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204
Computing Persistent Homology
 Discrete Comput. Geom
"... We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enabl ..."
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Cited by 101 (20 self)
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We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This results generalizes and extends the previously known algorithm that was restricted to subcomplexes of S and Z2 coefficients. Finally, our study implies the lack of a simple classification over nonfields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.
The type of the classifying space for a family of subgroups
 J. Pure Appl. Algebra
"... We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the GCWversion EF(G) and the numerable Gspace version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact su ..."
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Cited by 55 (28 self)
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We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the GCWversion EF(G) and the numerable Gspace version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra, for the FarrellJones Conjecture about the algebraic Kand Ltheory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.
Phase Transitions on Nonamenable Graphs
, 2000
"... We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees. ..."
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Cited by 49 (8 self)
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We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees.
Bounded cohomology of subgroups of mapping class groups
 Geom. Topol. 6 (2002) 69–89. MR1914565 (2003f:57003), Zbl 1021.57001
"... groups ..."
Asymptotic geometry of the mapping class group and Teichmüller space
 GEOMETRY & TOPOLOGY
, 2006
"... In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of “projection” maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. ..."
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Cited by 33 (6 self)
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In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of “projection” maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the mapping class group of any surface is treegraded in the sense of Dru¸tu and Sapir; this treegrading has several consequences including answering a question of Drutu and Sapir concerning relatively hyperbolic groups. Another application is a generalization of the result of Brock and Farb that for low complexity surfaces Teichmüller space, with the Weil–Petersson metric, is ı–hyperbolic. Although for higher complexity surfaces these spaces are not ı–hyperbolic, we establish the presence of previously unknown negative curvature phenomena in the mapping class group and Teichmüller space for arbitrary surfaces.
Homological sensor networks
 Notices of the American Mathematical Society
, 2007
"... A sensor is a device that measures some feature of a domain or environment and returns a signal from which information may be extracted. Sensors vary in scope, resolution, and ability. The information they return can be as simple as a binary flag, as with a metal detector that beeps to indicate a de ..."
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Cited by 33 (0 self)
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A sensor is a device that measures some feature of a domain or environment and returns a signal from which information may be extracted. Sensors vary in scope, resolution, and ability. The information they return can be as simple as a binary flag, as with a metal detector that beeps to indicate a detection threshold being crossed. A more complex sensor, such as a video camera, can return a signal requiring sophisticated analysis to extract relevant data. An increasingly common application for sensors is to scan a region for a particular object or substance. For example, one might wish to determine the existence and location of an outbreak of fire in a national forest. Questions of more interest to national security involve detection of radiological or biological hazards, hidden mines and munitions, or specific individuals in a crowd. All of these scenarios pose difficult and challenging data management problems. Numerous strategies exist, aided by the fact that sensor technology provides an expansive array of available hardware. A fundamental dichotomy exists in the approach to sensing an environment based on the number and complexity of sensors. For a fixed cost (monetary, or perhaps “total complexity”), one can deploy a small number of sophisticated “global ” sensors with high signal complexity and precise readings. In contrast, one can deploy a large number of small, coarse, “local ” devices that may
Bounds on exceptional Dehn filling
 Geom. Topol
"... Abstract. We show that for a hyperbolic knot complement, there are at most 12 Dehn fillings which are not irreducible with infinite wordhyperbolic fundamental group. 1. ..."
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Cited by 31 (1 self)
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Abstract. We show that for a hyperbolic knot complement, there are at most 12 Dehn fillings which are not irreducible with infinite wordhyperbolic fundamental group. 1.
Cannon–Thurston Maps for Trees of Hyperbolic Metric Spaces
 J. Differential Geometry
, 1998
"... Abstract. Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasiisometrically embedded condition. Let v be a vertex of T. Let (Xv, dv) denote the hyperbolic metric space corresponding to v. Then i: Xv → X extends continuously to a map î: ̂Xv → ̂X. This generalizes a Theorem of Can ..."
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Cited by 30 (1 self)
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Abstract. Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasiisometrically embedded condition. Let v be a vertex of T. Let (Xv, dv) denote the hyperbolic metric space corresponding to v. Then i: Xv → X extends continuously to a map î: ̂Xv → ̂X. This generalizes a Theorem of Cannon and Thurston. The techniques are used to give a new proof of a result of Minsky: Thurston’s ending lamination conjecture for certain Kleinian groups. Applications to graphs of hyperbolic groups and local connectivity of limit sets of Kleinian groups are also given. 1.
Thick metric spaces, relative hyperbolicity, and quasiisometric rigidity
, 2005
"... Abstract. We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasiisometric image of a nonrelatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group bei ..."
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Cited by 30 (9 self)
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Abstract. We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasiisometric image of a nonrelatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasiisometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasiisometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be nonhyperbolic relative to any nontrivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Nonuniform lattices in higher rank semisimple Lie groups are thick and hence nonrelatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples