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Operads in Algebra, Topology and Physics
Martin Markl, Czech Academy of Sciences, Prague, Czech Republic, Steve Shnider, Bar-Ilan University, Ramat-Gan, Israel, and Jim Stasheff, University of North Carolina, Chapel Hill, NC
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Mathematical Surveys and Monographs
2002; 349 pp; softcover
Volume: 96
ISBN-10: 0-8218-4362-1
ISBN-13: 978-0-8218-4362-8
List Price: US$98 Member Price: US$78.40
Order Code: SURV/96.S

Topology, $$C^*$$-Algebras, and String Duality - Jonathan Rosenberg

--Bulletin of the London Mathematical Society

Operads are mathematical devices that 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 from algebraic topology first appeared in the sixties, although the formal definition and appropriate generality were not forged until the seventies. In the nineties, a renaissance and further development of the theory were inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, symplectic and contact geometry, combinatorics, knot theory, moduli spaces, cyclic cohomology, and, last but not least, theoretical physics, especially string field theory and deformation quantization.

The book contains a detailed and comprehensive historical introduction describing the development of operad theory from the initial period when it was a rather specialized tool in homotopy theory to the present when operads have a wide range of applications in algebra, topology, and mathematical physics. Many results and applications currently scattered in the literature are brought together here along with new results and insights. The basic definitions and constructions are carefully explained and include many details not found in any of the standard literature.

Graduate students, research mathematicians, and mathematical physicists interested in homotopy theory, gauge theory, and string theory.

Reviews

-- Bulletin of the London Mathematical Society

"The first book whose main goal is the theory of operads per se ... a book such as this one has been long awaited by a wide scientific readership, including mathematicians and theoretical physicists ... Written in a way to stimulate thought and abundant in references, spanning from 1898 through 2001, the book under review is guaranteed to contribute to the constant quest of mathematics for novel ideas and effective applications ... a great piece of mathematical literature and will be helpful to anyone who needs to use operads, from graduate students to mature mathematicians and physicists."

-- Mathematical Reviews, Featured Review

Part I
• Introduction and history
Part II
• Operads in a symmetric monoidal category
• Topology-review of classical results
• Algebra
• Geometry