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The topology of the nodal sets of eigenfunctions and a problem of Michael Berry

Daniel Peralta-Salas
Institution: 
ICMAT
Schedule: 
Tuesday, October 27, 2020 - 16:00 to 17:00
Location: 
A-005
Location: 
room A-005 (maximum 25 people inside) and on zoom, sign in to get the link
Abstract: 

 In 2001, Sir Michael Berry conjectured that given any knot there should exist a (complex-valued) eigenfunction of the harmonic oscillator (or the hydrogen atom) whose nodal set contains a component of such a knot type. This is a particular instance of the following problem: how is the topology of the nodal sets of eigenfunctions of Schrodinger operators? In this talk I will focus on the flexibility aspects of the problem: either you construct a suitable Riemannian metric adapted to the submanifold you want to realize, or you consider operators with a large group of symmetries (e.g., the Laplacian on the round sphere, or the harmonic quantum oscillator), and exploit the large multiplicity of the high eigenvalues. In particular, I will show how to prove Berry's conjecture using an inverse localization property. This talk is based on different joint works with A. Enciso, D. Hartley and F. Torres de Lizaur. This series of seminars focuses on Real Geometry, touching topics from real algebraic geometry, control theory, convex geometry, nodal sets, random topology and subriemannian geometry. More information can be found here: https://sites.google.com/view/geometric-structures/.

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