Christopher Judge
From
In residence at
Institut Denis Poisson / CNRS, University of Orléans, University of Tours - FR
Host scientist
Luc Hillairet
BIOGRAPHY
Professor Chris Judge holds a permanent position in Mathematics at Indiana University Bloomington, where his research spans automorphic forms, PDEs, Teichmüller theory, and dynamical systems. He has established expertise in how eigenvalues and eigenfunctions of the Laplacian vary as one perturbed the geometry of the domain as well as expertise in the dynamics and geometry of translation surfaces. His work combines tools from euclidean and hyperbolic geometry, representation theory, PDEs and functional analysis.
Recent major achievements include a complete resolution of the `hot spots conjecture' for Euclidean triangles, published in the Annals of Mathematics, demonstrating that the maximum of the first nontrivial eigenfunction of the Laplacian occurs at a boundary point rather than an interior critical point. This resolution for Euclidean triangles had been the goal of the failed Polymath 7 project.
Judge has published extensively and has received support from the National Science Foundation, the Simons Foundation, and the Leverhulme Trust. His collaborative work has involved partnerships across Europe and North America, fostering connections between research communities in spectral geometry and Teichmuller dynamics.
PROJECT
Behavior of Laplace eigenfunctions under geometric degeneration
This research project aims to investigate fundamental questions in the spectral theory of hyperbolic surfaces, focusing on eigenvalue behavior under geometric deformation. The two main goals are to prove the existence of (1) a compact hyperbolic surface with simple spectrum and (2) a finite-area hyperbolic surface whose embedded eigenvalues do not dominate Weyl's law. The approach combines novel techniques including earthquake deformations along geodesic laminations and asymptotic separation of variables. This research builds on recent breakthroughs by Hillairet and Judge regarding spectral properties of hyperbolic triangles, while developing new methodological synthesis. Success would resolve long-standing conjectures and provide new tools for analyzing quantum chaos on curved surfaces. The collaboration between Professor Judge (Indiana University) and Professor Hillairet (Institut Denis Poisson) unites complementary expertise in geometric and analytic aspects of spectral theory.