Braids and Dynamics

This monograph uses braids to explore dynamics on surfaces, with an eye towards applications to mixing in fluids. The text uses the particular example of taffy pulling devices to represent pseudo-Anosov maps in practice. In addition, its final chapters also briefly discuss current applications in the emerging field of analyzing braids created from trajectory data. While written with beginning graduate students, advanced undergraduates, or practicing applied mathematicians in mind, the book is also suitable for pure mathematicians seeking real-world examples. Readers can benefit from some knowledge of homotopy and homology groups, but these concepts are briefly reviewed. Some familiarity with Matlab is also helpful for the computational examples.

Presents principal concepts of topological dynamics as applied to fluid dynamics Discusses current applications involving braids from data Suitable for graduates, advanced undergraduates, and researchers in pure and applied mathematics

Résumé

"This book is laudable for giving an account of the theory of braid groups and surface homeomorphisms that's down-to-earth, inviting, and accessible to its intended audience: beginning graduate students, advanced undergraduates, and applied mathematicians." (Bena Tshishiku, Mathematical Reviews, February, 2024)

"This book is a very pleasant to read, linking seemingly unrelated topic like braid and fluid dynamics, perhaps a bit easier for a pure mathematician than for an applied one, but still very much enjoyable." (Mauro Artigiani, zbMATH 1509.37003, 2023)

"This short book considers an unusual combination of algebraic topology, dynamics and applications to the mixing of fluids. Probably not many mathematics texts use a taffy puller in a primary example. This one does, and it incorporates interesting connections to dynamics on surfaces, the algebraic topology of braids and applications to mixing. ... Several intriguing ideas are presented here very quickly, and it takes some effort to make all the pieces fit together." (Bill Satzer, MAA Reviews, January 29, 2023)

Contenu
Introduction.- Topological dynamics on the torus.- Stretching with three rods.- Braids.- The Thurston-Nielsen classification.- Topological entropy.- Train tracks.- Dynnikov coordinates.- The braidlab library.- Braids and data analysis.- References.- Appendix: Derivation of Dynnikov update rules.