Given a nonconstant harmonic function, we obtain Minkowski bounds on its critical and almost critical set. The proof relies on a refined blow-up analysis for harmonic functions based on the properties of Almgren's frequency. With minor modifications, these estimates are valid also for solutions to a very general class of elliptic PDEs. Given the link between harmonic functions and eigenfunctions of the Laplacians, with the necessary modifications these results apply also to nodal and singular sets of eigenfunctions. This is joint work with Aaron Naber.

If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes saturating sup-norm estimates. The condition is that there exists a self-focal point x₀ in M for the geodesic flow at which the associated Perron-Frobenius operator U: L²(S^*_x0M) →L²(S^*_x0M) has a nontrivial invariant function. The proof is based on von Neumann's ergodic theorem and stationary phase. In two dimensions, the condition simplifies and is equivalent to the condition that there be a point through which the geodesic flow is periodic. This is joint work with Steve Zelditch.

Let (M,g) be a compact, closed, real-analytic Riemannian manifold and P(h) = -h² Δ_g + V be a Schrodinger operator with (g,V) real-analytic. Given a regular energy value E with consider L²-normalized eigenfunctions φ_h satisfying P(h) φ_h = ( E + o(1)) φ_h . We first prove that for any hypersurface H in the forbidden region Ω(E) = { V > E } there exists a constant c_H > 0 such that

(a) ∫_H | φ_h|² ≥ e^-c_H/h.

(b) # { Z _φh ∩ H } = O_H (h^-1).

The Steklov problem is an elliptic eigenvalue problem with the spectral parameter in the boundary conditions. While this problem shares some common properties with its more well known Dirichlet and Neumann cousins, the Steklov eigenvalues and eigenfunctions have a number of distinctive geometric features. We will discuss some recent advances in the subject, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry. The talk is based on a joint survey article with A. Girouard.