Scalar curvature, conformal geometry, and the Ricci flow with surgery

In this note we will review recent results concerning two geometric problems associated to the scalar curvature. In the first part we will review the solution to Schoen's conjecture about the compactness of the set of solutions to the Yamabe problem. It has been discovered, in a series of three papers, that the conjecture is true if and only if the dimension is less than or equal to 24. In the second part we will discuss the connectedness of the moduli space of metrics with positive scalar curvature in dimension three. In two dimensions this was proved by Weyl in 1916. This is a geometric application of the Ricci flow with surgery and Perelman's work on Hamilton's Ricci flow.