Abstract. I will explain what are associators (and why are they useful and natural) and what is the Grothendieck-Teichmüller group, and why it is completely obvious that the Grothendieck-Teichmüller group acts simply transitively on the set of all associators. Not enough will be said about how this can be used to show that "every bounded-degree associator extends", that "rational associators exist", and that "the pentagon implies the hexagon".
In a nutshell: the filtered tower of braid groups (with bells and whistles attached) is isomorphic to its associated graded, but the isomorphism is neither canonical nor unique - such an isomorphism is precisely the thing called "an associator". But the set of isomorphisms between two isomorphic objects *always* has two groups acting simply transitively on it - the group of automorphisms of the first object acting on the right, and the group of automorphisms of the second object acting on the left. In the case of associators, that first group is what Drinfel'd calls the Grothendieck-Teichmüller group GT, and the second group, isomorphic (though not canonically) to the first, is the "graded version" GRT of GT.
Almost everything I will talk about is in my old paper "On Associators and the Grothendieck-Teichmüller Group I, also at arXiv:q-alg/9606021.
Talk video. (also @Newton). Handout: GT.html, GT.pdf, GT.png. Sources: GT.zip. Pensieve: 2013-01
Abstract. I will define "meta-groups" and explain how one specific meta-group, which in itself is a "meta-bicrossed-product", gives rise to an "ultimate Alexander invariant" of tangles, that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that's a wonderful playground.
This will be a repeat of a talk I gave in Regina in August 2012 and in a number of other places, and I plan to repeat it a good further number of places. Though here at the Newton Institute I plan to make the talk a bit longer, giving me more time to give some further fun examples of meta-structures, and perhaps I will learn from the audience that these meta-structures should really be called something else.
Talk video. (also @Newton). Handout: beta.html, beta.pdf, beta.png. Sources: beta.zip. Pensieve: 2012-08.