Systolic and Diastolic Geometry - Spectral Optimization with Respect to the Metric on Surfaces

Explore a lecture by Romain Petrides from Paris Cité University on "Spectral optimization with respect to the metric on surfaces of any topology" as part of the "Systolic and Diastolic Geometry" event at IMPA (Instituto de Matemática Pura e Aplicada) in Rio de Janeiro. This 58-minute presentation delves into geometric problems comparing volume of spaces with lengths/areas of curves/surfaces they contain. The lecture is part of a broader conference examining fundamental questions in geometric analysis, such as: which billiards with fixed area have the largest minimum-period orbit, how large can the shortest periodic geodesic be on spheres of equal area, and how large can the first positive eigenvalue of the Laplacian operator be among surfaces of equal area. The event brought together specialists from Differential Geometry, Analysis, and Symplectic Geometry to discuss state-of-the-art approaches to these problems and foster innovative techniques for future research.