THIS IS A JOINT TOPOLOGY/ALGEBRAIC TOPOLOGY SEMINAR.   There are several distinct reasons to ask for the existence of an S_n-equivariant map from the configuration space F(R^d,n) of n labeled points in R^d to a certain S_n-representation sphere of dimension (d+1)(n-1)-1. We will describe some of these reasons and sketch several approaches towards such Borsuk-Ulam type problems. We obtain a complete answer using equivariant obstruction theory, based on regular cell complex models for the configuration spaces, and a tiny dose of number theory. Joint work with P. V. M. Blagojevic and W. Lück.