Coarse nodal topology and persistence barcodes
— Vukasin Stojisavljevic
6 octobre 2022 - 11:00Salle de conférences IRMA
The total number of nodal domains of a Laplace-Beltrami eigenfunction on a closed manifold is bounded from above by an appropriate power of the corresponding eigenvalue. This is a consequence of Courant's nodal domain theorem combined with Weyl's law. In general, bounds of this type do not exist for linear combinations of eigenfunctions. We will show how, by coarsely counting nodal domains, i.e. by ignoring small oscillations, we may obtain a similar upper bound for linear combinations as well. The proof uses the theory of persistence modules and barcodes combined with multiscale polynomial approximation of functions in Sobolev spaces. Using the same method, we may study coarse topology of a zero set of a function, as well as coarse topology of the set of common zeros of a number of different functions. This allows us to prove a coarse version of Bézout's theorem for linear combination of Laplace-Beltrami eigenfunctions. The talk is based on a joint work with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.