## 1. Braids and the Grothendieck-Teichmuller Group

In

Overview Days
for non-specialists.

**Abstract.** I will explain what are associators (and why are
they useful and natural) and what is the Grothendieck-Teichmuller group,
and why it is completely obvious that the Grothendieck-Teichmuller 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-Teichmuller 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-Teichmuller 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

## 2. Meta-Groups, Meta-Bicrossed-Products, and the Alexander Polynomial

On January 17, 2013.

**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.