Betti numbers of random nodal sets of elliptic pseudo-differential operators

TL;DR

This paper estimates the growth of Betti numbers of random nodal sets of elliptic pseudo-differential operators on manifolds, revealing their asymptotic behavior and equidistribution properties of critical points.

Contribution

It provides new upper bounds for Betti numbers of random nodal sets of elliptic operators, linking topological invariants to eigenvalue growth and critical point distribution.

Findings

Betti numbers grow at most polynomially with eigenvalue cutoff L

Critical points of Morse functions on nodal sets are equidistributed

Explicit estimates for Laplace-Beltrami and Dirichlet-to-Neumann operators

Abstract

Given an elliptic self-adjoint pseudo-differential operator $P$ bounded from below, acting on the sections of a Riemannian line bundle over a smooth closed manifold $M$ equipped with some Lebesgue measure, we estimate from above, as $L$ grows to infinity, the Betti numbers of the vanishing locus of a random section taken in the direct sum of the eigenspaces of $P$ with eigenvalues below $L$. These upper estimates follow from some equidistribution of the critical points of the restriction of a fixed Morse function to this vanishing locus. We then consider the examples of the Laplace-Beltrami and the Dirichlet-to-Neumann operators associated to some Riemannian metric on $M$.