In this talk, we investigate the role of numerical analysis and scientific computing in the construction of rigorous proofs of conjectures. We focus on eigenproblems, and present recent progress on three unusual, conceptually simple, eigenvalue problems. We explore how validated numerics and provable convergence and error estimates are helpful in proving theorems about the eigenvalue problems. The first of these problems concerns sharp bounds on the eigenvalue of the Laplace-Beltrami operator of closed Riemannian surfaces of genus higher than one. One may ask: for a fixed genus, and a given fixed surface area, which surface maximizes the first Laplace eigenvalue? The second of these concerns eigenvalue problems for the Laplacian, with mixed Dirichlet-Neumann data. If the Neumann and Dirichlet curves meet at an angle which is $\pi$ or larger, reflection strategies will not work. The third problem is about the famous Hot Spot conjecture: the extrema of the 2nd Neumann eigenfunction of the Laplacian in an acute triangle will be at the vertices.
Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, room 5340