Narrow Results

An approach to the quantization of gravity in the presence of matter is examined which starts from the classical Einstein–Hilbert action and matter approximated by “point” particles minimally coupled to the metric. Upon quantization, the Hamilton constraint assumes the form of the Schrödinger equation: it contains the usual Wheeler–DeWitt term and the term with the time derivative of the wave function. In addition, the wave function also satisfies the Klein–Gordon equation, which arises as the quantum counterpart of the constraint among particles’ momenta. Comparison of the novel approach with the usual one in which matter is represented by scalar fields is performed, and shown that those approaches do not exclude, but complement each other. In final discussion it is pointed out that the classical matter could consist of superparticles or spinning particles, described by the commuting and anticommuting Grassmann coordinates, in which case spinor fields would occur after quantization.

The robot path planning problem involves planning optimal paths for a robot to follow while ensuring it will not hit any obstacles or itself. In a state or perfectly known world, this has been addressed using the configuration space representation and the A* search algorithm. However, when movement, changes, or unexpected obstacles occur in the environment, a new method, Differential A*, can adapt the solution to the current situation. It updates only the fraction of space that is critically affected. This technique can provide significant speed improvements, with the same desired results, compared to complete space regeneration.

Given two convex polygons P and Q in the plane that are free to translate and rotate, a convex packing of them is the convex hull of a placement of P and a placement of Q whose interiors do not intersect. A minimum area convex packing of P and Q is one whose area is minimized. The problem of designing a deterministic algorithm for finding a minimum area convex packing of two convex polygons has remained open. We address this problem by first studying the contact configurations between P and Q and their algebraic structures. Crucial geometric and algebraic properties on the area function are then derived and analyzed which enable us to successfully discretize the search space. This discretization, together with a delicate algorithmic design and careful complexity analysis, allows us to develop an efficient O((n + m)nm) time deterministic algorithm for finding a true minimum area convex packing of P and Q, where n and m are the numbers of vertices of P and Q, respectively.

The Minkowski sum of two plane curves can be regarded as the area generated by sweeping one curve along the other. The boundary of the Minkowski sum consists of translated portions of the given curves and/or portions of a more complicated curve, the “envelope” of translates of the swept curve. We show that the Minkowski-sum boundary is describable as an algebraic curve (or subset thereof) when the given curves are algebraic, and illustrate the computation of its implicit equation. However, such equations are typically of high degree and do not offer a practical basis for tracing the boundary. For the case of polynomial parametric curves, we formulate a simple numerical procedure to address the latter problem, based on constructing the Gauss maps of the given curves and using them to identifying “corresponding” curve segments that are to be summed. This yields a set of discretely-sampled arcs that constitutes a superset of the Minkowski-sum boundary, and can be regarded as a planar graph. To extract the true boundary, we present a method for identifying and “trimming” away extraneous arcs by systematically traversing this graph.

We introduce the notion of expansiveness to characterize a family of robot configuration spaces whose connectivity can be effectively captured by a roadmap of randomly-sampled milestones. The analysis of expansive configuration spaces has inspired us to develop a new randomized planning algorithm. This new algorithm tries to sample only the portion of the configuration space that is relevant to the current query, avoiding the cost of precomputing a roadmap for the entire configuration space. Thus, it is well-suited for problems where only a single query is submitted for a given environment. The algorithm has been implemented and successfully applied to complex assembly maintainability problems from the automotive industry.

Given a suitable link map f into a manifold M, we constructed, in a previous publication, link homotopy invariants κ(f) and μ(f). In the present paper we study the case M=Sⁿ×ℝ^m-n in detail. Here μ(f) turns out to be the starting term of a whole sequence μ^(s)(f), s=0, 1,…, of higher μ-invariants which together capture all the information contained in κ(f). We discuss the geometric significance of these new invariants. In several instances we obtain complete classification results. A central ingredient of our approach is the homotopy theory of wedges of spheres.

For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of cycles in a projective space of arbitrary odd dimension and the self-linking number of a zero homologous knot in the 3-dimensional projective space.

Motivated in part by the study of Fadell–Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n ≥ 1, the class of m-string braid groups B_m(𝕊²\{x₁,…,x_n}) of the n-punctured sphere includes the usual Artin braid groups B_m (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type (for n = 3).

We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of B_m is completely determined for all m ∈ ℕ (respectively, for all m ≠ 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients.

When n ≥ 2, we prove that the lower central series (respectively, derived series) of B_m(𝕊²\{x₁,…,x_n}) is constant from the commutator subgroup onwards for all m ≥ 3 (respectively, m ≥ 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B₂(𝕊²\{x₁,x₂}) admits various interpretations, as the Baumslag–Solitar group BS(2,2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B₂(𝕊²\{x₁,x₂}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product ℤ₂ * ℤ. Further, its lower central series quotients Γ_i/Γ_{i + 1} are direct sums of copies of ℤ₂, the number of summands being determined explicitly. In the case m ≥ 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Γ_i/Γ_{i + 1} are 2-elementary finitely-generated groups.

We estimate from below the number of lines meeting each of given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space. We obtain also a similar lower bound for the number of lines meeting each of given 4 disjoint smooth closed curves in a given linear order in ℝ³. The estimates are formulated in terms of linking numbers of the curves and obtained by orienting of the corresponding configuration spaces and evaluating of their signatures. This involves a study of a surface swept by lines meeting 3 given disjoint smooth closed curves. Higher-dimensional generalizations of these results are outlined.

We describe a way of producing an $(n-3)$-dimensional manifold ${𝒦}({𝒢})$ starting with an Alexander self-dual simplicial complex${𝒢}$ on $n$ vertices (or, in another terminology, by a simple game with constant sum with $n$ players). The construction presents ${𝒦}({𝒢})$ explicitly, by describing its regular cellulation.

We construct maps from the fundamental groups of complements to some plane configurations to the groups $G_n^k$ for large $k$.

It had been proved by Birman and Goldberg that the normal closure of the pure braid group $P_n(D)$ in the pure braid group of the torus $P_n(T)$ is the commutator subgroup $[P_n(T),P_n(T)]$. In this paper, we are going to study the case of full braid groups: i.e. the normal closure of $B_n(D)$ in $B_n(T)$, which turns out to have an interesting geometric description.

We construct a group ${\mathrm{\Gamma }}_n^4$ corresponding to the motion of points in ${ℝ}^3$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of ${\mathrm{\Gamma }}_n^4$. We will also study the group of pure braids in ${ℝ}^3$, which is described by a fundamental group of the restricted configuration space of ${ℝ}^3$, and define the group homomorphism from the group of pure braids in ${ℝ}^3$ to ${\mathrm{\Gamma }}_n^4$. At the end of this paper, we give some comments about relations between the restricted configuration space of ${ℝ}^3$ and triangulations of the 3-dimensional ball and Pachner moves.

We develop a combinatorial version of harmonic analysis on configuration spaces over Riemannian manifolds. Our constructions are based on the use of a lifting operator which can be considered as a kind of (combinatorial) Fourier transform in the configuration space analysis. The latter operator gives us a natural lifting of the geometry from the underlying manifold onto the configuration space. Properties of correlation measures for given states (i.e. probability measures) on configuration spaces are studied including a characterization theorem for correlation measures.

The classical Dirichlet form given by the intrinsic gradient on Γ_ℝ^d is associated with a Markov process consisting of a countable family of interacting diffusions. By considering each diffusion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure valued diffusion.

The quasi-sure analysis of Dirichlet forms is used to find exceptional sets of configurations for this Markov process. We consider large scale properties of the configuration and show that, for quite general measures, the process never hits those unusual configurations that violate the law of large numbers. Furthermore, for certain Gibbs measures, which model random particles in ℝ^d that interact via a potential function, we show, for d=1, 2, that the process never hits those unusual configurations that violate the law of the iterated logarithm.

Some a priori bounds for measures on configuration spaces are considered. We establish relations between them and consequences for corresponding measures (such as support properties etc.). Applications to Gibbs measures are discussed.

We study the continuous version of the contact model. Using an analytic approach, we construct the non-equilibrium contact process as a Markov process on configuration space. The construction is based on the analysis of correlation functions evolution. The problem concerning invariant measures as well as asymptotics of correlation functions are also studied.

We show that for non-relativistic free particles, the (bosonic) many particle equations can be rewritten in geometric fashion in terms of a classical theory of conformally stretched spacetime. We further generalize the results for the particles subject to a potential.

In mechanics, both classical and quantum, one studies the profound interaction between two types of energy, namely, the kinetic energy and the potential energy. The former can be organized as the kinematic metric on the configuration space while the latter can be represented by a suitable potential function, such as the Newtonian potential in celestial mechanics and the Coulomb potential in quantum mechanics of atomic and molecular physics. In this paper, the author studies the kinematic geometry of n-body systems. The main results are (i) the introduction of a canonical coordinate system which reveals the total amount of kinematic symmetry by an SO(3)× O(n-1) action in such a canonical coordinate representation; (ii) an in depth analysis of the above kinematic system both in the setting of classical invariant theory and by the technique of equivariant Riemannian geometry; (iii) a remarkably simple formula for the potential function in such a canonical coordinate system which reveals the well-fitting between the kinematic symmetry and the potential energy.

In 1985 Kevin Walker in his study of topology of polygon spaces [15] raised an interesting conjecture in the spirit of the well-known question "Can you hear the shape of a drum?" of Marc Kac. Roughly, Walker's conjecture asks if one can recover relative lengths of the bars of a linkage from intrinsic algebraic properties of the cohomology algebra of its configuration space. In this paper we prove that the conjecture is true for polygon spaces in R³. We also prove that for planar polygon spaces the conjecture holds in several modified forms: (a) if one takes into account the action of a natural involution on cohomology, (b) if the cohomology algebra of the involution's orbit space is known, or (c) if the length vector is normal. Some of our results allow the length vector to be non-generic, the corresponding polygon spaces have singularities. Our main tool is the study of the natural involution and its action on cohomology. A crucial role in our proof plays the solution of the isomorphism problem for monoidal rings due to Gubeladze.