On a Conjecture of a Polya Functional for Triangles and Rectangles

This seminar talk from the "Spectral Geometry in the clouds" series explores a mathematical conjecture related to a Pólya functional for triangles and rectangles. Join Phanuel Mariano from Union College as he investigates a functional involving the product of the first Dirichlet eigenvalue and the torsional rigidity of a planar domain, normalized by its area. Learn about Pólya's original work showing this quantity is bounded by 1, and follow Mariano's exploration of the conjecture that this functional is bounded above by π²/12 and below by π²/24 for bounded planar convex domains. Discover the proof that this conjecture holds true for all triangles and rectangles, including precise estimates for triangles and a demonstrated monotonicity property for rectangles. The presentation covers joint work with Rodrigo Banuelos and was delivered as part of the Centre de recherches mathématiques (CRM) seminar series.