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

good, does not support this; the label ‘Example’ is given in a rather small font followed by a ‘PROOF,’ and the body of an example is nonitalic, utterly unlike other statements accompanied by demonstrations. The proofs of some theorems are continued after the statements and proofs of needed lemmas with no helpful demarcations (“Continuation of proof of Theorem . . . ”). Finally, at least one lemma (3.34) has a corollary with no theorem or proposition in sight; the status of Lemma 3.36 is also unclear. Lemma 1.16 on page 12 giving a closed formula for Stirling numbers of the second kind is important enough to figure in the proof of capstone Theorem 3.71 (Schlömilch’s formula) on page 114. This nonstandard approach to chapter organization may impede some readers’ comprehending the structure of the author’s highly purposeful discussion. Review of 3 Enumerative Combinatorics Author of Book: Charalambos A. Charalambides Publisher: Chapman & Hall/CRC ISBN L-58488-290-5, Hard Cover, 609 pages

Enumerative Combinatorics

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

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Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

Deformation quantization of Poisson manifolds, I

© Publications mathematiques de l’I.H.E.S., 1969, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

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The irreducibility of the space of curves of given genus

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories.The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

Introduction to Toric Varieties.

This volume represents a series of lectures which aims to introduce the beginner to the finite dimensional representations of Lie groups and Lie algebras. Following an introduction to representation theory of finite groups, the text explains how to work out the representations of classical groups.

Representation Theory: A First Course

Part I. Calculus Of Tableux: 1. Bumping and sliding 2. Words: the plactic monoid 3. Increasing sequences: proofs of the claims 4. The Robinson-Schensted-Knuth Correspondence 5. The Littlewood-Richardson rule 6. Symmetric polynomials Part II. Representation Theory: 7. Representations of the symmetric group 8. Representations of the general linear group Part III. Geometry: 9. Flag varieties 10. Schubert varieties and polynomials Appendix A Appendix B.

Young Tableaux: With Applications to Representation Theory and Geometry

A compactification of configuration spaces

We describe recent work of Klyachko, Totaro, Knutson, and Tao that characterizes eigenvalues of sums of Hermitian matrices and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Recent breakthroughs, primarily by A. Klyachko, B. Totaro, A. Knutson, and T. Tao, with contributions by P. Belkale and C. Woodward, have led to complete solutions of several old problems involving the various notions in the title. Our aim here is to describe this work and especially to show how these solutions are derived from it. Along the way, we will see that these problems are also related to other areas of mathematics, including geometric invariant theory, symplectic geometry, and combinatorics. In addition, we present some related applications to singular values of arbitrary matrices. Although many of the theorems we state here have not appeared elsewhere, their proofs are mostly “soft” algebra based on the hard geometric or combinatorial work of others. Indeed, this paper emphasizes concrete elementary arguments. We do give some new examples and counterexamples and raise some new open questions. We have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before. However, there is a very large literature, particularly for linear algebra problems about eigenvalues, singular values, and invariant factors. We have listed only a few of these articles, from whose bibliographies, we hope, an interested reader can trace the history; we apologize to the many whose work is not cited directly. We begin in the first five sections by describing each of the problems, together with some of their early histories, and we state as theorems the new solutions to these problems. In Section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs. Then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians. Sections 7, 8, 9, and 10 also contain variations and generalizations of some of the theorems stated in the first five sections, as well as attributions of the theorems to their authors. One of our fascinations with this subject, even now that we have proofs of the theorems, is the challenge to understand in a deeper way why all these subjects are Received by the editors in July 1999 and in revised form January 3, 2000. 2000 Mathematics Subject Classification. Primary 15A42, 22E46, 14M15; Secondary 05E15, 13F10, 14C17, 15A18, 47B07. The author was partly supported by NSF Grant #DMS9970435. c ©2000 American Mathematical Society

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William Fulton

Papers

Introduction to Toric Varieties.

Representation Theory: A First Course

Young Tableaux: With Applications to Representation Theory and Geometry

A compactification of configuration spaces

Eigenvalues, invariant factors, highest weights, and Schubert calculus