## Research

I am interested in all kind of geometry, in particular in algebraic topology and geometric group theory. I also like basic number theory.

My research concerns configuration spaces understood as the spaces of all possible placement of given finite number of points inside given topological space. It has many applications: in physics - configuration space of system of particles; in industry - motion planning, robotics. In mathematics it can be used to obtain invariants of manifolds - crucial in geometry, to investigate mapping class groups or to construct interesting groups such as braid groups, which are fundamental groups of configurations spaces of R^n.

Classically people considered configuration spaces of manifolds, which - due to their regularity - have configuration spaces, that can be described with use of well-known tools such as fibrations. Configuration spaces become richer for a spaces that posses some kind of singularities, for instance - cell complexes. Even for graphs description of configuration space may be quite difficult.

The aim of the research is to describe as many topological properties of configuration spaces as possible for widest possible class of spaces with particular focus on cube and simplicial complexes.

A note recording partial progress: