Center for Advanced Intelligence Project

Speaker: Eiko Kin (Osaka University, Japan)

Title: Braids and topological mixing

Abstract: In mathematics, the braids are important tools for the knot theory, hyperbolic geometry, and dynamical systems etc. In the last ten years, the braids have been used to study mixing in fluids. Various simple mixing devices (e.g. taffy machines) have been developed. These devices utilize a particular type of braid, so-called a pseudo-Anosov type. The notion of pseudo-Anosov braids comes from the Nielsen-Thurston theory on the surface automorphisms, and the theory says that the devices using pseudo-Anosov braids are “efficient” in some sense.

In this lecture, I will give a quick introduction of the Nielsen-Thurston theory and the classification of braids. I will give a picture of the “complexity” forced by pseudo-Anosov braids. In particular, I will explain why pseudo-Anosov braids are useful and why they can be used to build interesting mixing devices.

Reference:
(1) Boyland, Philip L.; Aref, Hassan; Stremler, Mark A.
Topological fluid mechanics of stirring. J. Fluid Mech. 403 (2000), 277–304.

(2) Jean-Luc Thiffeault, A mathematical history of taffy pullers,
preprint

https://arxiv.org/pdf/1608.00152.pdf

Cosponsored by RIKEN iTHEMS and AIP Mathematical Science Team