Oberwolfach Preprints (OWP)

Permanent URI for this collection

https://oa.tib.eu/renate/handle/123456789/14626

Browse

Recent Submissions

Now showing 1 - 5 of 269
  • Thumbnail Image
    Item
    Bredon cohomology and robot motion planning
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Farber, Michael; Grant, Mark; Lupton, Gregory; Oprea, John
    In this paper we study the topological invariant TC(X) reflecting the complexity of algorithms for autonomous robot motion. Here, X stands for the configuration space of a system and TC(X) is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in X. We focus on the case when the space X is aspherical; then the number TC(X) depends only on the fundamental group π=π1(X) and we denote it TC(π). We prove that TC(π) can be characterised as the smallest integer k such that the canonical π×π-equivariant map of classifying spaces E(π×π)→ED(π×π) can be equivariantly deformed into the k-dimensional skeleton of ED(π×π). The symbol E(π×π) denotes the classifying space for free actions and ED(πtimesπ) denotes the classifying space for actions with isotropy in a certain family D of subgroups of π×π. Using this result we show how one can estimate TC(π) in terms of the equivariant Bredon cohomology theory. We prove that TC(π)≤max{3,cdD(π×π)}, where cdD(π×π) denotes the cohomological dimension of π×π with respect to the family of subgroups D. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family D.
  • Thumbnail Image
    Item
    Z2-Thurston norm and complexity of 3-manifolds, II
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Jaco, William; Rubinstein, J. Hyam; Spreer, Jonathan; Tillmann, Stephan
    In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3-manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry SL2(R)˜.
  • Thumbnail Image
    Item
    Gradient canyons, concentration of curvature, and Lipschitz invariants
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Paunescu, Laurentiu; Tibăr, Mihai-Marius
    We find new bi-Lipschitz invariants of holomorphic functions of two variables by using the gradient canyons and by combining analytic and geometric viewpoints on the concentration of curvature.
  • Thumbnail Image
    Item
    The Varchenko determinant of a Coxeter arrangement
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Pfeiffer, Götz; Randriamaro, Hery
    The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relating determinants.
  • Thumbnail Image
    Item
    The colored Jones polynomial and Kontsevich-Zagier series for double twist knots
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Lovejoy, Jeremy; Osburn, Robert
    Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots K(−m,−p) and K(−m,p) where m and p are positive integers. In the (−m,−p) case, this leads to new families of q-hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of K(m,p) gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function for strongly unimodal sequences.