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New Publications Offered by the AMS New Publications Offered by the AMS appropriate generality waited for the seventies. These early Algebra and Algebraic occurrences were in algebraic topology in the study of (iter- ated) loop spaces and their chain algebras. In the nineties, Geometry there was a renaissance and further development of the theory inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, Almost Commuting derived categories, Morse theory, symplectic and contact EMOIRS M of the geometry, combinatorics, knot theory, moduli spaces, cyclic American Mathematical Society Elements in Compact cohomology, and, not least, theoretical physics, especially Volume 157 Number 747 string field theory and deformation quantization. The general- Almost Commuting Lie Groups Elements in ization of quadratic duality (e.g., Lie algebras as dual to Compact Lie Groups Armand Borel, Institute for commutative algebras) together with the property of Koszul- Armand Borel Advanced Study, Princeton, NJ, ness in an essentially operadic context provided an additional Robert Friedman computational tool for studying homotopy properties outside John W. Morgan and Robert Friedman and THEMAT A IC M A L N A S O C I C of the topological setting. R I E E T M Y A FO 8 U 88 John W. Morgan, Columbia NDED 1 University, New York City, NY The book contains a detailed and comprehensive historical American Mathematical Society introduction describing the development of operad theory Contents: Introduction; Almost from the initial period when it was a rather specialized tool in commuting N-tuples; Some characterizations of groups of homotopy theory to the present when operads have a wide type A; c-pairs; Commuting triples; Some results on diagram range of applications in algebra, topology, and mathematical automorphisms and associated root systems; The fixed physics. Many results and applications currently scattered in subgroup of an automorphism; C-triples; The tori S(k) and the literature are brought together here along with new results w S c (g,k) and their Weyl groups; The Chern-Simons invariant; and insights. The basic definitions and constructions are care- The case when C is not cyclic; Bibliography; Diagrams and fully explained and include many details not found in any of tables. the standard literature. Memoirs of the American Mathematical Society, Volume 157, There is a chapter on topology, reviewing classical results with Number 747 the emphasis on the W-construction and homotopy invariance. May 2002, 136 pages, Softcover, ISBN 0-8218-2792-8, Another chapter describes the (co)homology of operad alge- LC 2002018237, 2000 Mathematics Subject Classification: bras, minimal models, and homotopy algebras. A chapter on 22C05, 17B20; 57R20, 17B40, Individual member $31, List geometry focuses on the configuration spaces and their $52, Institutional member $42, Order code MEMO/157/747N compactifications. A final chapter deals with cyclic and modular operads and applications to graph complexes and moduli spaces of surfaces of arbitrary genus. Mathematical Surveys This item will also be of interest to those working in geometry and Operads in Algebra, Monographs Volume 96 Topology and and topology. Operads in Contents: Part I: Introduction and history; Part II: Operads in a Algebra, Physics symmetric monoidal category; Topology–review of classical Topology Martin Markl, Czech Academy results; Algebra; Geometry; Generalization of operads; Epilog; and Physics Bibliography; Glossary of notations; Index. of Sciences, Prague, Czech Martin Markl Mathematical Surveys and Monographs, Volume 96 Steve Shnider Republic, Steve Shnider, Bar- Jim Stasheff Ilan University, Ramat-Gan, June 2002, approximately 360 pages, Hardcover, ISBN 0-8218- 2134-2, LC 2002016342, 2000 Mathematics Subject THEMAT A IC M A L N ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ A S O C I C R I E E ΑΓΕΩΜΕ T Y M American Mathematical Society Israel, and Jim Stasheff, A F 8 O 88 UNDED 1 Classification: 18D50, 55P48, Individual member $53, List University of North Carolina, $89, Institutional member $71, Order code SURV/96N Chapel Hill Operads are mathematical devices which describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of “homotopy” where they play a key role in organizing hierarchies of higher homotopies. Significant examples first appeared in the sixties though the formal definition and APRIL 2002 NOTICES OF THE AMS 505 New Publications Offered by the AMS Some Generalized EMOIRS Analysis M of the American Mathematical Society Kac-Moody Algebras Volume 157 Number 746 Some Generalized with Known Root Approximation and Kac-Moody Algebras EMOIRS Multiplicities M of the with Known American Mathematical Society Entropy Numbers of Root Multiplicities Volume 157 Number 745 Peter Niemann, Logica UK Ltd, Approximation and Peter Niemann Volterra Operators Entropy Numbers THEMAT A IC London, UK M A L N A S O C I C R I E E T of Volterra Operators M Y A F OU 88 NDED 18 with Application with Application to Contents: Introduction; Generalized to Brownian Motion American Mathematical Society Kac-Moody algebras; Modular forms; Mikhail A. Lifshits Brownian Motion Werner Linde Lattices and their Theta-functions; The EM TH AT A IC M A L N A S O C I C R Mikhail A. Lifshits, Saint I E E T M Y A F OU 88 proof of Theorem 1.7; The real simple roots; Hyperbolic Lie NDED 18 algebras; Appendix A; Appendix B; Bibliography; Notation. Petersburg State University, St. American Mathematical Society Petersburg, Russia, and Memoirs of the American Mathematical Society, Volume 157, Number 746 Werner Linde, Friedrich- Schiller University, Jena, Germany May 2002, 119 pages, Softcover, ISBN 0-8218-2888-6, LC 2002018236, 2000 Mathematics Subject Classification: 17B65, This item will also be of interest to those working in proba- Individual member $30, List $50, Institutional member $40, bility. Order code MEMO/157/746N Contents: Introduction; Main results; Scale transformations; Upper estimates for entropy numbers; Lower estimates for entropy numbers; Approximation numbers; Small ball behav- EMOIRS The Based Ring of iour of weighted Wiener processes; Appendix; Bibliography. M of the American Mathematical Society Two-Sided Cells of Memoirs of the American Mathematical Society, Volume 157, Volume 157 Number 749 Number 745 The Based Ring Affine Weyl Groups of Two-Sided Cells May 2002, 87 pages, Softcover, ISBN 0-8218-2791-X, LC of Affine Weyl Groups 2002018235, 2000 Mathematics Subject Classification: 47G10; ~ of Type An−1 of Type An-1 47B06, 60G15, 47B38, Individual member $29, List $48, Nanhua Xi Nanhua Xi, University of Institutional member $38, Order code MEMO/157/745N THEMAT A IC M A L N A S O C I C R I E E T M Y A F O 88 Sydney, NSW, Australia U NDED 18 American Mathematical Society Contents: Cells in affine Weyl groups; Mathematical Surveys Operators, Functions, and Monographs Type An−1; Canonical left cells; The Volume 93 ∩ −1 Operators, and Systems: An group Fλ and its representation; A bijection between Γλ Γλ Functions, and Irr Fλ; A factorization formula in J ∩ −1; A multiplication and Systems: Easy Reading Γλ Γ λ An Easy Reading formula in J ∩ −1; The based rings − Γλ Γ JΓ ∩Γ 1 and Jc; Bibliog- λ λ λ Volume 2: Model raphy; Index; Notation. Volume 2: Model Operators Operators and Systems Memoirs of the American Mathematical Society, Volume 157, and Systems Number 749 Nikolai K. Nikolski Nikolai K. Nikolski, University May 2002, 95 pages, Softcover, ISBN 0-8218-2891-6, LC of Bordeaux I, Talence, France THEMAT A IC M A L N ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ A S O C I C R I E E ΑΓΕΩΜΕ T Y M American Mathematical Society A F 8 O 88 2002018239, 2000 Mathematics Subject Classification: 20G05, UNDED 1 This unique work combines together 18F25; 16S80, 20C07, Individual member $29, List $48, in two volumes four formally distinct Institutional member $38, Order code MEMO/157/749N topics of modern analysis and its applications: A. Hardy classes of holomorphic functions B. Spectral theory of Hankel and Toeplitz operators C. Function models for linear operators and free interpola- tions, and D. Infinite-dimensional system theory and signal processing Volume I contains parts A and B; this volume, Volume II, contains Parts C and D. Hardy classes of holomorphic functions: This topic is known to be the most powerful tool of complex analysis for a variety of applications, starting with Fourier series, through the Riemann zeta-function, all the way to Wiener’s theory of signal processing. Spectral theory of Hankel and Toeplitz operators: These now become the supporting pillars for a large part of harmonic and complex analysis and for many of their applications. In this book, moment problems, Nevanlinna-Pick and 506 NOTICES OF THE AMS VOLUME 49, NUMBER 4 New Publications Offered by the AMS Carathéodory interpolation, and best rational approximations tical models to random matrices, random permutations, and are considered to illustrate the power of Hankel and Toeplitz number theory. The theory of isomonodromic deformations of operators. systems of differential equations with rational coefficents, and Function models for linear operators and free interpolation: most notably, the related apparatus of the Riemann-Hilbert This is a universal topic and, indeed, is the most influential problem, underlie the analytic side of this striking develop- operator theory technique in the post-spectral-theorem era. In ment. this book, its capacity is tested by solving generalized The contributions in this volume are based on lectures given Carleson-type interpolation problems. by leading experts at the CRM workshop (Montreal, Canada). Infinite-dimensional system theory and signal processing: Included are both survey articles and more detailed exposi- This topic is the touchstone of the three previously developed tions relating to the theory of isomonodromic deformations, techniques. The presence of this applied topic in a pure math- the Riemann-Hilbert problem, and modern applications.
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