If M is a manifold (or even a topological space), the configuration space

F(M,k) is the space of k points in M. For example, the space F(R^3,k)

represents the space of possible configurations of k objects in space.

First we will recall some of the basic things from homotopy theory such as cohomology,

and then consider the special case of F(R^2, k) and describe its cohomology, and its connection with

the braid groups. We then move on to discuss the following two actively researched problems:

1) When is F(M,k) invariant under homotopy?

2) Find an algebraic model for F(M,k).

The present research is joint work with Pascal Lambrechts.