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ICTP Associates Seminars are part personal, part scientific, with speakers sharing their career trajectory and experiences, then introducing their research field and work. This seminar will be given by Marcos Petrúcio Cavalcante from Mathematics of the Federal University of Alagoas (UFAL), in Maceió, Brazil. Join on Zoom at: https://zoom.us/meeting/register/9hJpULyeTuOKKvck7sqCgg Abstract In the first part of this talk I will tell a little about my path in mathematics, from my training in differential geometry to building a research group and a graduate program in Maceió, in the Northeast of Brazil. In the second part I will discuss some problems I have worked on in recent years concerning the shape of geometric objects that arise as critical points of a functional: soap films, soap bubbles, and domains that are critical for the fundamental frequency of a vibrating membrane. In each of these problems the second variation of the functional defines a notion of stability, together with a Morse index that counts the number of independent directions in which the object can be improved. A general principle suggests that stable configurations are simple, both geometrically and topologically. I will illustrate this principle with pictures and examples, and then describe two recent results obtained with coauthors: a lower bound for the Morse index of a compact constant mean curvature surface in Euclidean space in terms of its genus, and a rigidity theorem for stable extremal domains of the first Dirichlet eigenvalue on the round sphere. Bio Marcos Petrúcio Cavalcante is a Full Professor at the Institute of Mathematics of the Federal University of Alagoas (UFAL), in Maceió, Brazil, where he currently coordinates the graduate program in Mathematics. He received his PhD in 2006 from IMPA (Rio de Janeiro) under the supervision of Manfredo do Carmo. He works in differential geometry and geometric analysis, with emphasis on minimal and constant mean curvature surfaces, eigenvalue problems and overdetermined elliptic equations on Riemannian manifolds. He has been a Regular Associate of ICTP since 2022 and is spending the autumn of 2026 at ICTP for an extended research stay. |
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ICTP Associates Seminar: Stability phenomena in geometry - soap films, bubbles and vibrating membranes
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