[CvGmt News] Course Announcement
depascal at dm.unipi.it
depascal at dm.unipi.it
Sun Jan 2 14:54:52 CET 2011
ERC School on Analysis in Metric Spaces and Geometric Measure Theory
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ERC School on Analysis in Metric Spaces and Geometric Measure Theory
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 d(l) such that any closed curve of length l is the
boundary of a disc of area d(l), 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, \mathbbR-trees,
and subsets of \mathbbR-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
Reference: http://crm.sns.it/hpp/events/event.html?id=181
You find this news in the cvgmt preprint server: http://cvgmt.sns.it/news/20110110/
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