ALEX LUBOTSKY, Hebrew University and ETH-Zurich COLLOQUIUM DIPARTIMENTO DI MATEMATICA E APPLICAZIONI, MILANO-BICOCCA con SEMINARIO MATEMATICO E FISICO DI MILANO: "High dimensional expanders: from Ramanujan graphs to Ramanujan complexes" Thursday, May 12 2016, at 15:30 Aula 3014, edificio U5 del Dipartimento di Matematica e Applicazioni, Universita' di Milano - Bicocca, Via R. Cozzi 55 |
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Alessandro Giuliani, Università di Roma 3 Height fluctuations and universality relations in interacting dimer models Wednesday, May 04 2016, at 16:00 Sala di rappresentanza, Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50 |
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Ben Moonen, Universita'di Nijmegen On the Tate and Mumford-Tate conjectures for varieties with h^{2,0}=1 Wednesday, April 06 2016, at 17:00 Sala di rappresentanza, Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50 |
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Enzo Mitidieri, Università degli Studi di Trieste Liouville Theorems in PDE’s: old and new Tuesday, March 15 2016, at 16:30 Sala Consiglio, 7 piano, Dipartimento di Matematica, Via Bonardi 9, Milano |
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Camillo De Lellis, Universitaet Zuerich Regularity and singularity of area-minimizing surfaces Friday, February 26 2016, at 11:00 Sala Consiglio, 7 piano, Dipartimento di Matematica, Via Bonardi 9, Milano |
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Abstract
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The Plateau's problem, named after the Belgian physicist J. Plateau, is a classic in the calculus of variations and regards minimizing the area among all surfaces spanning a given contour. A successful existence theory, that of integral currents, was developed by De Giorgi in the case
of hypersurfaces in the fifties and by Federer and Fleming in the general case in the sixties.
When dealing with hypersurfaces, the minimizers found in this way are rather regular: the corresponding regularity theory has been the achievement of several mathematicians in the 60es,70es and 80es (De Giorgi, Fleming, Almgren, Simons, Bombieri, Giusti, Simon among others).
In codimension higher than one, a phenomenon which is absent for hypersurfaces, namely that of branching, causes very serious problems: a famous theorem of Wirtinger and Federer shows that any holomorphic subvariety in $\mathbb C^n$ is indeed an area-minimizing current. A celebrated monograph of Almgren solved the issue at the beginning of the 80es, proving that the singular set of a general area-minimizing (integral) current has (real) codimension at least 2.
However, his original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry. |
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