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Discrete Morse theory for Filtrations
, 2012
"... The central result of this dissertation is an extension of discrete Morse theory to filtered cell complexes. Discrete Morse theory imposes a pairing on some cells of a complex X and uses that pairing to introduce a new complex M – called the Morse complex – consisting of the unpaired cells of X alon ..."
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Cited by 1 (1 self)
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The central result of this dissertation is an extension of discrete Morse theory to filtered cell complexes. Discrete Morse theory imposes a pairing on some cells of a complex X and uses that pairing to introduce a new complex M – called the Morse complex – consisting of the unpaired cells of X
Equivariant Discrete Morse Theory
, 2007
"... In this paper, we study Forman's discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain theory that can be used to simplify the description of the Ghomotopy type of a simplicial complex. As an application, we det ..."
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Cited by 3 (1 self)
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In this paper, we study Forman's discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain theory that can be used to simplify the description of the Ghomotopy type of a simplicial complex. As an application, we
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 233 (8 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
Discrete Morse Theory for Manifolds with Boundary
, 2012
"... We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities ” relating the homology of the manifold to the number of interior critical cells. We also ..."
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Cited by 14 (8 self)
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We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities ” relating the homology of the manifold to the number of interior critical cells. We also
Discrete Morse Theory for Cellular Resolutions
 J. Reine Angew. Math
, 2000
"... We develop an analog of Forman's discrete Morse theory for cell complexes in the setting of cellular resolutions of multigraded monomial modules. In particular, using discrete Morse theory for cellular resolutions of multigraded ideals we are able to give minimal cellular resolutions for gen ..."
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Cited by 34 (2 self)
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We develop an analog of Forman's discrete Morse theory for cell complexes in the setting of cellular resolutions of multigraded monomial modules. In particular, using discrete Morse theory for cellular resolutions of multigraded ideals we are able to give minimal cellular resolutions
Discrete Morse Theory and the Cohomology Ring
 Transactions of the American Mathematical Society
, 2002
"... Abstract. In Morse Theory for Cell Complexes, we presented a discrete Morse theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built ..."
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Cited by 6 (0 self)
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Abstract. In Morse Theory for Cell Complexes, we presented a discrete Morse theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built
Parameterized complexity of discrete Morse theory
 SCG ’13: Proceedings of the 29th Annual Symposium on Computational Geometry, ACM
"... Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3manifolds. However, such matchings are known to be NPhard to compute on 3manifolds, through a reduction to the erasability problem. Her ..."
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Cited by 2 (2 self)
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. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand we prove that the erasability problem is W [P]complete on the natural parameter. On the other hand we propose an algorithm for computing optimal Morse
Towards optimality in discrete Morse theory
 Experimental Mathematics
, 2003
"... Morse theory is a fundamental tool for investigating the topology of smooth manifolds. This tool has been extended to discrete structures by Forman, which allows combinatorial analysis and direct computation. This theory relies on discrete gradient vector fields, whose critical elements describe the ..."
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Cited by 7 (3 self)
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Morse theory is a fundamental tool for investigating the topology of smooth manifolds. This tool has been extended to discrete structures by Forman, which allows combinatorial analysis and direct computation. This theory relies on discrete gradient vector fields, whose critical elements describe
BIRTH AND DEATH IN DISCRETE MORSE THEORY
, 808
"... Abstract. Suppose M is a finite simplicial complex and that for 0 = t0, t1,..., tr = 1 we have a discrete Morse function Fti: M → R. In this paper, we study the births and deaths of critical cells for the functions Fti and present an algorithm for pairing the cells that occur in adjacent slices. We ..."
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Cited by 2 (0 self)
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Abstract. Suppose M is a finite simplicial complex and that for 0 = t0, t1,..., tr = 1 we have a discrete Morse function Fti: M → R. In this paper, we study the births and deaths of critical cells for the functions Fti and present an algorithm for pairing the cells that occur in adjacent slices. We
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