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235
Integrability of Lie brackets
 Ann. of Math
"... In this paper we present the solution to a longstanding problem of differential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we es ..."
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Cited by 153 (19 self)
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In this paper we present the solution to a longstanding problem of differential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigmamodel for Poisson manifolds, and we describe other geometrical applications.
Computable elastic distances between shapes
 SIAM J. of Applied Math
, 1998
"... Abstract. We define distances between geometric curves by the square root of the minimal energy required to transform one curve into the other. The energy is formally defined from a left invariant Riemannian distance on an infinite dimensional group acting on the curves, which can be explicitly comp ..."
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Cited by 146 (19 self)
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Abstract. We define distances between geometric curves by the square root of the minimal energy required to transform one curve into the other. The energy is formally defined from a left invariant Riemannian distance on an infinite dimensional group acting on the curves, which can be explicitly computed. The obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed. An analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem, which can efficiently be implemented, as illustrated by experiments.
Hamiltonian torus actions on symplectic orbifolds and toric varieties
, 1995
"... Abstract. In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with co ..."
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Cited by 125 (7 self)
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Abstract. In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each facet and that all such orbifolds are algebraic toric varieties. Contents
Multiscale representations for manifoldvalued data
 SIAM J. MULTISCALE MODEL. SIMUL
, 2005
"... We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere S 2, the special orthogonal group SO(3), the positive definite matrices SPD(n), and the Grassmann manifolds G(n, k). The representations are based on the deployment of Desl ..."
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Cited by 75 (4 self)
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We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere S 2, the special orthogonal group SO(3), the positive definite matrices SPD(n), and the Grassmann manifolds G(n, k). The representations are based on the deployment of Deslauriers–Dubuc and averageinterpolating pyramids “in the tangent plane” of such manifolds, using the Exp and Log maps of those manifolds. The representations provide “wavelet coefficients ” which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as S n−1, SO(n), G(n, k), where the Exp and Log maps are effectively computable. Applications to manifoldvalued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper.
Richardson: Cohomology and deformations in graded Lie algebras
 Bull. Amer. Math. Soc
, 1966
"... authors gave an outline of the similarities between the deformations of complexanalytic structures on compact manifolds on one hand, and the deformations of associative algebras on the other. The first theory had been stimulated in 1957 by a paper [7] by Nijenhuis ..."
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Cited by 75 (0 self)
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authors gave an outline of the similarities between the deformations of complexanalytic structures on compact manifolds on one hand, and the deformations of associative algebras on the other. The first theory had been stimulated in 1957 by a paper [7] by Nijenhuis
FeigenbaumCoulletTresser universality and Milnor's Hairiness Conjecture
, 1999
"... We prove the FeigenbaumCoulletTresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computerfree proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor’s c ..."
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Cited by 72 (8 self)
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We prove the FeigenbaumCoulletTresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computerfree proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor’s conjectures on selfsimilarity and “hairiness ” of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N> 1 has Hausdorff dimension strictly between 0 and 1. In the course of getting these results we supply the space of quadraticlike germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimensionone foliation of the connectedness locus.
ALMOST EVERY REAL QUADRATIC MAP IS EITHER REGULAR OR STOCHASTIC
, 1997
"... We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family Pc: x ↦ → x² + c has zero measure. This yields the statement in the title (where “ regul ..."
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Cited by 53 (4 self)
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We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family Pc: x ↦ → x² + c has zero measure. This yields the statement in the title (where “ regular ” means to have an attracting cycle and “stochastic” means to have an absolutely continuous invariant measure). An application to the MLC problem is given.
Optimal Matching Between Shapes Via Elastic Deformations
, 1999
"... We describe an elasting matching procedure between plane curves based on computing a minimal deformation cost between the curves. The design of the deformation cost is based on a geodesic distance defined on an infinite dimensional group acting on the curves. The geodesic paths also provide an optim ..."
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Cited by 47 (10 self)
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We describe an elasting matching procedure between plane curves based on computing a minimal deformation cost between the curves. The design of the deformation cost is based on a geodesic distance defined on an infinite dimensional group acting on the curves. The geodesic paths also provide an optimal deformation process, which allows to interpolate between any plane curves. Key words: Time warping. Elastic matching. Infinite dimensional group action. Geodesic distance. 1 Introduction Deformable shapes arise naturally in image analysis, especially for applications in medical or biological imaging. To detect, recognize, or simply organize such shapes (for example in a database), there is an obvious need for comparison tools. In this paper, we address the problem of comparing and matching silhouettes (ie. plane curves) with the help of a variational approach on an infinite dimensional deformation group acting on them. When comparing nonrigid objects, there cannot be any natural notion o...
Einstein metrics with prescribed conformal infinity on 4manifolds
, 2008
"... This paper considers the existence of conformally compact Einstein metrics on 4manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that gen ..."
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Cited by 41 (13 self)
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This paper considers the existence of conformally compact Einstein metrics on 4manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability depends on the topology of the filling manifold. The obstruction to extending these results to arbitrary boundary values is also identified. While most of the paper concerns dimension 4, some general results on the structure of the space of such metrics hold in all dimensions.