Symmetry in Newcastle

Friday, 5th Apr 2019

W 238, Behavioural Sciences

Schedule:

12-1: Talk 1
1-2: Lunch
2-3: Talk 2
3-3.30: Tea
3.30-4.30: Talk 3

Dr Arnaud Brothier

(University of NSW)

Jones' actions of the Thompson's groups: applications to group theory and mathematical physics

Motivating in constructing conformal field theories Jones recently discovered a very general process that produces actions of the Thompson groups $F$,$T$ and $V$ such as unitary representations or actions on $C^{\ast}$-algebras. I will give a general panorama of this construction along with many examples and present various applications regarding analytical properties of groups and, if time permits, in lattice theory (e.g. quantum field theory).

Dr Lawrence Reeves

(The University of Melbourne)

An irrational-slope Thompson's group

Let $t$ be the the multiplicative inverse of the golden mean. In 1995 Sean Cleary introduced the irrational-slope Thompson's group $F_t$, which is the group of piecewise-linear maps of the interval $[0,1]$ with breaks in $Z[t]$ and slopes powers of $t$. In this talk we describe this group using tree-pair diagrams, and then demonstrate a ﬁnite presentation, a normal form, and prove that its commutator subgroup is simple. This group is the first example of a group of piecewise-linear maps of the interval whose abelianisation has torsion, and it is an open problem whether this group is a subgroup of Thompson's group $F$.

Dr Richard Garner

(Macquarie University)

Topos-theoretic aspects of self-similarity

A Jonsson-Tarski algebra is a set X endowed with an isomorphism $X\to XxX$. As observed by Freyd, the category of Jonsson-Tarski algebras is a Grothendieck topos - a highly structured mathematical object which is at once a generalised topological space, and a generalised universe of sets.
In particular, one can do algebra, topology and functional analysis inside the Jonsson-Tarski topos, and on doing so, the following objects simply pop out: Cantor space; Thompson's group V; the Leavitt algebra L2; the Cuntz semigroup S2; and the reduced \$C^{\ast}-algebra of S2. The first objective of this talk is to explain how this happens.
The second objective is to describe other "self-similar toposes" associated to, for example, self-similar group actions, directed graphs and higher-rank graphs; and again, each such topos contains within it a familiar menagerie of algebraic-analytic objects. If time permits, I will also explain a further intriguing example which gives rise to Thompson's group F and, I suspect, the Farey AF algebra.
No expertise in topos theory is required; such background as is necessary will be developed in the talk.