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  • 1.
    Björner, Anders
    KTH, Superseded Departments, Mathematics.
    Face numbers of Scarf complexes2000In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 24, no 3-Feb, p. 185-196Article in journal (Refereed)
    Abstract [en]

    Let A be a (d + 1) x d real matrix whose row vectors positively span R-d and which is generic in the sense of Barany and Scarf [BS1]. Such a matrix determines a certain infinite d-dimensional simplicial complex Sigma, as described by Barany et al. [BHS]. The group Z(d) acts on Sigma with finitely many orbits. Let f(i) be the number of orbits of (i + 1)-simplices of Sigma. The sequence f = (f(0), f(1),..., f(d-1)) is the f-vector of a certain triangulated (d - 1)-ball T embedded in Sigma. When A has integer entries it is also, as shown by the work of Peeva and Sturmfels [PS], the sequence of Betti numbers of the minimal free resolution of k[x(1),...,x(d+1)]/I, where I is the lattice ideal determined by A. In this paper we study relations among the numbers f(i). It is shown that f(0), f(1),..., f([(d-3)/2]) determine the other numbers via linear relations, and that there are additional nonlinear relations. In more precise (and more technical) terms, our analysis shows that f is linearly determined by a certain M-sequence (g(0), g(1),..., g([(d-1)/2])). namely, the g-vector of the (d - 2)-sphere bounding T. Although T is in general not a cone over its boundary, it turns out that its f-vector behaves as if it were.

  • 2.
    Björner, Anders.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Paffenholz, A.
    Sjöstrand, J.
    Ziegler, G. M.
    Bier spheres and posets2005In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 34, no 1, p. 71-86Article in journal (Refereed)
    Abstract [en]

    In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n -2)-spheres on 2n vertices, as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that cut across an ideal. Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular generalized Bier spheres. In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields many shellable spheres, most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.

  • 3.
    Björner, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tancer, Martin
    Note: Combinatorial Alexander Duality-A Short and Elementary Proof2009In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 42, no 4, p. 586-593Article in journal (Refereed)
    Abstract [en]

    Let X be a simplicial complex with ground set V. Define its Alexander dual as the simplicial complex X* = {sigma subset of V vertical bar V \ sigma is not an element of X}. The combinatorial Alexander duality states that the ith reduced homology group of X is isomorphic to the (vertical bar V vertical bar - i - 3) th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof from first principles accessible to a nonexpert.

  • 4.
    Bucher-Karlsson, Michelle
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On Minimal Triangulations of Products of Convex Polygons2009In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 41, no 2, p. 328-347Article in journal (Refereed)
    Abstract [en]

    We give new lower bounds for the minimal number of simplices needed in a triangulation of the product of two convex polygons, improving the lower bounds in Bowen et al. (Topology 44:321-339, 2005).

  • 5. De Silva, Vin
    et al.
    Vejdemo-Johansson, Mikael
    Stanford University, USA.
    Morozov, Dmitriy
    Persistent cohomology and circular coordinates2011In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 45, p. 737-759Article in journal (Refereed)
    Abstract [en]

    Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional, but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise NLDR analysis of a broader range of realistic data sets.

  • 6.
    Engström, Alexander
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Set Partition Complexes2008In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 40, p. 357-364Article in journal (Refereed)
    Abstract [en]

    The Hom complexes were introduced by Lovasz to study topological obstructions to graph colorings. The vertices of Hom(G,K-n ) are the n-colorings of the graph G, and a graph coloring is a partition of the vertex set into independent sets. Replacing the independence condition with any hereditary condition defines a set partition complex. We show how coloring questions arising from, for example, Ramsey theory can be formulated with set partition complexes.

    It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that Hom(G,K-n ) is (n - d - 2)-connected, where d is the maximal degree of a vertex of G. We generalize this to set partition complexes.

  • 7.
    Håstad, Johan
    et al.
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Linusson, Svante
    Wastlund, J.
    A smaller sleeping bag for a baby snake2001In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 26, no 1, p. 173-181Article in journal (Refereed)
    Abstract [en]

    By a sleeping bag for a baby snake in d dimensions we mean a subset of R-d which can cover, by rotation and translation, every curve of unit length. We construct sleeping bags which are smaller than any previously known in dimensions 3 and higher. In particular, we construct a three-dimensional sleeping bag of volume approximately 0.075803. For large d we construct d-dimensional sleeping bags with volume less than (c root logd)(d)/d(3d/2) for some constant c. To obtain the last result, we show that every curve of unit length in R-d lies between two parallel hyperplanes at distance at most c(1)d(-3/2)root logd, for some, constant c(1).

  • 8.
    Jonsson, Jakob
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Certain Homology Cycles of the Independence Complex of Grids2010In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 43, no 4, p. 927-950Article in journal (Refereed)
    Abstract [en]

    Let G be an infinite graph such that the automorphism group of G contains a subgroup K congruent to Z(d) with the property that G/K is finite. We examine the homology of the independence complex Sigma(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as "cross-cycles," the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G, K) of rational numbers r such that there is a group I with the property that Sigma(G/I) contains cross-cycles of degree exactly r . |G/I| - 1; |G/I| denotes the size of the vertex set of G/I. In each of the three cases, Q( G, K) turns out to be an interval of the form [a, b] boolean AND Q = {r is an element of Q : a <= r <= b}. For example, for the square grid, we obtain the interval [1/5, 1/4] boolean AND Q.

  • 9.
    Kozlov, Dimitrij
    KTH, Superseded Departments, Mathematics.
    Directed trees in a string, real polynomials with triple roots, and chain mails2004In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 32, no 3, p. 373-382Article in journal (Refereed)
    Abstract [en]

    This paper starts with an observation that two infinite series of simplicial complexes, which a priori do not seem to have anything to do with each other, have the same homotopy type. One series consists of the complexes of directed forests on a double directed string, while the other one consists of Shapiro-Welker models for the spaces of hyperbolic polynomials with a triple root. We explain this coincidence in the more general context by finding an explicit homotopy equivalence between complexes of directed forests on a double directed tree, and doubly disconnecting complexes of a tree.

  • 10.
    Tancer, Martin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Department of Applied Mathematics, Charles University in Prague, Czech Republic.
    Recognition of Collapsible Complexes is NP-Complete2016In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 55, no 1, p. 21-38Article in journal (Refereed)
    Abstract [en]

    We prove that it is NP-complete to decide whether a given (3-dimensional) simplicial complex is collapsible. This work extends a result of Malgouyres and Francés showing that it is NP-complete to decide whether a given simplicial complex collapses to a 1-complex.

  • 11. Tancer, Martin
    et al.
    Vorwerk, Kathrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Non-embeddability of geometric lattices and buildings2014In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 51, no 4, p. 779-801Article in journal (Refereed)
    Abstract [en]

    A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show that order complexes of thick geometric lattices as well as several classes of finite buildings, all of which are order complexes, are hard to embed. That means that such -dimensional complexes require -dimensional Euclidean space for an embedding. (This dimension is always sufficient for any -complex.) We develop a method to show non-embeddability for general order complexes of posets.

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