Parametrised topological complexity was recently defined by Cohen, Farber and Weinberger as a generalisation of Farber’s topological complexity. It quantifies the complexity of the motion planning problem in situations where the topology of the configuration space is not fixed, but instead varies in a family parametrized by some topological space, e.g. the fibres of a fibration. After reviewing the basic definitions, we will discuss some bounds which are group-theoretic in nature and arise when the spaces involved are all aspherical. Throughout the talk we will be motivated by two examples involving configurations of points in the plane, which model swarms of obstacle avoiding robots and swarms of robots with controls.

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