Given a space X, the configuration space of n points on X is the space of ways to distribute those points in X without allowing any pair to coincide.
But what if we allow some subsets of the points to coincide? The allowed coincidences form a simplicial complex S; we call the resulting simplicial configuration space X_S. In many ways the theory of simplicial configuration spaces generalizes the theory of configuration spaces, but is richer because of the combinatorial structure S.
I will discuss some tools for computing the homology of X_S, as it reflects the topology of X and combinatorial properties of S. Along the way, we'll discover a novel polynomial invariant of simplicial complexes which generalizes the chromatic polynomial of graphs.
Time permitting, I will also mention a few ideas for applying this structure to social choice problems ranging from romantic comedy to the descent into war.