Narrow Results

In this paper, we consider the ordered configuration space of $n$ open unit-diameter disks in the infinite strip of width $w$. In the spirit of Arnol’d and Cohen, we provide a finite presentation for the rational homology groups of this ordered configuration space as a twisted algebra. We use this presentation to prove that the ordered configuration space of open unit-diameter disks in the infinite strip of width $w$ exhibits a notion of first-order representation stability similar to Church–Ellenberg–Farb and Miller–Wilson’s first-order representation stability for the ordered configuration space of points in a manifold. In addition, we prove that for large $w$ this disk configuration space exhibits notions of second- (and higher) order representation stability.

We present a new method to prove existence and uniform a priori estimates for Gibbs measures associated with classical particle systems in a continuum. The method is based on the choice of appropriate Lyapunov functionals and on corresponding exponential bounds for the local Gibbs specification. Extensions to infinite range and multibody interactions are included.

Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct “Vassiliev classes” in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in ${ℝ}^d$ for $d>3$. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configuration spaces. Our classes form the integer lattice among the previously discovered real cohomology classes. Thus we obtain nontrivial classes from trivalent graph cocycles. Our methods generalize to yield mod-$p$ classes out of mod-$p$ graph cocycles, which need not be reductions of classes over the integers.

This paper gives a description of the diagonal ${\mathrm{\text{GL}}}_3$-orbits on the quadruple projective space ${({\mathbb{P}}^2)}^4$. We give explicit representatives of orbits, and describe the closure relations of orbits. A distinguished feature of our setting is that it is the simplest case, where diag$({\mathrm{\text{GL}}}_n)$ has infinitely many orbits but has also an open orbit in the multiple projective space ${({\mathbb{P}}^{n-1})}^m$.

An implementation of a computational geometry algorithm is robust if the combinatorial output is correct for every input. Robustness is achieved by ensuring that the predicates in the algorithm are evaluated correctly. A predicate is the sign of an algebraic expression whose variables are input parameters. The hardest case is detecting degenerate predicates where the value of the expression equals zero. We encounter this case in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to a stationary polyhedron. Each predicate involved in the construction is expressible as the sign of a univariate polynomial $f$ evaluated at a zero $t$ of a univariate polynomial $g$, where the coefficients of $f$ and $g$ are polynomials in the coordinates of the polyhedron vertices. A predicate is degenerate when $t$ is a zero of a common factor of $f$ and $g$. We present an efficient degeneracy detection algorithm based on a one-time factoring of all the univariate polynomials over the ring of multivariate polynomials in the vertex coordinates. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.

We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal Vassiliev link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel anomaly -that groups the Bott and Taubes anomalous terms- is a combination of diagrams with two univalent vertices and we explicitly define the isomorphism of which transforms the Kontsevich integral into the Poirier limit of the perturbative expression of the Chern-Simons theory for framed links, as a function of α

We define algebraic structures on graph cohomology and prove that they correspond to algebraic structures on the cohomology of the spaces of imbeddings of S¹ or ℝ into ℝⁿ. As a corollary, we deduce the existence of an infinite number of nontrivial cohomology classes in Imb(S¹, ℝⁿ) when n is even and greater than 3. Finally, we give a new interpretation of the anomaly term for the Vassiliev invariants in ℝ³.

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math.191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.

We investigate the space $C(X)$ of images of linearly embedded finite simplicial complexes in ${ℝ}^n$ isomorphic to a given complex $X$, focusing on two special cases: $X$ is the $(n-2)$-skeleton $K$ of an $n$-simplex, and $X$ is the $(n-2)$-skeleton $L$ of an $(n+1)$-simplex, so $X$ has codimension 2 in ${ℝ}^n$, in both cases. The main result is that for $n>2$, $C(X)$ (for either $X=K,L$) deformation retracts to a subspace homeomorphic to the double mapping cylinder

It has been seen elsewhere how elementary topology may be used to construct representations of the Iwahori-Hecke algebra associated with two-row Young diagrams, and how these constructions are related to the production of the same representations from the monodromy of n-point correlation functions in the work of Tsuchiya & Kanie and to the construction of the one-variable Jones polynomial. This paper investigates the extension of these results to representations associated with arbitrary multi-row Young diagrams and a functorial description of the two-variable Jones polynomial of links in S³.

We establish sharp upper bounds for the topological complexity TC(X) of motion planning algorithms in topological spaces X such that the fundamental group is "small", i.e. when π₁(X) is cyclic of order ≤ 3 or has small cohomological dimension.

We consider the moduli spaces ${{ℳ}}_d(\ell )$ of a closed linkage with $n$ links and prescribed lengths $\ell \in {ℝ}^n$ in $d$-dimensional Euclidean space. For $d>3$ these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold.

We use intersection homology to assign a ring to these spaces that can be used to distinguish the homeomorphism types of ${{ℳ}}_d(\ell )$ for a large class of length vectors in the case of $d$ even. This result is a high-dimensional analogue of the Walker conjecture which was proven by Farber, Hausmann and the author.

We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions. Our algorithms are optimal in a very concrete sense, namely, they have the minimal possible number of local planners. Our algorithms are motivated by those presented by Mas-Ku and Torres-giese (as streamlined by Farber), and are developed within the more general context of the multitasking (a.k.a. higher) motion planning problem. In addition, an eventual implementation of our algorithms is expected to work more efficiently than previous ones when applied to systems with a large number of moving objects.

Invariant quantities of the classical motion of an ideal incompressible fluid in a two-dimensional bounded domain are used to construct a family {Π_α}_α of probability measures of the Gibbs form, which are invariant under the flow. The Gibbs exponent H is given by the renormalized energy. These measures are supported by the space of configuration Γ, i.e. the fluid vorticity is concentrated in a finite number of distinct points. Properties of a deterministic vortex dynamics having Π_α as invariant measure are investigated; in particular Markov uniqueness is proven. The classical (pre-)Dirichlet form associated to Π_α is also introduced and analyzed.