**Monday 8 May 2023**

**15:00–16:30***

**Fiona Torzewska (University of Vienna)** — *Motion groupoids and mapping class groupoids*

The braiding statistics of point particles in 2-dimensional topological phases are given by representations of the braid groups. One approach to the study of generalised particles in topological phases, loop particles in 3-dimensions for example, is to generalise (some of) the several different realisations of the braid group.

In this talk I will construct for each manifold M its motion groupoid Mot_{M}, whose object class is the power set of M. I will discuss several different, but equivalent, quotients on motions leading to the motion groupoid. In particular that the congruence relation used in the construction Mot_{M} can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows – worldlines (e.g. monotonic 'tangles').

I will also give a construction of a mapping class groupoid MCG_{M} associated to a manifold M with the same object class. For each manifold M I will construct a functor F : Mot_{M} → MCG_{M}, and prove that this is an isomorphism if π_{0} and π_{1} of the appropriate space of self-homeomorphisms of M is trivial. In particular, there is an isomorphism in the physically important case M=[0,1]^{n} with fixed boundary, for any n∈**N**.

I will give several examples throughout.

* Eastern European Time, i.e. UTC+2 in winter and UTC+3 in summer.