What I (Don’t) Know About Critical Points of Eigenfunctions
Organizers
Speaker
Time
Thursday, October 23, 2025 2:15 PM - 3:30 PM
Venue
A7-301
Online
Zoom 388 528 9728
(BIMSA)
Abstract
Abstract:
This talk will be an overview of what is known—and what isn’t—about critical points of Laplace eigenfunctions, motivated by a recent question of Prof. Yau.
I will first recall the naive bounds that follow from Courant’s nodal domain theorem, then review Yau’s contributions and conjectural perspectives on the structure of critical sets. We will then look at several “pathological” examples that challenge our conjectural picture and discuss why, for generic metrics, there is still hope that the conjectural picture holds.
(Disclaimer: I am by no means an expert on this topic—so don’t expect any new theorems, and some of the questions I’ll ask may already be known to experts!)
This talk will be an overview of what is known—and what isn’t—about critical points of Laplace eigenfunctions, motivated by a recent question of Prof. Yau.
I will first recall the naive bounds that follow from Courant’s nodal domain theorem, then review Yau’s contributions and conjectural perspectives on the structure of critical sets. We will then look at several “pathological” examples that challenge our conjectural picture and discuss why, for generic metrics, there is still hope that the conjectural picture holds.
(Disclaimer: I am by no means an expert on this topic—so don’t expect any new theorems, and some of the questions I’ll ask may already be known to experts!)