This proposal concerns geometric questions about various types of spaces, namely smooth Riemannian manifolds, measured metric spaces, singular spaces and noncommutative spaces. The questions about smooth Riemannian manifolds concern curvature conditions, such as when a compact manifold-with-boundary can admit a metric having nonnegative scalar curvature on the interior and nonnegative mean curvature on the boundary, and when a closed manifold can admit a metric with positive p-curvature tensor. The questions about measured metric spaces concern possible extensions of the Ricci tensor to such spaces. The questions about singular spaces concern elliptic analysis. The questions about noncommutative geometry concern local proofs of the general foliation index theorem.
As mathematics and physics have evolved, the concept of space has also evolved. Some evident examples from physics are the evolution from flat space to curved space, and the evolution from the classical phase space to the quantum mechanical Hilbert space. There have also been reasons intrinsic to mathematics that lead one to extend the notion of space. Of course, one does not want to make arbitrary definitions, but one rather wants to consider concepts that arise naturally from real problems. A more recent notion of space falls under the heading of ``noncommutative geometry'', in which a traditional space is first considered to be defined in terms of the functions on it, and then the ring of functions is generalized from the commutative case to allow for a noncommutative ring of ``functions''. Noncommutative geometry is rooted in the branch of geometric analysis called index theory. Sometimes problems about ordinary ``commutative'' spaces, such as foliations, can be translated into problems about noncommutative spaces and attacked from that angle. Many of the questions considered in this proposal have their roots in physics. Throughout the years there has been much fruitful interaction between mathematics and physics, and the proposed work will hopefully contribute to this interaction. In particular, questions about positive scalar curvature arise in general relativity theory, and noncommutative geometry touches on many areas of modern theoretical physics.
