Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently,
the applicant and collaborators have developed new methods for studying the regularity of solutions to these
equations, and applied them to a different problem, that of estimating quantities depending on the smaller
eigenvalues of a Schroedinger operator. This project builds on the early success of this program: it will
produce new understanding of the behaviour of eigenvalues, establish sharp estimates for spectral
quantities, particularly on manifolds with curvature bounds, and find optimal conditions under which noncompact
solutions to curvature flows are stable.