Given a smooth manifold, does it admit a Riemannian metric with positive Ricci curvature? The list of examples for which the answer is affirmative is not very long: Kähler manifolds with positive definite first Chern class, homogeneous spaces (e.g. spheres and projective spaces), and a handful of others. One general approach to this question is to ask if the existence of a Ricci-positive metric is stable under elementary topological modification, e.g. connected sum, boundary union, or surgery. There are a few results in this direction, from which we have some further examples of Ricci-positive metrics on: connected sums of products of two spheres, any connected sum of products of projective spaces with a single sphere, or any connected sums of projective spaces.
In our previous work, we gave a sufficient condition in terms of the individual Riemannian manifolds (M,g) and (N,h) to ensure that M#N also admits a Ricci-positive metric. In this talk I will explain how it is possible to extend this result to construct Ricci-positive metrics on connected sums of products of M with arbitrarily many spheres under the same hypotheses. The idea of the proof is straightforward, given (M,g) which satisfies the sufficient condition, show that there is a metric on M times a sphere that also satisfies the condition. I hope to indicate the major difficulties involved in this construction and how to overcome them.