Stability phenomena in geometry - soap films, bubbles and vibrating membranes

Marcos Petrúcio Cavalcante connects geometric stability to physical examples involving soap films, bubbles and vibrating membranes. He first describes his route through differential geometry and the development of a research group and graduate programme in Maceió, Brazil. The scientific discussion concerns shapes that are critical points of a functional. Its second variation defines stability, while the Morse index counts independent directions of improvement. Through examples and pictures, the talk explains why stable configurations tend to have simple geometry and topology. Two results make this principle precise: a genus-dependent lower bound on the Morse index of compact constant-mean-curvature surfaces in Euclidean space, and a rigidity theorem for stable extremal domains of the first Dirichlet eigenvalue on the round sphere.