Emerging Topics Workshop on Nodal sets of Eigenfunctions

Abstract: We present a sequence of positive quadratic forms associated with harmonic functions on Abelian groups. We show how the positivity property recovers the polynomial Liouville property and we prove a three spheres theorem in terms of random...

Abstract: I will discuss recent results regarding the problem of counting intersections of eigenfunction nodal sets with real analytic curves H and the connection to eigenfunction restriction bounds over H.

Abstract: The classical Liouville theorem says that if a harmonic function on the plane is bounded then it is a constant. At the same time for any angle on the plane, there exist non-constant harmonic functions that are bounded outside the angle...

Abstract: In the talk I will describe what is known and (mostly) unknown about asymptotic statistical topology and geometry of zero sets of random spherical harmonics of large degree. I plan to discuss (a) several provoking open questions and (b)...

Abstract: An open problem is to prove that for any (or at least any generic) Riemannian metric, there is some sequence of eigenfunctions of the Laplacian for which the number of nodal domains tends to infinity. It sounds easy but as yet there are...

Abstract: In the 1970s Berry conjectured that the behavior of high energy, quantum-chaotic billiard systems could be well modeled by random waves. That is random combinations of the plane waves e^{ik ·x}. On manifolds it is more natural to...