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24
NONSMOOTH CALCULUS
"... Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singu ..."
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Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.
The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity
 ICM Proceedings
, 2006
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GLOBAL CONFORMAL ASSOUAD DIMENSION IN THE HEISENBERG GROUP
"... Abstract. We study global conformal Assouad dimension in the Heisenberg group H n. For each α ∈ {0} ∪ [1, 2n + 2], there is a bounded set in H n with Assouad dimension α whose Assouad dimension cannot be lowered by any quasiconformal map of H n. On the other hand, for any set S in H n with Assouad ..."
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Abstract. We study global conformal Assouad dimension in the Heisenberg group H n. For each α ∈ {0} ∪ [1, 2n + 2], there is a bounded set in H n with Assouad dimension α whose Assouad dimension cannot be lowered by any quasiconformal map of H n. On the other hand, for any set S in H n with Assouad dimension strictly less than one, the infimum of the Assouad dimensions of sets F (S), taken over all quasiconformal maps F of H n, equals zero. We also consider dilatationdependent bounds for quasiconformal distortion of Assouad dimension. The proofs use recent advances in selfsimilar fractal geometry and tilings in H n and regularity of the CarnotCarathéodory distance from smooth hypersurfaces. 1.
Quasisymmetric structures on surfaces
"... We show that a locally Ahlfors 2regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces embedded in some Euclidean space that are locally biLipschi ..."
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We show that a locally Ahlfors 2regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces embedded in some Euclidean space that are locally biLipschitz equivalent to a ball in the plane.
SURVEY ON GEOMETRIC GROUP THEORY
, 2008
"... This article is a survey article on geometric group theory from the point of view of a nonexpert who likes geometric group theory and uses it in his own research. ..."
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This article is a survey article on geometric group theory from the point of view of a nonexpert who likes geometric group theory and uses it in his own research.
ON THE CONFORMAL GAUGE OF A COMPACT METRIC SPACE
, 2012
"... Abstract. In this article we study the Ahlfors regular conformal gauge of a compact metric space (X, d), and its conformal dimension dimAR(X, d). Using a sequence of finite coverings of (X, d), we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain ..."
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Abstract. In this article we study the Ahlfors regular conformal gauge of a compact metric space (X, d), and its conformal dimension dimAR(X, d). Using a sequence of finite coverings of (X, d), we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to biLipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute dimAR(X, d) using the critical exponent QN associated to the combinatorial modulus.