2018, Carrasquel, José, Lupton, Gregory, Oprea, John

Topological complexity is a number that measures how hard it is to plan motions (for robots, say) in terms of a particular space associated to the kind of motion to be planned. This is a burgeoning subject within the wider area of Applied Algebraic Topology. Surprisingly, the same mathematics gives insight into the question of creating social choice functions, which may be viewed as algorithms for making decisions by artificial intelligences.

2015, Macías-Virgós, Enrique, Oprea, John, Strom, Jeff, Tanré, Daniel

In this note, we study height functions on quaternionic Stiefel manifolds and prove that all these height functions are Morse-Bott. Among them, we characterize the Morse functions and give a lower bound for their number of critical values. Relations with the Lusternik-Schnirelmann category are discussed.

2017, Farber, Michael, Grant, Mark, Lupton, Gregory, Oprea, John

In this paper we study the topological invariant TC(X) reflecting the complexity of algorithms for autonomous robot motion. Here, X stands for the configuration space of a system and TC(X) is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in X. We focus on the case when the space X is aspherical; then the number TC(X) depends only on the fundamental group π=π1(X) and we denote it TC(π). We prove that TC(π) can be characterised as the smallest integer k such that the canonical π×π-equivariant map of classifying spaces E(π×π)→ED(π×π) can be equivariantly deformed into the k-dimensional skeleton of ED(π×π). The symbol E(π×π) denotes the classifying space for free actions and ED(πtimesπ) denotes the classifying space for actions with isotropy in a certain family D of subgroups of π×π. Using this result we show how one can estimate TC(π) in terms of the equivariant Bredon cohomology theory. We prove that TC(π)≤max{3,cdD(π×π)}, where cdD(π×π) denotes the cohomological dimension of π×π with respect to the family of subgroups D. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family D.