Topological inference for Morse indices

The Morse index associated to a critical point of a smooth function is a local quantity that is equal to the number of negative eigenvalues of the Hessian evaluated at that point. As second derivatives might be difficult to compute or unavailable in real world contexts, one can use the fundamental results of Morse theory to bypass the need for second derivatives. Unfortunately, this involves computing the relative homology of pairs of sub-level sets, which is no longer a local quantity. In this talk, we propose a new algorithm which combines the best of both worlds by reducing the computation of the Morse index to a homology inference problem. The key ingredient is the theory of Gromoll-Meyer pairs, which facilitates this transition from global sublevel sets to local submanifolds with corners. Finally, we describe an upper bound on the density of sample points needed in order to recover the Morse index.