Date:
Thu, 27/03/202512:15-14:00
Location:
Ross 70
Title: Nodal volume of harmonic functions
Abstract: Nadirashvili conjectured that the nodal volume (i.e., the appropriate Hausdorff measure of the zero set) of a non-constant harmonic function on \mathbb{R}^n is infinite, for n\geq 2. While the conclusion is straightforward in two dimensions, it proved to be a very difficult question in higher dimensions. Nearly a decade ago, in a remarkable feat, Logunov resolved this conjecture in all higher dimensions. Recently, together with Logunov and Sartori, we obtained almost sharp local estimates for the nodal volume of harmonic functions. This talk aims to provide an overview of a closely related topic: nodal volume of Laplace eigenfunctions on smooth compact manifolds, delve into the main ideas of Logunov that helped resolve Nadirashvili's conjecture, and conclude with a discussion of the recent result on local estimates.
Abstract: Nadirashvili conjectured that the nodal volume (i.e., the appropriate Hausdorff measure of the zero set) of a non-constant harmonic function on \mathbb{R}^n is infinite, for n\geq 2. While the conclusion is straightforward in two dimensions, it proved to be a very difficult question in higher dimensions. Nearly a decade ago, in a remarkable feat, Logunov resolved this conjecture in all higher dimensions. Recently, together with Logunov and Sartori, we obtained almost sharp local estimates for the nodal volume of harmonic functions. This talk aims to provide an overview of a closely related topic: nodal volume of Laplace eigenfunctions on smooth compact manifolds, delve into the main ideas of Logunov that helped resolve Nadirashvili's conjecture, and conclude with a discussion of the recent result on local estimates.