One of the main problems in robotics is that of motion planning. It consists of finding an algorithm which takes pairs of positions as an input and outputs a path between them. It is not always possible to find such an algorithm which depends continuously on the inputs. Studying this problem from a topological perspective, in 2003 Michael Farber introduced the topological complexity of a space, which measures the minimal (unavoidable) discontinuity of all motion planners on a given topological space. The topological complexity TC(X) turns out to be a homotopy invariant of the space X.

In this talk we will determine (or narrow down to a few values in some cases) the topological complexity of the unordered configuration spaces of aspherical surfaces (including surfaces with boundary and non-orientable surfaces). We will also see how this can be understood as the topological complexity of the surface braid groups.

This is joint work with Andrea Bianchi.