All teachers and students are warmly welcome!

School of Mathematics

June 10, 2026

Abstract： 

The mathematical description of many physical phenomena, such as quantum mechanics, wave propagation, or diffusion processes is governed by the same object, the Laplace operator. This lets us, in many classical settings, use physical intuition to interpret the properties of the Laplace operator by considering the simplest constant coefficient version of the physical phenomenon at hand, and exploiting our physical intuition. 

This talk takes a complementary point of view: we will show multiple scenarios where studying this simplest Laplace operator set-up is in fact exactly equivalent to studying the more general phenomenon. We study the behavior of the first few eigenfunctions of the Laplace operator in extreme geometric or asymptotic regimes. In the limit, the Laplacian itself undergoes a qualitative change, recovering variable coefficient physical equations. Depending on the regime, the limiting behavior may be described by an effective Schrödinger operator, a drift–diffusion equation, or a heat equation with an emergent time variable. 

Several classical problems will serve as guiding examples, including questions related to the Hot Spots and KLS conjectures, homogenization for perforated domains, and rigidity phenomena for eigenfunctions of convex sets. Although these topics are not governed by a single unifying mechanism, they share a common feature: informative spectral behavior becomes visible only in limiting regimes where an effective operator emerges. 

Biography:

  Jaume de Dios Pont is currently a Faculty Fellow at the NYU Center for Data Science. His research focuses on spectral theory and convex geometry, especially in high dimensions, and, more recently, AI for Mathematics. He received his PhD in Mathematics from UCLA in 2023, under the supervision of Terence Tao, followed by a Postdoctoral position at ETH where he worked with Svitlana Mayboroda. He is best known for his work on counterexamples to the Hot Spots Conjecture.