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Michael Farber

  1. The topology of spaces of polygons.

    Ср, 05/04/2011 - 15:01 — SomNambul
    Authors: Michael Farber, Viktor Fromm
    Subjects: Algebraic Topology
    Abstract

    Let $E_{d}(\ell)$ denote the space of all closed $n$-gons in $\R^{d}$ (where
    $d\ge 2$) with sides of length $\ell_1,..., \ell_n$, viewed up to translations.
    The spaces $E_d(\ell)$ are parameterized by their length vectors
    $\ell=(\ell_1,..., \ell_n)\in \R^n_{>}$ encoding the length parameters.
    Generically, $E_{d}(\ell)$ is a closed smooth manifold of dimension
    $(n-1)(d-1)-1$ supporting an obvious action of the orthogonal group ${O}(d)$.
    However, the quotient space $E_{d}(\ell)/{O}(d)$ (the moduli space of shapes of
    $n$-gons) has singularities for a generic $\ell$, assuming that $d>3$;

  2. Homology of planar telescopic linkages.

    Чт, 09/17/2009 - 10:33 — SomNambul
    Authors: Michael Farber, Viktor Fromm
    Subjects: Algebraic Topology
    Abstract

    We study topology of configuration spaces of planar linkages having one leg
    of variable length. Such telescopic legs are common in modern robotics where
    they are used for shock absorbtion and serve a variety of other purposes. Using
    a Morse theoretic technique, we compute explicitly, in terms of the metric
    data, the Betti numbers of configuration spaces of these mechanisms.

  3. Homology of planar telescopic linkages.

    Чт, 09/17/2009 - 10:33 — SomNambul
    Authors: Michael Farber, Viktor Fromm
    Subjects: Algebraic Topology
    Abstract

    We study topology of configuration spaces of planar linkages having one leg
    of variable length. Such telescopic legs are common in modern robotics where
    they are used for shock absorbtion and serve a variety of other purposes. Using
    a Morse theoretic technique, we compute explicitly, in terms of the metric
    data, the Betti numbers of configuration spaces of these mechanisms.

  4. Topology of Random Right Angled Artin Groups.

    Пнд, 09/07/2009 - 10:33 — SomNambul
    Authors: Armindo Costa, Michael Farber
    Subjects: Algebraic Topology
    Abstract

    In this paper we study topological invariants of a class of random groups.
    Namely, we study right angled Artin groups associated to random graphs and
    investigate their Betti numbers, cohomological dimension and topological
    complexity. The latter is a numerical homotopy invariant reflecting complexity
    of motion planning algorithms in robotics. We show that the topological
    complexity of a random right angled Artin group assumes, with probability
    tending to one, at most three values.

  5. Topology of Random Right Angled Artin Groups.

    Пнд, 09/07/2009 - 10:33 — SomNambul
    Authors: Armindo Costa, Michael Farber
    Subjects: Algebraic Topology
    Abstract

    In this paper we study topological invariants of a class of random groups.
    Namely, we study right angled Artin groups associated to random graphs and
    investigate their Betti numbers, cohomological dimension and topological
    complexity. The latter is a numerical homotopy invariant reflecting complexity
    of motion planning algorithms in robotics. We show that the topological
    complexity of a random right angled Artin group assumes, with probability
    tending to one, at most three values.

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