will give 2 or 3 lectures presenting the basic theory of Geodesic Currents (see the article by Francis Bonahon cited in the course syllabus). Caglar will lecture at the usual time on Monday, Wednesday, and possibly Friday if needed.
- There will be an extra lecture on Wednesday April 25, from 12:10–2:00pm in the usual classroom. This will be the final lecture for the course.
Notes:
Here my (possibly illegible) notes for the course. They are likely riddled with typos, inaccuracies, and gross oversimplifications.- I – Overview and Background
- II – The Hyperbolic Plane
- III – Teichmüller Space, the Mapping Class Group, and Moduli Space
- IV – Generalized McShane Identity
- V – Volume Recursion
- VI – Measured Laminations
- VII – Growth of Simple Closed Curves
- Guest lecture – Geodesic Currents
- VIII – Counting with Geodesic Currents
Schedule:
Here is a breif summary of the topics covered in each class. For more details, consult the course notes above (which indicate the starting point of each lecture).
Mon Jan 8 — Discussion of Syllabus. Overview of Mirzakhani's results and the broad goals of the course. Wed Jan 10 — Smooth manifolds; Diffeomorphisms; Tangent vectors & bundles; Derivatives. Fri Jan 12 — Snow Day. Mon Jan 15 — MLK Day. Wed Jan 17 — Operations on vector bundles (dual, tensor product); Cotangent bundle; Riemannian Metrics. Fri Jan 19 — Isometries; Isometry group of \(\mathbb{R}^n\); Covers & metrics; Curves; Lengths; Geodesics. Mon Jan 22 — Uniqueness of geodesics; Hyperbolic plane: metric & geodesics; Möbius transformations. Wed Jan 24 — Action of \(\mathrm{SL}(2,\mathbb{R})\) on tangent vectors and geodesics of \(\mathbb{H}^2\); Examples of isometries. Fri Jan 26 — Calculate \(\mathrm{Isom}(\mathbb{H}^2)\); Classify isometries (elliptic, parabolic, hyperbolic); Translation length. Mon Jan 29 — 3 Definitions of Hyperbolic surfaces Curves & conjugacy classes; Geodesic curve length. Wed Jan 31 — Classification of surfaces; Mapping class group; Examles; Dehn twists; Flaring vs cusps. Fri Feb 2 — Finite-volume hyperbolic structures; Teichmüller space; Topology; Mapping class group action. Mon Feb 5 — Classify right-angled hyperbolic hexagons; Teichmüller space of pair of pants. Wed Feb 7 — MCG of pair of pants; Alexander Lemma; Fenchel–Nielsen length and twist parameters. Fri Feb 9 — FN parameters give global coordinates on Teich; Tensor & Exterior product; differential forms. Mon Feb 12 — Fenchel-Nielsen 2-form & Wolpert's magic formula; Riemann Surfaces; Uniformization. Wed Feb 14 — Quadratic differentials; Weil–Petersson pairing, metric, and 2-form; Sketch Wolpert's formula Fri Feb 16 — Moduli Space; Finiteness of WP-volume; McShane Identity; Calculate \(\mathrm{vol}(\mathcal{M}_{1,1}) = \pi^2/6\). Mon Feb 19 — Towards Generalized McShane Identity: boundary-cuff pants and ortho-boundary geodesics. Wed Feb 21 — Ortho-emanating rays via gaps for boundary pants; Complement of gaps has measure zero. Fri Feb 23 — The generalized Mirzakhani-McShane identity; Behavior of basic summand functions. Mon Feb 26 — Notation and setup for the Covolume Formula; Accounting for Half-Twists. Wed Feb 28 — Proof of the Covolume Formula. Fri Mar 2 — Statement and proof of Volume Recursion; Polynomial nature of moduli space volumes. Mon Mar 12 — Polynomial nature of volumes; Coefficients are intersections of tautological bundles. Wed Mar 14 — Geodesic Laminations; Laminations as subsets of double boundary. Fri Mar 16 — Class cancelled. Mon Mar 19 — Topology of geodesics laminations; Minimal Laminations. Measured Laminations Wed Mar 21 — Length of a measured lamination; Independence of metric. Fri Mar 23 — Intersection pairing, Topology of \(\mathcal{M}\mathcal{L}\) as Thurston boundary of Teichüller space; Train tracks. Mon Mar 26 — Carrying tracks and laminations; Transverse weights; Recurrence; Linearity of carrying. Wed Mar 28 — Maximal tracks; Train track charts on \(\mathcal{M}\mathcal{L}\) and resulting Piecewise Linear structure. Fri Mar 30 — Thurston measure on \(\mathcal{M}\mathcal{L}\); Ergodicity; Unit ball volume \(B(X)\); Counting multicurves. Mon April 2 — Dehn Coordinates for multicurves; Combinatorial Length for multicurves. Wed April 4 — Bounding \(B(X)\) in terms of short curves on \(X\); \(B\) is proper and integrable on moduli space. Fri April 6 — Counting curves in an orbit: asymptoics, volume integration, convergence of discrete measures. Mon April 9 — No Class. Wed April 11 — No Class. Fri April 13 — No Class. Mon April 16 — Caglar Uyanik lecturing on Geodesic Currents. Wed April 18 — Caglar Uyanik lecturing on Geodesic Currents. Fri April 20 — No Class. Mon April 23 — Proof of convergene of the discrete measures. Wed April 25 — Counting with Geodesic Currents! Applications to conjugacy length and lattice counting.