Course Information for Math 595 MCG, Fall 2009

  • Instructor: Chris Leininger, (pronunciation).

  • Textbook: None. The following papers will be the primary focus of the class (our library has a subscription to each of these journals, so just go to MathSciNet to get a copy).
    • Bowditch, B., Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006), 105--129.
    • Masur, H. and Minsky, Y., Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103--149.
    • Masur, H. and Minsky, Y., Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal. 10 (2000), no. 4, 902--974.

  • Lecture Room and Times: Mondays, Wednesdays and Friday from 9AM until 9:50AM in 243 Altgeld Hall (I may be a few minutes late each day as I teach at 8AM pretty far away).

  • Prerequisites: I'm going to assume you are very familiar with fundamental group and covering spaces and the classification of surfaces. Differential topology and some Riemannian geometry would be helpful, but not a necessity.

  • Plan: We will start out with some background on surfaces, curves in surfaces and mapping class groups. One unified source for much of this material is Farb and Margalit's "primer on the mapping class group". You can download it at www.math.utah.edu/~margalit/primer/ We will not spend a great deal of time on this, however. The main objective of the class will be to go through the proof of hyperbolicity of the curve complex and the construction and structure of heirarchies of tight geodesics. These have proven to be some of the most useful tools in studying not only mapping class groups, but also Teichmuller spaces and hyperbolic 3-manifolds. The goal is thus to develop an understanding of the theorems and techniques of the papers listed above.

    There are notes by Saul Schleimer on the curve complex which might be helpful that you can find here.

    A nice survey-type discussion of Masur and Minsky's approach is in Minsky's paper: "A geometric approach to the complex of curves on a surface", available on his website . This also has a discussion of the fact that when ξ(S) =1, that C(S) is 3/2--hyperbolic.

  • Grades: This is based on classroom attendence and participation. I expect anyone who comes all semester to get an A (of course, if you're sick you should stay home...). There will be no exams, quizzes or homework collected.

  • Problems: I will give some problems and exercises throughout the semester, usually during the lecture. The range of difficulty will be dramatic: some exercises will merely require you to finish a proof, others may be open problems in the field. I'm happy to discuss these problems whenever.