Mathematics Colloquium: Svitlana Mayboroda (Minnesota), ' Localization of eigenfunctions and associated free boundary problems'

Event Date: 

Thursday, April 17, 2014 -
3:30pm to 4:30pm

Event Location: 

  • 6635 South Hall

Abstract:

The phenomenon of wave localization permeates acoustics, quantum physics, elasticity, energy engineering. It was used in construction of the noise abatement walls, LEDs, optical devices. Anderson localization of quantum states of electrons has become one of the prominent subjects in quantum physics, as well as harmonic analysis and probability. Yet, no methods predict specific spatial location of the localized waves.

In this talk I will present recent results revealing a universal mechanism of spatial localization of the eigenfunctions of an elliptic operator and emerging operator theory/analysis/geometric measure theory approaches and techniques. We prove that for any operator on any domain there exists a ``landscape" which splits the domain into disjoint subregions and indicates location, shapes, and frequencies of the localized eigenmodes. In particular, the landscape connects localization to a certain multi-phase free boundary problem, regularity of minimizers, and geometry of free boundaries.

This is joint work with D. Arnold, G. David, M. Filoche, and D. Jerison.