"The Plateau's problem consists in finding the surface of least area spanning a given contour. This question has attracted the attention of many mathematicians in the last two centuries, providing a prototypical problem for several fields of research in mathematics. For hypersurfaces a lot is known about the existence and regularity thanks to the classical works of De Giorgi, Almgren, Fleming, Federer, Simons, Allard, Simon, Schoen and several other authors.
In higher codimension a quite powerful existence theory, the ``theory of currents'', was developed by Federer and Fleming in 1960. The success of this theory relies on its homological flavor and indeed it has found several applications to problems in differential geometry. Many geometric objects which are widely studied in the modern literature are naturally area-minimizing currents: two examples among many are special lagrangians and holomorphic subvarieties. However the understanding of the regularity issues is, compared to the case of hypersurfaces, much poorer. Aside from its intrinsic interest, a good regularity theory is likely to provide more insightful geometric applications. A quite striking example is Taubes' proof of the equivalence between the Gromov and Seiberg-Witten invariants.
A very complicated and far reaching regularity theory has been developed by Almgren thirty years ago in a monumental work of almost 1000 pages. The first part of this project aims at reaching the same conclusions of Almgren with a more flexible and accessible theory. In the second part I wish to go beyond Almgren's work and attack some of the many open questions which still remain in the field."
Call for proposal
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