It is perhaps not universally acknowledged that an outstanding place to find interesting topological objects is within the walls of an automated warehouse or factory. The examples of topological spaces constructed in this exposition arose simultaneously from two seemingly disparate fields: the first author, in his thesis [1], discovered these spaces after working with the group of H. Landau, Z. Landau, J. Pommersheim, and E. Zaslow [2] on problems about multiple random walks on graphs. The second author [8, 7] discovered these same spaces while collaborating with D. Koditschek in the Artificial Intelligence Lab at the University of Michigan. The net result makes evident the abundance of topological objects within the physical world. Topology seeks to describe, as one author puts it, the "shape of space" [15], with "shape" being interpreted as appropriate for the context at hand. We will begin with thinking about spaces up to homeomorphism (continuous maps with continuous inverse), but will quickly need to abandon this class in favor of a looser form of equivalence: homotopy type. Although few topological prerequisites are necessary for fully appreciating the examples discussed here, the class of spaces we consider gives an earthly incarnation of several intricate ideas from topology, such as K(π, 1) spaces (a.k.a. Eilenberg-MacLane spaces of type K(π, 1)) and NPC (or non-positively curved) spaces. 1. Configurations and Braids Our story begins with a classical construction: that of a configuration space of points. We consider first the configuration space of N distinct labelled points RG supported in part by NSF Grant # DMS-9971629. The authors wish to thank Margaret Symington for her careful reading of the manuscript.

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.

We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of Lükő on average chord lengths of closed curves.

Abstract. We define a family of quasi-isometry invariants of groups called higher divergence functions, which measure isoperimetric properties "at infinity." We give sharp upper and lower bounds on the divergence functions for right-angled Artin groups, using different pushing maps on the associated cube complexes. In the process, we define a class of RAAGs we call orthoplex groups, which have the property that their Bestvina-Brady subgroups have hard-to-fill spheres. Our results give sharp bounds on the higher Dehn functions of Bestvina-Brady groups, a complete characterization of the divergence of geodesics in RAAGs, and an upper bound for filling loops at infinity in the mapping class group.

It is perhaps not universally acknowledged that an outstanding place to find interesting topological objects is within the walls of an automated warehouse or factory.The examples of topological spaces constructed in this exposition arose simultaneously from two seemingly disparate fields: the first author, in his thesis [1], discovered these spaces after working with the group of H. Landau, Z. Landau, J. Pommersheim, and E. Zaslow [2] on problems about multiple random walks on graphs. The second author [8,7] discovered these same spaces while collaborating with D. Koditschek in the Artificial Intelligence Lab at the University of Michigan. The net result makes evident the abundance of topological objects within the physical world.Topology seeks to describe, as one author puts it, the "shape of space" [15], with "shape" being interpreted as appropriate for the context at hand. We will begin with thinking about spaces up to homeomorphism (continuous maps with continuous inverse), but will quickly need to abandon this class in favor of a looser form of equivalence: homotopy type.Although few topological prerequisites are necessary for fully appreciating the examples discussed here, the class of spaces we consider gives an earthly incarnation of several intricate ideas from topology, such as K(π, 1) spaces (a.k.a. Eilenberg-MacLane spaces of type K(π, 1)) and NPC (or non-positively curved) spaces. Configurations and BraidsOur story begins with a classical construction: that of a configuration space of points. We consider first the configuration space of N distinct labelled points RG supported in part by NSF Grant # DMS-9971629. The authors wish to thank Margaret Symington for her careful reading of the manuscript.