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About this book

In 2013, a school on Geometric Measure Theory and Real Analysis, organized by G. Alberti, C. De Lellis and myself, took place at the Centro De Giorgi in Pisa, with lectures by V. Bogachev, R. Monti, E. Spadaro and D. Vittone.

The book collects the notes of the courses. The courses provide a deep and up to date insight on challenging mathematical problems and their recent developments: infinite-dimensional analysis, minimal surfaces and isoperimetric problems in the Heisenberg group, regularity of sub-Riemannian geodesics and the regularity theory of minimal currents in any dimension and codimension.

Table of Contents


Sobolev classes on infinite-dimensional spaces

Sobolev classes of functions of generalized differentiability belong to the major analytic achievements in the XX century and have found impressive applications in the most diverse areas of mathematics. So it does not come as a surprise that their infinite-dimensional analogs attract considerable attention. It was already at the end of the 60s and the beginning of the 70s of the last century that in the works of N. N. Frolov, Yu. L. Daletskiĭ, L. Gross, M. Krée, and P. Malliavin Sobolev classes with respect to Gaussian measures on infinite-dimensional spaces were introduced and studied. Their first triumph came with the development of the Malliavin calculus since the mid of the 70s. At present, such classes and their generalizations have become a standard tool of infinite-dimensional analysis. They find applications in stochastic analysis, optimal transportation, mathematical physics, and mathematical finance.
Vladimir I. Bogachev

Isoperimetric problem and minimal surfaces in the Heisenberg group

The 2n +1-dimensional Heisenberg group is the manifold ℍ n = ℂ n × ℝ, n ∊ ℕ, endowed with the group product
Roberto Monti

Regularity of higher codimension area minimizing integral currents

This lecture notes are an expanded and revised version of the course Regularity of higher codimension area minimizing integral currents that I taught at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013.
The lectures aim to explain partially without proofs the main steps of a new proof of the partial regularity of area minimizing integer rectifiable currents in higher codimension, due originally to F. Almgren, which is contained in a series of papers in collaboration with C. De Lellis (University of Zürich).
Emanuele Spadaro

The regularity problem for sub-Riemannian geodesics

We study the regularity problem for sub-Riemannian geodesics, i.e., for those curves that minimize length among all curves joining two fixed endpoints and whose derivatives are tangent to a given, smooth distribution of planes with constant rank. We review necessary conditions for optimality and we introduce extremals and the Goh condition. The regularity problem is nontrivial due to the presence of the so-called abnormal extremals, i.e., of certain curves that satisfy the necessary conditions and that may develop singularities. We focus, in particular, on the case of Carnot groups and we present a characterization of abnormal extremals, that was recently obtained in collaboration with E. Le Donne, G. P. Leonardi and R. Monti, in terms of horizontal curves contained in certain algebraic varieties. Applications to the problem of geodesics’ regularity are provided.
Davide Vittone


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