Given a space $X$, the configuration space $F(X,n)$ is the space of possible ways to place $n$ points on $X$, so that no two occupy the same position. But what if we allow some of the points to coincide?
The natural way to encode the allowed coincidences is as a simplicial complex $S$. I will describe how the configuration space $M(S,X)$ obtained in this way gives rise to polynomial and homological invariants of $S$, how those invariants are related to the cohomology ring $H^*(X)$, and what this has to do with the topology of spaces of maps into $X$.
I will also mention some potential applications of this structure to problems arising from international relations and economics.
This is joint work with Vin de Silva, Radmila Sazdanovic, and Robert J Carroll