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Explore recent advancements in constructing embedded minimal surfaces in S³ and free boundary minimal surfaces in B³ with prescribed topology through a seminar on spectral geometry. Delve into the work of Stern, Karpukhin, Kusner, and McGrath, focusing on intrinsic shape optimization problems for Laplace and Steklov eigenvalues in the presence of symmetries. Examine key analytic challenges in the existence theory for metrics maximizing Laplace and Steklov eigenvalues on surfaces, both with and without prescribed symmetry. Discover new techniques developed to overcome these challenges and consider lingering open questions in the field. This 48-minute talk, presented by Daniel Stern from Cornell University, is part of the Seminar Spectral Geometry in the clouds series organized by the Centre de recherches mathématiques (CRM).
