Aaron Abrams

Finding topology in a factory: Configuration spaces

In searching the world for examples of interesting topological objects, it may not be obvious that an outstanding place to look is within the walls of an automated warehouse or factory. In this exposition we describe a class of topological spaces that arise naturally in this very context. The examples we construct actually arose simultaneously in two seemingly disparate fields: the first author, in his thesis [1], discovered these spaces after working on problems about random walks on graphs with H. Landau, Z. Landau, J. Pommersheim, and E. Zaslow [2]. The second author discovered these same spaces while collaborating with D. Koditschek in the Artificial Intelligence Lab at the University of Michigan [7], [8]. In Sections 1 and 2 we give some motivations arising from robotics, as well as a little background on configuration spaces. For the remainder of the paper we focus on a fascinating class of topological spaces related to motion-planning on graphs.

State Complexes for Metamorphic Robots

A metamorphic robotic system is an aggregate of homogeneous robot units which can individually and selectively locomote in such a way as to change the global shape of the system. We introduce a mathematical framework for defining and analyzing general metamorphic robots. With this formal structure, combined with ideas from geometric group theory, we define a new type of configuration space for metamorphic robots—the state complex—which is especially adapted to parallelization. We present an algorithm for optimizing an input reconfiguration sequence with respect to elapsed time. A universal geometric property of state complexes—non-positive curvature—is the key to proving convergence to the globally timeoptimal solution obtainable from the initial path.

Distances of Heegaard splittings

J. Hempel [Topology, 2001] showed that the set of distances of the Heegaard splittings (S;V;h n (V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V ‰PML(S). Here h is a pseudo-Anosov homeomorphism of a surface S while V is the set of isotopy classes of simple closed curves in S bounding essential disks in a flxed handlebody. With the same hypothesis we show that the distance of the splitting (S;V;h n (V)) grows linearly with n, answering a question of A. Casson. In addition we prove the converse of Hempel’s theorem. Our method is to study the action of h on the curve complex associated to S . We rely heavily on the result, due to H. Masur and Y. Minsky [Invent. Math., 1999], that the curve complex is Gromov hyperbolic.

Configuration Spaces of Colored Graphs

This paper is intended to provide concrete examples of concepts discussed elsewhere in this volume, especially splittings of groups and nonpositively curved cube complexes but also other things. The idea of the construction (configuration spaces) is not new, but this family of examples does not seem to be well known. Nevertheless they arise in a variety of contexts; applications are discussed in the last section. Most proofs are omitted.

Homological and homotopical Dehn functions are different

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, whereas the homotopical Dehn function measures fillings of curves by disks. Because the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions; however, before this work, there were no known examples of finitely presented groups for which the two functions differ. This paper gives such examples, constructed by amalgamating a free-by-cyclic group with several Bestvina–Brady groups.

Discretized configurations and partial partitions

We show that the discretized configuration space of

Distributions of Order Patterns of Interval Maps

k

Optimal Estimators for Threshold-Based Quality Measures

points in the

An Iterated Random Function with Lipschitz Number One

n

The number of possibilities for random dating

-simplex is homotopy equivalent to a wedge of spheres of dimension