For any topological space $X$, the configuration space $\operatorname{Conf}_{n}(X)$ of $n$ points in $X$ is the subspace of the iterated product $X^{\times n}$ consisting of $n$-tuples of distinct points in $X$.  Configuration spaces play an important role in many areas of mathematics, in particular low-dimensional topology and homotopy theory.  For example, the fundamental group of $\operatorname{Conf}_{n}(\mathbb R^{2})$ is the pure $n$-stranded braid group, while the orbit space of the natural action of the symmetric group, $\operatorname{Conf}_{n}(\mathbb R^{2})/\Sigma_{n}$, has fundamental group isomorphic to the entire $n$-stranded braid group.  Moreover, $\operatorname{Conf}_{n}(\mathbb R^{2})/\Sigma_{n}$ is homeomorphic to the space of complex monic polynomials of degree $n$ with exactly $n$ roots.  

In this talk I will provide a brief overview of the theory of configuration spaces, then describe the connection between configuration spaces and little disks operads, which encode operations and relations among operations in iterated loop spaces.  To conclude I will explain a new method for computing homotopy invariants of the configuration space of a product of two closed manifolds in terms of the configuration spaces of each factor separately that exploits this relationship with operads.

Joint work with Bill Dwyer and Ben Knudsen.