Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, February 19, 2020
1:00 p.m.

PhD Candidate:  Travis Ens
Supervisor:   Dror Bar-Natan
Thesis title:   On Braidors: An Analogue of the Theory of Drinfel’d Associators for Braids in
an Annulus


We develop the theory of braidors, an analogue of Drinfel’d’s theory of associators in which braids in an annulus are considered rather than braids in a disk.  After defining braidors and showing they exist, we prove that a braidor is defined by a single equation, an analogue of a well-known theorem of Furusho [Furusho (2010)] in the case of associators. Next some progress towards an analogue of another key theorem, due to Drinfel’d [Drinfel’d (1991)] in the case of associators, is presented. The desired result in the annular case is that braidors can be constructed degree be degree. Integral to these results are annular versions \textbf{GT}$_a$ and \textbf{GRT}$_a$ of the Grothendieck-Teichm\”uller groups \textbf{GT} and \textbf{GRT} which act faithfully and transitively on the space of braidors.

We conclude by providing surprising computational evidence that there is a bijection between the space of braidors and associators and that the annular versions of the Grothendieck-Teichm\”uller groups are in fact isomorphic to the usual versions potentially providing a new and in some ways simpler description of these important groups, although these computations rely on the unproven result to be meaningful.

A copy of the thesis can be found here:  ens_thesis


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