Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

Metric Spaces Length Spaces Constructions Spaces of Bounded Curvature Smooth Length Structures Curvature of Riemannian Metrics Space of Metric Spaces Large-scale Geometry Spaces of Curvature Bounded Above Spaces of Curvature Bounded Below Bibliography Index.

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A Course in Metric Geometry

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Positive Definite Matrices

We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound

 $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$
 (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$
 and dim(M) ⩽ N. The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact. Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincare inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.

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On the geometry of metric measure spaces. II

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries. The theory of these objects is developed in a manner accessible to anyone familiar with the rudiments of topology and group theory: non-trivial theorems are proved by concatenating elementary geometric arguments, and many examples are given. Part I is an introduction to the geometry of geodesic spaces. In Part II the basic theory of spaces with upper curvature bounds is developed. More specialized topics, such as complexes of groups, are covered in Part III. The book is divided into three parts, each part is divided into chapters and the chapters have various subheadings. The chapters in Part III are longer and for ease of reference are divided into numbered sections.

Metric Spaces of Non-Positive Curvature

Geodesics and curvature in metric simplicial complexes

The closure of the set of isoperimetric exponents for finitely presented groups is 
$ \{1\} \cup [2,\infty $
 . For each pair of positive integers 
$ p \ge q $
 , one can construct groups with aspherical presentations for which the Dehn function is 
$ \simeq n^{2\alpha} $
 , where 
$ \alpha\,=\{\rm log}_{2}(2p/q) $
 .

There is only one gap in the isoperimetric spectrum

The optimal isoperimetric inequality for torus bundles over the circle

We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup. A useful structure theory for subgroups of finite direct products of groups has yet to be developed. To begin to study such subgroups it is natural to assume one knows about the subgroups of the direct factors and to concentrate on subdirect products. Recall that G is termed a subdirect product of the groups A1, . . . , An if G ⊆ A1 × · · · × An is a subgroup that projects surjectively to each factor. Work by various authors has exposed the surprisingly rich structure to be found amongst the subdirect products of superficially-tame groups. For example, in contrast to the fact that subdirect products of abelian or nilpotent groups are again in the specified class, nonabelian free groups harbour a great diversity of subdirect products, including some with unsolvable decision problems [21]. This diversity has long been known, but it is only as a result of more recent work by Baumslag-Roseblade [5] and Bridson-Howie-Miller-Short [9] that it has been understood as a phenomenon that is intimately tied to the failure of various homological finiteness conditions. From this background we pick out the three strains of thought to be developed in this article: the subtlety of subdirect products in general; 1991 Mathematics Subject Classification. 20F05, 20E07, 20F67, 20J05, 20F10.

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Martin R. Bridson

Papers

Metric Spaces of Non-Positive Curvature

Geodesics and curvature in metric simplicial complexes

There is only one gap in the isoperimetric spectrum

The optimal isoperimetric inequality for torus bundles over the circle

Structure and finiteness properties of subdirect products of groups