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117
A discrete Morse theory for cell complexes
 in ‘‘Geometry, Topology 6 Physics for Raoul Bott
, 1995
"... In this paper we will present a very simple discrete Morse theory for CW complexes. In addition to proving analogues of the main theorems of Morse theory, we also present discrete analogues of such (seemingly) intrinsically smooth notions as the gradient vector field and the gradient ..."
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Cited by 241 (9 self)
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In this paper we will present a very simple discrete Morse theory for CW complexes. In addition to proving analogues of the main theorems of Morse theory, we also present discrete analogues of such (seemingly) intrinsically smooth notions as the gradient vector field and the gradient
Topologybased Simplification for Feature Extraction from 3D Scalar Fields
"... This paper describes a topological approach for simplifying continuous functions defined on volumetric domains. The MorseSmale complex provides a segmentation of the domain into monotonic regions having uniform gradient flow behavior. We present a combinatorial algorithm that simplifies the Morse ..."
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Cited by 46 (21 self)
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This paper describes a topological approach for simplifying continuous functions defined on volumetric domains. The MorseSmale complex provides a segmentation of the domain into monotonic regions having uniform gradient flow behavior. We present a combinatorial algorithm that simplifies the MorseSmale complex by repeated application of two atomic operations that removes pairs of critical points. The simplification procedure leaves important critical points untouched, and is therefore useful for extracting features. We present a visualization of the simplified topology.
A Practical Approach to MorseSmale Complex Computation: Scalability and Generality
"... Abstract—The MorseSmale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalarvalued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for co ..."
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Cited by 36 (9 self)
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Abstract—The MorseSmale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalarvalued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for computing MS complexes for large scale data of any dimension where scalar values are given at the vertices of a closurefinite and weak topology (CW) complex, therefore enabling computation on a wide variety of meshes such as regular grids, simplicial meshes, and adaptive multiresolution (AMR) meshes. A new divideandconquer strategy allows for memoryefficient computation of the MS complex and simplification onthefly to control the size of the output. In addition to being able to handle various data formats, the framework supports implementationspecific optimizations, for example, for regular data. We present the complete characterization of critical point cancellations in all dimensions. This technique enables the topology based analysis of large data on offtheshelf computers. In particular we demonstrate the first full computation of the MS complex for a 1 billion/1024 3 node grid on a laptop computer with 2Gb memory. Index Terms—Topologybased analysis, MorseSmale complex, large scale data. 1
Efficient Computation of MorseSmale Complexes for ThreeDimensional Scalar Functions
, 2007
"... The MorseSmale complex is an efficient representation of the gradient behavior of a scalar function, and critical points paired by the complex identify topological features and their importance. We present an algorithm that constructs the MorseSmale complex in a series of sweeps through the data, ..."
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Cited by 32 (14 self)
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The MorseSmale complex is an efficient representation of the gradient behavior of a scalar function, and critical points paired by the complex identify topological features and their importance. We present an algorithm that constructs the MorseSmale complex in a series of sweeps through the data, identifying various components of the complex in a consistent manner. All components of the complex, both geometric and topological, are computed, providing a complete decomposition of the domain. Efficiency is maintained by representing the geometry of the complex in terms of point sets.
A topological approach to simplification of threedimensional scalar functions
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS (SPECIAL ISSUE IEEE VISUALIZATION
, 2006
"... This paper describes an efficient combinatorial method for simplification of topological features in a 3D scalar function. The MorseSmale complex, which provides a succinct representation of a function’s associated gradient flow field, is used to identify topological features and their significance ..."
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Cited by 28 (13 self)
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This paper describes an efficient combinatorial method for simplification of topological features in a 3D scalar function. The MorseSmale complex, which provides a succinct representation of a function’s associated gradient flow field, is used to identify topological features and their significance. The simplification process, guided by the MorseSmale complex, proceeds by repeatedly applying two atomic operations that each remove a pair of critical points from the complex. Efficient storage of the complex results in execution of these atomic operations at interactive rates. Visualization of the simplified complex shows that the simplification preserves significant topological features and removes small features and noise.
Sasakian geometry, hypersurface singularities, and Einstein
, 2005
"... This review article has grown out of notes for the three lectures the second author presented during the XXIVth Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed ov ..."
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Cited by 21 (3 self)
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This review article has grown out of notes for the three lectures the second author presented during the XXIVth Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed over the last 5 years
The comparison geometry of Ricci curvature, in Comparison geometry
 Fakultät für Mathematik, Universität
, 1997
"... Abstract. We survey comparison results that assume a bound on the manifold’s Ricci curvature. 1. ..."
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Cited by 18 (0 self)
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Abstract. We survey comparison results that assume a bound on the manifold’s Ricci curvature. 1.
The local linearization problem for smooth SL(n)actions
 Enseign. Math
, 1997
"... Abstract. This paper considers SL(n, R)actions on Euclidean space fixing the origin. We show that all C 1actions on R n are linearizable. We give C ∞actions of SL(2, R) on R 3 and of SL(3, R) on R 8 which are not linearizable. We classify the C 0actions of SL(n, R) on R n. Finally, the paper con ..."
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Cited by 16 (0 self)
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Abstract. This paper considers SL(n, R)actions on Euclidean space fixing the origin. We show that all C 1actions on R n are linearizable. We give C ∞actions of SL(2, R) on R 3 and of SL(3, R) on R 8 which are not linearizable. We classify the C 0actions of SL(n, R) on R n. Finally, the paper concludes with a study of the linearizability of SL(n, Z)actions. RÉSUMÉ. Dans cet article, on considère les actions de SL(n, R) sur l’espace euclidien qui fixent l’origine. On montre que les actions C1 sur Rn sont linéarisables. On donne des actions C ∞ de SL(2, R) sur R3 et de SL(3, R) sur R8 qui ne sont pas linéarisables. On classifie les actions C0 de SL(n, R) sur Rn. L’article s’achève par une étude de la linéarisabilité des actions de SL(n, Z). 1.
Homotopically equivalent smooth manifolds
 I, Izv. Akad. Nauk SSSR Ser. Mat
, 1964
"... In this paper we introduce a method for the investigation of smooth simply connected manifolds of dimension n ≥ 5 that permits a classification of them with exactness up to orientationpreserving diffeomorphisms. This method involves a detailed investigation of the properties of the socalled Thom c ..."
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Cited by 16 (3 self)
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In this paper we introduce a method for the investigation of smooth simply connected manifolds of dimension n ≥ 5 that permits a classification of them with exactness up to orientationpreserving diffeomorphisms. This method involves a detailed investigation of the properties of the socalled Thom complexes of normal bundles and is based on a theorem of Smale concerning the equivalence of the concepts of “hcobordism ” and “orientationpreserving diffeomorphism. ” In the last chapter we work out some simple examples. Appendices are given in which the results of this paper are applied to certain other problems.