The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
William P. Thurston - One of the best experts on this subject based on the ideXlab platform.
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Finite covers of random 3-manifolds
Inventiones mathematicae, 2006Co-Authors: Nathan M. Dunfield, William P. ThurstonAbstract:A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π_1(Σ)→ Q . If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.
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Finite covers of random 3-manifolds
Inventiones Mathematicae, 2006Co-Authors: Nathan M. Dunfield, William P. ThurstonAbstract:A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show that if the genus of S is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman's theorem that the action of the mapping class group on the SU(2) character variety is ergodic.
G. Elber - One of the best experts on this subject based on the ideXlab platform.
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Contouring 1- and 2-Manifolds in arbitrary dimensions
International Conference on Shape Modeling and Applications 2005 (SMI' 05), 2005Co-Authors: Joon-kyung Seong, G. ElberAbstract:We propose an algorithm for contouring k-manifolds (k = 1,2) embedded in an arbitrary n-dimensional space. We assume (n -k) geometric constraints are represented as polynomial equations in n variables. The common zero-set of these (n-k) equations is computed as an 1-or 2-manifold, respectively, for k = 1 or k = 2. In the case of 1-manifolds, this framework is a generalization of techniques for contouring regular intersection curves between two implicitly-defined surfaces of the form F(x,y,z) = G(x,y,z) = 0. Moreover, in the case of 2-Manifolds, the algorithm is similar to techniques for contouring iso-surfaces of the form F(x, y, z) = 0, where n = 3 and only one (=3 -2) constraint is provided. By extending the Dual Contouring technique to higher dimensions, we approximate the simultaneous zero-set as a piecewise linear 1or 2-manifold. There are numerous applications for this technique in data visualization and modeling, including the processing of various geometric constraints for freeform objects, and the computation of convex hulls, bisectors, blendings and sweeps.
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SMI - Contouring 1- and 2-Manifolds in arbitrary dimensions
International Conference on Shape Modeling and Applications 2005 (SMI' 05), 2005Co-Authors: Joon-kyung Seong, G. ElberAbstract:We propose an algorithm for contouring k-manifolds (k = 1,2) embedded in an arbitrary n-dimensional space. We assume (n -k) geometric constraints are represented as polynomial equations in n variables. The common zero-set of these (n-k) equations is computed as an 1-or 2-manifold, respectively, for k = 1 or k = 2. In the case of 1-manifolds, this framework is a generalization of techniques for contouring regular intersection curves between two implicitly-defined surfaces of the form F(x,y,z) = G(x,y,z) = 0. Moreover, in the case of 2-Manifolds, the algorithm is similar to techniques for contouring iso-surfaces of the form F(x, y, z) = 0, where n = 3 and only one (=3 -2) constraint is provided. By extending the Dual Contouring technique to higher dimensions, we approximate the simultaneous zero-set as a piecewise linear 1or 2-manifold. There are numerous applications for this technique in data visualization and modeling, including the processing of various geometric constraints for freeform objects, and the computation of convex hulls, bisectors, blendings and sweeps.
Nathan M. Dunfield - One of the best experts on this subject based on the ideXlab platform.
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Finite covers of random 3-manifolds
Inventiones mathematicae, 2006Co-Authors: Nathan M. Dunfield, William P. ThurstonAbstract:A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π_1(Σ)→ Q . If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.
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Finite covers of random 3-manifolds
Inventiones Mathematicae, 2006Co-Authors: Nathan M. Dunfield, William P. ThurstonAbstract:A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show that if the genus of S is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman's theorem that the action of the mapping class group on the SU(2) character variety is ergodic.
Xiaochun Rong - One of the best experts on this subject based on the ideXlab platform.
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homeomorphism classification of positively curved manifolds with almost maximal symmetry rank
Mathematische Annalen, 2005Co-Authors: Xiaochun Rong, Fuquan FangAbstract:We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m≥5) of positive sectional curvature on which an (m−1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By [Wi], these results imply a homeomorphism classification for positively curved n-manifolds (n≥8) of almost maximal symmetry rank Open image in new window
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homeomorphism classification of positively curved manifolds with almost maximal symmetry rank
arXiv: Differential Geometry, 2003Co-Authors: Xiaochun Rong, Fuquan FangAbstract:We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m>4) of positive sectional curvature on which a (m-1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By a result of Wilking, these results imply a homeomorphism classification for positively curved n-manifolds (n>7) of almost maximal symmetry rank [\frac{n-1}2].
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positively curved manifolds with almost maximal symmetry rank
Geometriae Dedicata, 2002Co-Authors: Xiaochun RongAbstract:The symmetry rank of a Riemannian manifold is the rank of the isometry group. We determine precisely which closed simply connected 5-manifolds admit positively curved metrics with (almost maximal) symmetry rank two. We also determine the precise Euler characteristic and the fundamental groups of all closed positively curved n-manifolds with almost maximal symmetry rank [(n−1)/2] (n≠ 6, 7).
Colin Guillarmou - One of the best experts on this subject based on the ideXlab platform.
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lens rigidity for manifolds with hyperbolic trapped set
Journal of the American Mathematical Society, 2016Co-Authors: Colin GuillarmouAbstract:For a Riemannian manifold (M, g) with strictly convex boundary ∂M , the lens data consists in the set of lengths of geodesics γ with endpoints on ∂M , together with their endpoints (x − , x +) ∈ ∂M × ∂M and tangent exit vectors (v − , v +) ∈ T x− M × T x+ M. We show deformation lens rigidity for manifolds with hyperbolic trapped set and no conjugate points, a class which contains all manifolds with negative curvature and strictly convex boundary, including those with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension 2, we prove that the set of endpoints and exit vectors of geodesics (ie. the scattering data) determines the Riemann surface up to conformal diffeomorphism.
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lens rigidity for manifolds with hyperbolic trapped set
arXiv: Analysis of PDEs, 2014Co-Authors: Colin GuillarmouAbstract:For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consists in the set of lengths of geodesics $\gamma$ with endpoints on $\partial M$, together with their endpoints $(x_-,x_+)\in \partial M\times \partial M$ and tangent exit vectors $(v_-,v_+)\in T_{x_-} M\times T_{x_+} M$. We show deformation lens rigidity for a large class of manifolds which includes all manifolds with negative curvature and strictly convex boundary, possibly with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension $2$, we prove that the set of endpoints and exit vectors of geodesics (ie. the scattering data) determines the topology and the conformal class of the surface.