Maurer-Cartan Methods in Deformation Theory: The Twisting Procedure

This text provides a comprehensive overview of the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics, with a focus on the twisting procedure. It includes a novel approach to operads and applications in higher category theory and deformation theory.

Format: Paperback / softback
Length: 150 pages
Publication date: 07 September 2023
Publisher: Cambridge University Press

This comprehensive text offers a unique perspective on the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics, covering a wide range of topics. It presents a novel conceptual approach to the twisting procedure, guiding readers through various versions with the aid of ample motivating examples for both graduate students and researchers. The book delves into various subjects, including a novel approach to the twisting procedure for operads, which leads to Kontsevich graph homology. Additionally, it provides a detailed description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras, utilizing the largest deformation gauge group ever considered. The book concludes with concise overviews of recent applications in areas such as higher category theory and deformation theory.

Introduction:
The Maurer-Cartan methods have played a pivotal role in the development of modern mathematics, particularly in the fields of algebra, geometry, topology, and mathematical physics. These methods provide a powerful tool for studying the properties and structures of various mathematical objects, and have been applied to a wide range of problems throughout the mathematical community.

Scope of the Text:
This text aims to provide a comprehensive overview of the Maurer-Cartan methods in these four areas. It covers a wide range of topics, including the twisting procedure, operads, Kontsevich graph homology, (homotopy) associative algebras, and Lie algebras. The book is designed to be accessible to both graduate students and researchers, with a focus on providing clear explanations and examples that help to deepen the reader's understanding of these methods.

Novel Approach to the Twisting Procedure:
One of the key contributions of this text is its novel approach to the twisting procedure. The twisting procedure is a fundamental tool in the study of operads, which are a generalization of linear algebraic structures. The book presents a new conceptual treatment of the twisting procedure, guiding readers through various versions with the help of plentiful motivating examples. These examples are particularly useful for graduate students who are new to the field, as they help to illustrate the practical applications of the twisting procedure and its relationship to other areas of mathematics.

Applications to Kontsevich Graph Homology:
Another important topic covered in this text is Kontsevich graph homology. Kontsevich graph homology is a branch of mathematics that studies the homology of certain types of graphs, particularly those associated with operads. The book provides a detailed description of the twisting procedure for operads, leading to Kontsevich graph homology. This description is based on the use of the largest deformation gauge group ever considered, which allows for a more precise and efficient computation of homology.

Applications to (Homotopy) Associative Algebras and Lie Algebras:
In addition to Kontsevich graph homology, the text also discusses the twisting procedure for (homotopy) associative algebras and Lie algebras. These algebras are a generalization of associative algebras and Lie algebras, respectively, and are important in various areas of mathematics, including algebraic topology, mathematical physics, and higher category theory. The book provides a concise description of the twisting procedure for these algebras, utilizing the largest deformation gauge group ever considered.

Conclusion:
This text offers a comprehensive and up-to-date overview of the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics. It provides a novel conceptual treatment of the twisting procedure, guiding readers through various versions with the aid of ample motivating examples. The book also covers important applications to Kontsevich graph homology, (homotopy) associative algebras, and Lie algebras, utilizing the largest deformation gauge group ever considered. By providing a clear and concise explanation of these methods, this text is an invaluable resource for both graduate students and researchers in these fields.

Weight: 278g
Dimension: 152 x 228 x 11 (mm)
ISBN-13: 9781108965644