Poisson Geometry, Deformation Quantisation and Group Representations by J. Rawnsley – An Advanced Mathematics and Physic

Introduction

Poisson geometry stands as one of the most elegant bridges between classical mechanics and modern quantum theory. For Indian students and researchers delving into advanced mathematics and theoretical physics, this discipline offers a rich tapestry of ideas that connect differential geometry, noncommutative algebra, and representation theory. Poisson Geometry, Deformation Quantisation and Group Representations, edited by J. Rawnsley and published by Cambridge University Press, presents a masterful collection of essays from leading researchers. This hardbound volume is an essential resource for graduate students, mathematicians, and physicists seeking a rigorous yet accessible entry point into these interconnected fields.

Book Overview

This book is not a standard textbook but a carefully curated set of five comprehensive contributions, each written by an expert in the field. The chapters cover Poisson geometry and Morita equivalence, formality and star products, Lie groupoids and sheaves, geometric methods in representation theory, and deformation theory in physics modelling. The editor has ensured that each chapter stands alone while collectively building a coherent picture of the subject. The result is a volume that serves both as an introduction for newcomers and a reference for seasoned researchers. The language is clear, the mathematics is precise, and the examples are illuminating, making it suitable for Indian university libraries and personal collections alike.

Key Highlights

• Expert Contributions: Each chapter is authored by a world-renowned mathematician or physicist, offering unique perspectives and deep insights.
• Interdisciplinary Appeal: Bridges pure mathematics (differential geometry, algebra) with applied domains (quantum mechanics, deformation quantisation).
• Self-Contained Chapters: Readers can focus on specific topics without needing to read the entire book sequentially.
• Rigorous Yet Accessible: Written at a level suitable for graduate students, with careful explanations of advanced concepts.
• Hardcover Durability: A sturdy, premium binding from Cambridge University Press, ideal for long-term academic use.

Inside the Book

The volume opens with a chapter on Poisson geometry and Morita equivalence by Bursztyn and Weinstein, which lays the foundational structures. Next, Cattaneo explores formality and star products, connecting deformation quantisation to modern mathematical physics. Moerdijk and Mrcun then present Lie groupoids, sheaves, and cohomology, offering tools for understanding symmetry and singular spaces. Schmid’s contribution on geometric methods in representation theory reveals how Poisson geometry informs the study of Lie groups and their representations. Finally, Sternheimer discusses deformation theory as a powerful tool in physics modelling, tying the mathematical framework to real-world applications. Each chapter includes detailed proofs, examples, and references for further reading.

Key Topics

• Poisson manifolds and their geometric structures
• Morita equivalence for Poisson algebras
• Deformation quantisation and star products
• Formality theorems and Kontsevich’s approach
• Lie groupoids and Lie algebroids
• Sheaf cohomology in differential geometry
• Orbit method and geometric quantisation
• Representation theory of Lie groups
• Deformation theory in classical and quantum physics
• Applications to field theory and string theory

Reader Benefits

Indian students preparing for competitive exams or research in mathematics and physics will find this book a treasure trove of advanced ideas. It helps build intuition for abstract concepts through concrete examples and geometric pictures. Researchers will appreciate the up-to-date references and the clarity with which complex theories are presented. The hardcover format ensures longevity, making it a worthwhile investment for any serious academic library. Moreover, the book encourages cross-disciplinary thinking, a skill increasingly valued in modern science.

Learning Outcomes

By engaging with this book, readers will be able to: understand the foundational principles of Poisson geometry and its role in noncommutative algebra; apply deformation quantisation techniques to problems in quantum mechanics; analyse Lie groupoids and their cohomological properties; use geometric methods to study representations of Lie groups; and recognise how deformation theory models physical phenomena. These outcomes prepare students for advanced research in areas such as symplectic geometry, algebraic geometry, and mathematical physics.

Who Should Read

This book is ideal for graduate students and researchers in mathematics and theoretical physics. It is particularly suited for those specialising in differential geometry, algebra, representation theory, or quantum mechanics. Faculty members teaching advanced courses will find it a valuable resource for lecture preparation. Indian students pursuing PhDs in these fields will benefit from the clarity and depth of the contributions. Additionally, professionals in related areas like string theory or geometric quantisation will find the content directly applicable to their work.

About the Author

J. Rawnsley is a distinguished mathematician known for his contributions to symplectic geometry, representation theory, and mathematical physics. He has held academic positions at leading institutions and has published extensively in top-tier journals. His editorial work on this volume brings together some of the most influential voices in the field, ensuring a cohesive and high-quality presentation. Rawnsley’s own research has helped shape modern understanding of geometric quantisation and Poisson structures.

About the Publisher

Cambridge University Press is one of the oldest and most respected academic publishers in the world. With a history spanning over four centuries, it is renowned for producing authoritative works in science, mathematics, and the humanities. This hardcover edition reflects Cambridge’s commitment to excellence, featuring clear typesetting, durable binding, and meticulous editing. For Indian readers, Cambridge University Press books are a hallmark of quality and scholarly rigour.

Conclusion

Poisson Geometry, Deformation Quantisation and Group Representations is an indispensable addition to any serious mathematics or physics library. It captures the vibrant interplay between geometry, algebra, and physics at a level that is both challenging and rewarding. Whether you are a graduate student embarking on research or an experienced scholar seeking a comprehensive reference, this volume offers lasting value. Order your copy from Bookshops.in today and explore the deep connections that shape modern theoretical science.