The Plateau problem, named by Henry Lebesgue after the Belgian physicist, consists in finding the surface of least area which spans a given contour: Plateau discovered that soap films obey that simple energy principle. In order to investigate the question, generations of mathematicians have investigated the very fundamental notions of "surface", "boundary" and "area", proposing a variety of different theories. In this talk I will give a brief exposition of the so-called theory of currents, introduced by Federer and Fleming in the 60es after the pioneering work of De Giorgi in the case of hypersurfaces. I will then discuss an open question relating the shapes of the contour and that of the minimizer, posed by Almgren in the early eighties and recently solved in a joint work with Guido de Philippis, Jonas Hirsch and Annalisa Massaccesi.
Camillo De Lellis was born in San Benedetto del Tronto (AP), and received his Ph.D. in Mathematics from the Scuola Normale Superiore at Pisa, in 2002. He is Full Professor of Mathematics at University of Zürich since 2005 and is going to be appointed on July 2018 as Professor of the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey.
Camillo De Lellis is a world-renowned geometric analyst with a broad expertise in fluid dynamics, geometric regularity theory and calculus of variations. Camillo has given a number of remarkable contributions in central problems of these fields, introducing modern tools and innovative approaches that have resulted in new and monumental insights in these areas and have advanced fundamental understanding within the mathematical community. In particular, Camillo has worked on various aspects of the theory of hyperbolic systems of conservation laws and of incompressible fluid dynamics, where, together with Lászlo Székelyhidi, he has introduced the use of Mikhael Gromov's convex integration methods, combined with the embedding theorems of John Nash, to analyse non-uniqueness issues for weak solutions to the Euler equation and the Onsager conjecture (about dissipative solutions of the Euler equation). Camillo, in collaboration with Emanuele Spadaro, has also utilized pioneering techniques, coupled with a novel approach, to craft an accessible proof of the 1.000-page result on partial regularity of minimal surfaces obtained by Frederick J. Almgren, thus providing a concise modern version of the theory that not only strengthened Almgren's original assertion, but also presented many new insights opening new lines to inquiry.
Camillo's groundbreaking achievements in fluid dynamics and in the regularity of minimal surfaces earned him the prestigious Fermat Prize in 2013. His original and transformative work in the field has also resulted in the Stampacchia Gold Medal in 2009; the SIAG/APDE Prize in 2013 (shared with Lászlo Székelyhidi); the Cacciopoli Prize in 2014 and the Amerio Gold Medal Prize in 2015. Camillo De Lellis has been invited speaker at the International Congress of Mathematicians (ICM) at Hyderabad, in 2010, and plenary speaker at the European Congress of Mathematics (ECM) at Krakow, in 2012. He has also been awarded an ERC grant in 2012. Camillo serves in the editorial boards of many leading publications in the field, including: Annals of PDE, the Archive for Rational Mechanics and Analysis, Calculus of Variations and Partial Differential Equations, Inventiones Mathematicae and Journal of Differential Geometry.
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