Efficient algorithms for computing Reeb graphs

The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(nlogn+nlogg(loglogg)3) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O(nlogn(loglogn)3) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O(n2) algorithm. Finally, we present experimental results of our implementation and demonstrate that our algorithm for 3-manifolds performs efficiently in practice.

Combinatorial manifolds with complementarity

A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a face of the complex.

A Triangulation of CP3 as Symmetric Cube of S-2

The symmetric group acts on the Cartesian product (S (2)) (d) by coordinate permutation, and the quotient space is homeomorphic to the complex projective space a'',P (d) . We used the case d=2 of this fact to construct a 10-vertex triangulation of a'',P (2) earlier. In this paper, we have constructed a 124-vertex simplicial subdivision of the 64-vertex standard cellulation of (S (2))(3), such that the -action on this cellulation naturally extends to an action on . Further, the -action on is ``good'', so that the quotient simplicial complex is a 30-vertex triangulation of a'',P (3). In other words, we have constructed a simplicial realization of the branched covering (S (2))(3)-> a'',P (3).

On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce k-stellated spheres and consider the class W-k(d) of triangulated d-manifolds, all of whose vertex links are k-stellated, and its subclass W-k*; (d), consisting of the (k + 1)-neighbourly members of W-k(d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of W-k(d) for d >= 2k. As a consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, and we determine the integral homology type of members of W-k*(d) for d >= 2k + 2. As another application, we prove that, when d not equal 2k + 1, all members of W-k*(d) are tight. We also characterize the tight members of W-k*(2k + 1) in terms of their kth Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for homology manifolds in which the members of W-1(d) provide the equality case. This generalizes a result (the d = 4 case) due to Walkup and Kuhnel. As a consequence, it is shown that every tight member of W-1 (d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuhnel and Lutz asserting that tight homology manifolds should be strongly minimal. (C) 2013 Elsevier Ltd. All rights reserved.

Estimating persistent Betti numbers for discrete shape analysis

Persistent Topology is an innovative way of matching topology and geometry, and it proves to be an effective mathematical tool in shape analysis. In order to express its full potential for applications, it has to interface with the typical environment of Computer Science: It must be possible to deal with a finite sampling of the object of interest, and with combinatorial representations of it. Following that idea, the main result claims that it is possible to construct a relation between the persistent Betti numbers (PBNs; also called rank invariant) of a compact, Riemannian submanifold X of R^m and the ones of an approximation U of X itself, where U is generated by a ball covering centered in the points of the sampling. Moreover we can state a further result in which, this time, we relate X with a finite simplicial complex S generated, thanks to a particular construction, by the sampling points. To be more precise, strict inequalities hold only in "blind strips'', i.e narrow areas around the discontinuity sets of the PBNs of U (or S). Out of the blind strips, the values of the PBNs of the original object, of the ball covering of it, and of the simplicial complex coincide, respectively.

Territorial assemblages and Governance : The example of the Taonaba project [Town of Abymes, Guadeloupe]

This article reports on an action research to support the urban community of Cap Excellence in Guadaloupe in its local sustainable development project. After summarizing the terms of the debate around sustainable development, and presenting the region, the search will be put back into the context of a more general approach of territorial* intelligence (TI). The limits of a local Agenda 21 in the form of a 'programmed action plan' is the chance to enhance the concept of TI with that of territorial assemblage. Our study area is the natural reserve of the Grand Cul-de-Sac Marin of Guadeloupe, the second largest biosphere reserve designated by UNESCO in the archipelago of the Petites Antilles, more specifically the implementation of the Taonaba project, whose goal is to launch an ecotourism visitors' centre, operational at the end of 2012. Based on the analysis of a large amount of data, the article describes an evaluation tool for territorial assemblages for participative territorial governance. Our results were presented to local government officials in the Urban Sustainable Development Forum, which our group organised from 2 to 4 April 2012, in the district of Abymes/Pointe-à-Pitre

Territorial assemblages and Governance : The example of the Taonaba project [Town of Abymes, Guadeloupe]

This article reports on an action research to support the urban community of Cap Excellence in Guadaloupe in its local sustainable development project. After summarizing the terms of the debate around sustainable development, and presenting the region, the search will be put back into the context of a more general approach of territorial* intelligence (TI). The limits of a local Agenda 21 in the form of a 'programmed action plan' is the chance to enhance the concept of TI with that of territorial assemblage. Our study area is the natural reserve of the Grand Cul-de-Sac Marin of Guadeloupe, the second largest biosphere reserve designated by UNESCO in the archipelago of the Petites Antilles, more specifically the implementation of the Taonaba project, whose goal is to launch an ecotourism visitors' centre, operational at the end of 2012. Based on the analysis of a large amount of data, the article describes an evaluation tool for territorial assemblages for participative territorial governance. Our results were presented to local government officials in the Urban Sustainable Development Forum, which our group organised from 2 to 4 April 2012, in the district of Abymes/Pointe-à-Pitre

Territorial assemblages and Governance : The example of the Taonaba project [Town of Abymes, Guadeloupe]

This article reports on an action research to support the urban community of Cap Excellence in Guadaloupe in its local sustainable development project. After summarizing the terms of the debate around sustainable development, and presenting the region, the search will be put back into the context of a more general approach of territorial* intelligence (TI). The limits of a local Agenda 21 in the form of a 'programmed action plan' is the chance to enhance the concept of TI with that of territorial assemblage. Our study area is the natural reserve of the Grand Cul-de-Sac Marin of Guadeloupe, the second largest biosphere reserve designated by UNESCO in the archipelago of the Petites Antilles, more specifically the implementation of the Taonaba project, whose goal is to launch an ecotourism visitors' centre, operational at the end of 2012. Based on the analysis of a large amount of data, the article describes an evaluation tool for territorial assemblages for participative territorial governance. Our results were presented to local government officials in the Urban Sustainable Development Forum, which our group organised from 2 to 4 April 2012, in the district of Abymes/Pointe-à-Pitre

On f-vectors of polytopes and matroids

Thesis (Ph.D.)--University of Washington, 2016-08

3-MANIFOLD GROUPS, KÄHLER GROUPS AND COMPLEX SURFACES

Let G be a Kahler group admitting a short exact sequence 1 -> N -> G -> Q -> 1 where N is finitely generated. (i) Then Q cannot be non-nilpotent solvable. (ii) Suppose in addition that Q satisfies one of the following: (a) Q admits a discrete faithful non-elementary action on H-n for some n >= 2. (b) Q admits a discrete faithful non-elementary minimal action on a simplicial tree with more than two ends. (c) Q admits a (strong-stable) cut R such that the intersection of all conjugates of R is trivial. Then G is virtually a surface group. It follows that if Q is infinite, not virtually cyclic, and is the fundamental group of some closed 3-manifold, then Q contains as a finite index subgroup either a finite index subgroup of the three-dimensional Heisenberg group or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. As a corollary, we obtain a new proof of a theorem of Dimca and Suciu in Which 3-manifold groups are Kahler groups? J. Eur. Math. Soc. 11 (2009) 521-528] by taking N to be the trivial group. If instead, G is the fundamental group of a compact complex surface, and N is finitely presented, then we show that Q must contain the fundamental group of a Seifert-fibered 3-manifold as a finite index subgroup, and G contains as a finite index subgroup the fundamental group of an elliptic fibration. We also give an example showing that the relation of quasi-isometry does not preserve Kahler groups. This gives a negative answer to a question of Gromov which asks whether Kahler groups can be characterized by their asymptotic geometry.

Raman spectroscopy of some complex arsenate minerals—implications for soil remediation

The application of spectroscopy to the study of contaminants in soils is important. Among the many contaminants is arsenic, which is highly labile and may leach to non-contaminated areas. Minerals of arsenate may form depending upon the availability of specific cations for example calcium and iron. Such minerals include carminite, pharmacosiderite and talmessite. Each of these arsenate minerals can be identified by its characteristic Raman spectrum enabling identification.