The space of cycles in a compact Riemannian manifold has very rich topological structure. The space of hypercycles, for instance, taken with coefficients modulo 2, is weakly homotopically equivalent to the infinite dimensional real projective space. This reveals the existence of non-trivial k-parameter sweepouts for every k. We will discuss a proof of a Weyl’s law conjectured by Gromov (joint work with Liokumovich and Neves) in which the eigenvalues of the Laplacian are replaced by the areas of minimal hypersurfaces constructed by minimax methods. We will also discuss current work with Neves about Morse index bounds in the min-max theory of minimal surfaces and the problem of multiplicity.
This video is part of Harvard University‘s conference JDG 2017.
