Higer order Cheeger inequalities give lower bounds on the $n$-th eigenvalue of the Laplacian on a compact Riemannian manifold $M$
in terms of isoperimetric ratios associated to some optimal partitions of $M$ into $n$ subsets.
They generalizes the minimal cut of $M$ into two parts to get a lower bound of the spectral gap.
The talk will show how to extend such bounds to Steklov operators on the boundary of $M$
(also known as Dirichlet to Neumann operators).
Similar results will also presented in the corresponding probabilist contexts of finite and measurable state spaces.
Work in collaboration with Asma Hassannezhad (Bristol).