Scuola Normale Superiore, Pisa
date: January 10-14, 2011

Organizers: Luigi Ambrosio, Valentino Magnani, Stefan Wenger

Lecturers:

Dmitri Burago (Penn. State) ``From asymptotic volume of tori to minimal surfaces
in normed spaces and boundary rigidity, with a few digressions''

Robert Hardt (Rice) ``Rectifiable and flat chains and charges in a
metric space''

Steven Keith TBA

Emanuele Spadaro (Bonn) ``The role of multi-valued functions in the
regularity theory of minimal currents''

Robert Young (New York University) ``Asymptotics of filling problems''

Please register on the home-page of the school at the address:

http://crm.sns.it/hpp/events/event.html?id=181

ABSTRACTS:
Dmitri Burago (Penn. State) ``From asymptotic volume of tori to minimal surfaces in normed spaces and boundary rigidity, with a few digressions''

Robert Hardt (Rice) ``Rectifiable and flat chains and charges in a metric space''

Abstract. Rectifiability and compactness properties for Euclidean-space chains
having coefficients in a finite group G were studied by W.Fleming (1966). This allowed for the modeling of unorientable
least-area surfaces including a minimal Mobius band in 3-space. These properties were optimally extended by Brian White (1999) to any complete normed abelian group which contains no nonconstant Lipschitz curves. Independently L.Ambrosio and B.Kirchheim (2000) also generalized some basic rectifiability theorems of Federer and Fleming to the new notion of currents in a general metric space. Our recent work with T. De Pauw shares features and results with all these works, includes new definitions of a flat G chains in a metric space, and a proof that such a chain is determined by its 0 dimensional slices. Some classes of such chains give homology theories. Related dual cohomology theories involve the charges, introduced by De Paul, Moonens, and Pfeffer, which are dual to normal currents, suitably topologized. We will review all these works.

Steven Keith TBA

Emanuele Spadaro (Bonn) ``The role of multi-valued functions in the regularity theory of minimal currents''

Abstract. Almost 30 years ago Almgren wrote his by now famous "Big regularity paper" on the partial regularity of higher codimension minimizing currents. In this course I present some recent progress in collaboration with C. De Lellis in the direction of a new, simpler derivation of some of Almgren's results, recast in a more manageable framework. I will in particular talk about the theory of multiple valued minimizing functions and the approximation of minimal currents. Finally, time permitting, I will also discuss some of the issues on Almgren's center manifold construction.

Robert Young (New York University) ``Asymptotics of filling problems''

Abstract. Filling problems are an important class of problems in quantitative
geometry. They arise in geometric measure theory and geometric group
theory, but often with different motivations; geometric measure theory
partly arose from problems about the existence and regularity of
minimal surfaces, while geometric group theory uses filling problems
to study the large-scale geometry of a space.

The standard filling problem in geometric group theory is to find the
asymptotic growth of the Dehn function. The Dehn function of a space
is the minimal function $\delta(\ell)$ such that any closed curve of
length $\ell$ is the boundary of a disc of area $\delta(\ell)$, and
its rate of growth can reflect aspects of the geometry of the space,
such as negative or nonpositive curvature.

In this course, we will explore some of the connections between a
geometric measure theory approach to filling problems and a geometric
group theory approach. One of our main tools will be the asymptotic
cone of a space, a way of viewing a space ``from infinity'' which
captures the large-scale geometry of a space. Asymptotic cones often
have complicated geometry; they include Carnot spaces,
$\mathbb{R}$-trees, and subsets of $\mathbb{R}$-buildings, and
geometric measure theory gives us powerful tools to study such strange
spaces.

Topics covered will include:
Asymptotic cones and filling problems in negatively-curved and
non-positively curved spaces --- connections between the asymptotic
rank of a space and its filling problems.
Symmetric spaces and buildings --- applications to arithmetic groups
and the geometry of symmetric spaces.
Nilpotent groups and Carnot spaces --- filling cycles by using
approximations at many scales

http://cvgmt.sns.it/event/191/