The (un)stable topology of configuration spaces

This project concerns the mathematics of multiple occupancy. Collecting the possible positions of a fixed number of agents in a given environment, and omitting those in which two or more agents collide, one obtains a space of inexhaustible theoretical and, in a present-future of automated factories and autonomous vehicles, practical interest. This project approaches configuration spaces from the vantage of algebraic topology, the mathematical inquiry into the global character of space, enriched and deepened by contemporary techniques and ideas from topological robotics, mathematical physics, representation stability, homotopical algebra, and higher category theory. In addition, the project funds will bring in seminar speakers, enriching the mathematical environment of local graduate students and contributing to the advancement of early career researchers. The project has two main components. In the first, the environment or background space is a graph. In this arena, the aim is to understand asymptotic phenomena in Betti numbers and multiplicities of irreducible representations, identify universal generators and relations, investigate torsion phenomena, and calculate topological complexity in the unstable regime. In the second, the background space is manifold, and the aim is to compute positive characteristic homology by leveraging a connection to spectral Lie algebras, to study the invariance properties of configuration spaces, and to calculate the homology of the ordered configuration spaces of the torus using the representation theory of combinatorial categories. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.