Math 801: Topics in Applied Mathematics
Braids (Spring 2008)
Course description
All lectures in
a single
file
[pdf]
[djvu]
(djvu format is much smaller)
Lecture 1:
Introduction.
Lecture 2: Definitions
of Braids.
Lecture 3: Artin
Braids Groups.
Lecture 4: Fundamental Groups.
Lecture 5: Configuration
Spaces.
Lecture 6: The Presentation
Theorem.
Lectures 7–8:
The Presentation Theorem II: The Pure Braid Group.
Lecture 9: The Dirac
String Trick. [
Lectures
10–12: Mapping Class Groups.
Lecture 12*:
Mapping Class Groups of General Surfaces. [incomplete]
Lectures 13–14: The
Mapping Class Group of the Torus.
Lectures 15–16: The
Thurston–Nielsen Classification.
Lectures 17–18:
Topological Stirring.
Lecture 19:
Singularities of Foliations.
Lecture 20:
Representations of Bn.
Lecture 21:
Burau and Homology.
Lecture 22:
Topological Entropy.
Lectures 23–24:
Entropy and the Fundamental Group.
Lecture 25:
Action on π1(M) for the Torus; Manning's Theorem.
Lecture 26:
Subshifts of Finite Type.
Lectures 27–29:
Entropy of pseudo-Anosov Diffeomorphisms.
Lecture 30:
Markov Partition for pseudo-Anosovs.
Lecture 31:
From Markov Partitions to Train Tracks.
Lecture 32:
Train Track Graphs.
Lecture 33:
Normal Train Tracks and Folding.
Lecture 34:
Measured Train Tracks and Fibered Neighbouroods.
Lecture 35:
Train Track Automata.
Lecture 36:
Train Track Automata, part II: D4 and Culs-de-sac.
Lecture 37:
Minimising the Dilatation.
Lecture 38:
Maximising the Dilatation.
Lecture 39:
Computer Implementation of Train Track Automata.
Bibliography and Resources
- J. S. Birman, Braids, Links and Mapping Class Groups, Annals of
Mathematical Studies 82, Princeton University Press,
1975.
- V. L. Hansen, Braids and Coverings, London Mathematical Society
Student Texts 18, Cambridge University Press, 1989.
- Denis Auroux's
lecture
notes.
- D. Rolfsen, "New
developments in the theory of Artin's braid groups," Topology and
its Applications 127, 2003.
- J. S. Birman and T. E. Brendle, "Braids: A Survey,"
2004.
- M. Epple, "Orbits of asteroids, a braid,
and the first link invariant," Mathematical Intelligencer 20, 45,
1998.
- E. Artin, "Theory of braids," Annals of Mathematics 48, 101,
1947.
- B. Farb and D. Margalit, A Primer on Mapping
Class Groups, version 2.95, August 2007.
- Lee Mosher's web
site has several long works on mapping class groups. See also his
Notices article "What is a
train track?".
- W. P. Thurston, The Geometry and
Topology of Three-Manifolds, Electronic version 1.1 — March
2002.
- J. Milnor,
Foliations and Foliated Vector Bundles, MIT lecture notes,
1969.
- P. L. Boyland, "Isotopy Stability of Dynamics
on Surfaces," 1999.
- P. L. Boyland and J. Franks, Notes
on Dynamics of Surface Homeomorphisms, University of Warwick,
1989.
- G. Band and P. L. Boyland, "The Burau estimate for the
entropy of a braid," 2006.
- A. Fathi, F. Laundenbach, and V. Poénaru, Travaux de
Thurston sur les surfaces, Astérisque 66–67,
1979.
- A. J. Casson and S. A. Bleiler, Automorphisms of Surfaces after
Nielsen and Thurston, London Mathematical Society Student
Texts 9, Cambridge University Press, 1988.
- J.-Y. Ham and W. T. Song, "The
Minimum Dilatation of Pseudo-Anosov 5-Braids,"
Experiment. Math. 16, 167–180, 2007.
- R. C. Penner and J. L. Harer, Combinatorics of Train
Tracks, Annals of Mathematical Studies 125, Princeton
University Press, 1992.
Thurston's famous cartoon of a train track.
