[CvGmt News] Course Announcement

[depascal at dm.unipi.it]

ERC School on Analysis in Metric Spaces and Geometric Measure Theory
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ABSTRACTS ARE NOW AVAILABLE!!!

   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/