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Explore innovative methods for generating families of closed minimal surfaces in $S^3$ and free boundary minimal surfaces in $B^3$ through intrinsic shape optimization problems from spectral geometry in this hour-long conference talk. Delve into applications that resolve the topological realization problem for free boundary minimal surfaces in $B^3$ and demonstrate how the number of closed minimal surfaces in $S^3$ with prescribed topology and low area increases at least linearly with genus. Examine new techniques for establishing the existence of extremal metrics in broader contexts and consider persistent open problems in the field. Learn about joint research conducted with M. Karpukhin, R. Kusner, and P. McGrath, advancing understanding of minimal surfaces and their properties in spherical and ball geometries.
