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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

Thurston's famous cartoon of a train track.