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82
Volume and shape preservation via moving frame manipulation
 ACM Trans. Graph
, 2007
"... This paper introduces a method for mesh editing, aimed at preserving shape and volume. We present two new developments: the first is a minimization of a functional expressing a geometric distance measure between two isometric surfaces. The second is a local volume analysis linking the volume of an o ..."
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Cited by 28 (2 self)
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This paper introduces a method for mesh editing, aimed at preserving shape and volume. We present two new developments: the first is a minimization of a functional expressing a geometric distance measure between two isometric surfaces. The second is a local volume analysis linking the volume of an object to its surface curvature. Our method is based upon the moving frames representation of meshes. Applying a rotation field to the moving frames defines an isometry. Given rotational constraints, the mesh is deformed by an optimal isometry defined by minimizing the distance measure between the original and the deformed meshes. The resulting isometry nicely preserves the surface details, but, when large rotations are applied, the volumetric behavior of the model may be unsatisfactory. Using the local volume analysis, we define a scalar field by which we scale the moving frames. The scaled and rotated moving frames restore the volumetric properties of the original mesh, while properly maintaining the surface details. Our results show that even extreme deformations can be applied to meshes, with only minimal distortion of surface details and object volume.
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 22 (3 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Differential Invariants of Conformal and Projective Surfaces
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2007
"... We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in threedimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames. ..."
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Cited by 15 (13 self)
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We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in threedimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.
The border rank of the multiplication of 2 × 2 matrices is seven
 J. Amer. Math. Soc
"... One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, ..."
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Cited by 11 (3 self)
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One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and
Classical and quantum integrability of 2D dilaton gravities in Euclidean space
"... Euclidean dilaton gravity in two dimensions is studied exploiting its representation as a complexified first order gravity model. All local classical solutions are obtained. A global discussion reveals that for a given model only a restricted class of topologies is consistent with the metric and the ..."
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Cited by 9 (8 self)
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Euclidean dilaton gravity in two dimensions is studied exploiting its representation as a complexified first order gravity model. All local classical solutions are obtained. A global discussion reveals that for a given model only a restricted class of topologies is consistent with the metric and the dilaton. A particular case of string motivated Liouville gravity is studied in detail. Path integral quantisation in generic Euclidean dilaton gravity is performed nonperturbatively by analogy to the Minkowskian case.
On Legendrian varieties
"... Abstract. We investigate the geometry of Legendrian complex projective manifolds X ⊂ PV. By definition, this means V is a complex vector space of dimension 2n + 2, endowed with a symplectic form, and the affine tangent space to X at each point is a maximal isotropic subspace. We establish basic fact ..."
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Cited by 8 (2 self)
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Abstract. We investigate the geometry of Legendrian complex projective manifolds X ⊂ PV. By definition, this means V is a complex vector space of dimension 2n + 2, endowed with a symplectic form, and the affine tangent space to X at each point is a maximal isotropic subspace. We establish basic facts about their geometry and exhibit examples of inhomogeneous smooth Legendrian varieties, the first examples of such in dimension greater than one. 1.
GriffithsHarris Rigidity of Compact Hermitian Symmetric Spaces
"... I prove that any complex manifold that has a projective second fundamental form isomorphic to one of a rank two compact Hermitian symmetric space (other than a quadric hypersurface) at a general point must be an open subset of such a space. This contrasts the nonrigidity of all other compact Hermit ..."
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Cited by 7 (3 self)
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I prove that any complex manifold that has a projective second fundamental form isomorphic to one of a rank two compact Hermitian symmetric space (other than a quadric hypersurface) at a general point must be an open subset of such a space. This contrasts the nonrigidity of all other compact Hermitian symmetric spaces observed in [12, 13]. A key step is the use of higher order Bertini type theorems that may be of interest in their own right.
On tangential varieties of rational homogeneous varieties
 Jour. Lond. Math. Soc
"... Abstract. We determine which tangential varieties of homogeneously embedded rational homogeneous varieties are spherical. We determine the homogeneous coordinate rings and rings of covariants of the tangential varieties of homogenously embedded compact Hermitian symmetric spaces (CHSS). We give boun ..."
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Cited by 6 (1 self)
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Abstract. We determine which tangential varieties of homogeneously embedded rational homogeneous varieties are spherical. We determine the homogeneous coordinate rings and rings of covariants of the tangential varieties of homogenously embedded compact Hermitian symmetric spaces (CHSS). We give bounds on the degrees of generators of the ideals of tangential varieties of CHSS and obtain more explicit infomation about the ideals in certain cases. 1.