636 research outputs found
Invertible harmonic mappings, beyond Kneser
We prove necessary and sufficient criteria of invertibility for planar
harmonic mappings which generalize a classical result of H. Kneser, also known
as the Rad\'{o}-Kneser-Choquet theorem.Comment: One section added. 15 page
Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds
We give a sufficient condition for a metric (homology) manifold to be locally
bi-Lipschitz equivalent to an open subset in \rn. The condition is a Sobolev
condition for a measurable coframe of flat 1-forms. In combination with an
earlier work of D. Sullivan, our methods also yield an analytic
characterization for smoothability of a Lipschitz manifold in terms of a
Sobolev regularity for frames in a cotangent structure. In the proofs, we
exploit the duality between flat chains and flat forms, and recently
established differential analysis on metric measure spaces. When specialized to
\rn, our result gives a kind of asymptotic and Lipschitz version of the
measurable Riemann mapping theorem as suggested by Sullivan
Should we solve Plateau's problem again?
After a short description of various classical solutions of Plateau's
problem, we discuss other ways to model soap films, and some of the related
questions that are left open. A little more attention is payed to a more
specific model, with deformations and sliding boundary conditions.Comment: Lecture for the conference in Honor of E. Stein, 201
Real Analysis, Quantitative Topology, and Geometric Complexity
Contents
1 Mappings and distortion
2 The mathematics of good behavior much of the time, and the BMO frame of
mind
3 Finite polyhedra and combinatorial parameterization problems
4 Quantitative topology, and calculus on singular spaces
5 Uniform rectifiability
Appendices
A Fourier transform calculations
B Mappings with branching
C More on existence and behavior of homeomorphisms
D Doing pretty well with spaces which may not have nice coordinates
E Some simple facts related to homologyComment: 161 pages, Latex2
Gauge Invariant Framework for Shape Analysis of Surfaces
This paper describes a novel framework for computing geodesic paths in shape
spaces of spherical surfaces under an elastic Riemannian metric. The novelty
lies in defining this Riemannian metric directly on the quotient (shape) space,
rather than inheriting it from pre-shape space, and using it to formulate a
path energy that measures only the normal components of velocities along the
path. In other words, this paper defines and solves for geodesics directly on
the shape space and avoids complications resulting from the quotient operation.
This comprehensive framework is invariant to arbitrary parameterizations of
surfaces along paths, a phenomenon termed as gauge invariance. Additionally,
this paper makes a link between different elastic metrics used in the computer
science literature on one hand, and the mathematical literature on the other
hand, and provides a geometrical interpretation of the terms involved. Examples
using real and simulated 3D objects are provided to help illustrate the main
ideas.Comment: 15 pages, 11 Figures, to appear in IEEE Transactions on Pattern
Analysis and Machine Intelligence in a better resolutio
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Mini-Workshop: Topology of Real Singularities and Motivic Aspects
This workgroup focusses on some recent issues in real singularities, concerning the topology of the Milnor fibre of a singular map and several motivic aspects of singularities of sets definable in some structures over the reals or even over some valued field, with the ambition to develop the interplay between the two domains
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