PIRSA:20090022

On the geometry of nodal domains for random eigenfunctions on compact surfaces

Eswarathasan, S. (2020). On the geometry of nodal domains for random eigenfunctions on compact surfaces. Perimeter Institute. https://pirsa.org/20090022

Eswarathasan, Suresh. On the geometry of nodal domains for random eigenfunctions on compact surfaces. Perimeter Institute, Oct. 01, 2020, https://pirsa.org/20090022 

          @misc{ pirsa_PIRSA:20090022,
            doi = {10.48660/20090022},
            url = {https://pirsa.org/20090022},
            author = {Eswarathasan, Suresh},
            keywords = {Mathematical physics},
            language = {en},
            title = {On the geometry of nodal domains for random eigenfunctions on compact surfaces},
            publisher = {Perimeter Institute},
            year = {2020},
            month = {oct},
            note = {PIRSA:20090022 see, \url{https://pirsa.org}}
          }

Suresh Eswarathasan

Dalhousie University

Talk number
PIRSA:20090022
Collection
Talk Type
Subject
Abstract

A classical result of R. Courant gives an upper bound for the count of nodal domains (connected components of the complement of where a function vanishes) for Dirichlet eigenfunctions on compact planar domains.  This can be generalized to Laplace-Beltrami eigenfunctions on compact surfaces without boundary. When considering random linear combinations of eigenfunctions, one can make this count more precise and pose statistical questions on the geometries appearing amongst the nodal domains: what percentage have one hole? ten holes? what percentage have their boundary being tangent 100 times to a fixed non-zero vector field? The first 20-25 minutes will give a survey on some fundamental results of Nazarov-Sodin, Sarnak-Wigman, and Gayet-Welschinger before presenting some joint works with I. Wigman (King's College London) and Matthew de Courcy-Ireland (École Polytechnique Fédérale de Lausanne) answering these questions in the last 25-30 minutes.