Intended Audience: College students, and high school students who think they may be potential math geniuses.
In this episode we meet Jeremy Van Horn-Morris, a mathematician from the University of Arkansas, who talks to us about some geometric and visual tools mathematicians use to understand questions in classical physics concerning the motion of particles.
Jeremy discusses some of the motivation behind his paper with Kenneth Baker and John Etnyre, Cabling, contact structures and mapping class monoids. Here is the abstract to their paper:
In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.
While the video above is for a general audience, Jeremy Van Horn-Morris’s paper is not (it’s written for other mathematicians). However, high school students who think they are potential math geniuses may still enjoy looking at it to see what advanced theorems and proofs look like.
Students and mathematicians alike will also enjoy visiting Kenneth Baker’s blog, Sketches of Topology. The post, Its full of surfaces, provides a stunning visualization and description of the open book decomposition coming from the trefoil knot, which was mentioned by Jeremy the end of our interview. Some additional posts containing phenomenal depictions of open book decompositions can be found here and here.
As always, comments are welcome!
CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708