Configuration spaces of mechanical systems appear at the cross roads of mathematics and engineering, with applications in robotics, chemistry and biology. In mathematics, configuration spaces have been studied from a topological viewpoint and some of their models have been related to certain classical topics like moduli spaces. In the last two decades, thanks to the pioneering work of Farber, navigational complexity and motion planning have been at the forefront of algebraic topology. He modeled these concepts as invariants of some fibrations, resulting in an intense research activity.
In this talk, I will present my work on these two topics. In particular, I will first explain a new notion called the equivariant parametrized topological complexity. It is a measure of the complexity of designing parametrized motion planning algorithms that have symmetries. Next, I will present a result which shows that the configuration space of closed polygonal linkage can be obtained iteratively by performing cellular surgery on the projective Coxeter complex of type A. This result can be considered analogous to the seminal result of Kapranov, that shows how the Deligne-Mumford-Knudson compactification of the moduli space of genus zero curves can be obtained from the projective Coxeter complex of type A by blowing up along the minimal building set.
