Centre de recherches mathématiques - CRM Courses

The Steklov Problem with Piecewise Smooth Boundary

Delve into the Steklov problem on domains with piecewise smooth, Lipschitz boundaries through scattering theory, exploring eigenvalues and eigenfunctions in dimension two and potential extensions to higher dimensions.

• YouTube
• 1 hour 3 minutes
• Self-Paced
• Free Video

Steklov Spectral Inverse Problem in a Conformal Class

Delve into the Steklov spectral inverse problem in conformal classes, exploring how to recover a Riemannian manifold's metric from its Steklov spectrum using wave operator traces and geodesic X-ray transforms.

• YouTube
• 1 hour 10 minutes
• Self-Paced
• Free Video

Young Researchers: 5 Short Talks on Spectral Geometry 2025

Explore five short talks by young researchers on advanced mathematical topics including Steklov eigenvalues, isoperimetric properties, and spectral gaps in this seminar from the Spectral Geometry in the Clouds series.

• YouTube
• 57 minutes
• Self-Paced
• Free Video

Free Boundary Minimal Annuli in the Ball

Explore the theory of free boundary minimal surfaces in the unit ball of R³, focusing on minimal annuli and the Critical Catenoid Conjecture, with insights from recent developments in the field.

• YouTube
• 1 hour 2 minutes
• Self-Paced
• Free Video

On a Conjecture of a Polya Functional for Triangles and Rectangles

Delve into Phanuel Mariano's exploration of a Pólya functional involving Dirichlet eigenvalue and torsional rigidity, examining the conjectured bounds for triangles and rectangles with precise mathematical estimates.

• YouTube
• 56 minutes
• Self-Paced
• Free Video

Convex Sets Can Have Interior Hot Spots

Explore the Hot Spots conjecture in thermal equilibrium as Jaume de Dios Pont disproves the theory that maximum temperatures approach boundaries in convex sets, particularly in high dimensions.

• YouTube
• 1 hour 8 minutes
• Self-Paced
• Free Video

Precise Eigenvalue Location for Random Regular Graphs

Delve into the spectral theory of regular graphs as Theo McKenzie demonstrates that most regular graphs are Ramanujan, with a 69% probability, through rigorous analysis of the Green's function of the adjacency operator.

• YouTube
• 1 hour 8 minutes
• Self-Paced
• Free Video

Spectral Statistics of Negatively Curved Surface Covers

Delve into spectral statistics of negatively curved surface covers, exploring the BGS conjecture, Random Matrix Theory, and recent developments in quantum systems with Julien Moy from Orsay.

• YouTube
• 51 minutes
• Self-Paced
• Free Video

Isoperimetric and Geometric Inequalities for the Magnetic Laplacian

Explore isoperimetric and geometric inequalities for the magnetic Laplacian with Luigi Provenzano, focusing on eigenvalues, ground state energy bounds, and reverse Faber-Krahn inequality for simply connected domains.

• YouTube
• 50 minutes
• Self-Paced
• Free Video

Random Perturbation of Toeplitz Matrices

Explore the spectral analysis of Toeplitz matrices with random perturbations, examining outlier eigenvalues and their relationship to random analytic matrix fields and deterministic matrices capturing resonance structures.

• YouTube
• 1 hour 5 minutes
• Self-Paced
• Free Video

2D Pure Magnetic Tunneling Formula

Delve into the mathematical analysis of 2D magnetic laplacian, exploring non-degenerate wells, symmetry, and the exponential splitting between first two eigenvalues in this advanced spectral geometry seminar.

• YouTube
• 53 minutes
• Self-Paced
• Free Video

Asymptotic Optimization of Riesz Means of Laplace Eigenvalues on Convex Sets

Delve into the asymptotic optimization of Riesz means of Laplace eigenvalues on convex sets, exploring mathematical proofs for convergence in Hausdorff distance to a disk in 2D and similar results in higher dimensions.

• YouTube
• 1 hour 7 minutes
• Self-Paced
• Free Video

Upper Bounds on the Second Laplacian Eigenvalue on the Projective Space

Delve into the investigation of isoperimetric inequality of the second Laplacian eigenvalue on real projective space, exploring upper bounds, trial functions, and topological degree methods.

• YouTube
• 51 minutes
• Self-Paced
• Free Video

Weyl Formulae for Some Singular Metrics with Application to Acoustic Modes in Gas Giants

Explore Weyl formulae for singular metrics and their application to acoustic modes in gas giants, with insights on eigenvalue asymptotics of the Laplace-Beltrami operator and soundwave propagation in gas planets.

• YouTube
• 1 hour
• Self-Paced
• Free Video

Foliations of Bubblesheet Singularities: A Spectral Geometric Approach

Explore the concept of foliations in differential geometry, focusing on bubblesheet singularities through a spectral geometric approach and the new Quasi-Parallel Mean Curvature condition.

• YouTube
• 51 minutes
• Self-Paced
• Free Video