Derived Algebraic Geometry: An Elemental Approach by Renaud Gauthier – A Comprehensive Advanced Mathematics Textbook on

Introduction

Derived Algebraic Geometry: An Elemental Approach by Renaud Gauthier is a rigorous and comprehensive guide to one of the most advanced and exciting fields in modern mathematics. Published by de Gruyter, this second edition offers an updated and expanded treatment of derived algebraic geometry, spectral algebraic geometry, and their connections to theoretical physics. Designed for serious students and researchers, this hardcover volume bridges abstract theory with concrete applications, making it an essential resource for Indian mathematicians and physicists working at the cutting edge of algebraic geometry, homotopy theory, and mathematical physics.

Book Overview

This book presents a self-contained journey through the foundational concepts of derived algebraic geometry. Starting from schemes and simplicial sets, the author builds up to higher categories, model categories, and the rich structures of derived and spectral algebraic geometry. The second edition includes a new chapter on derived motivic spectra, an expanded introduction to infinity categories, and a revised chapter on stacks. The text also delves into motives, Goodwillie calculus, higher Galois theory, supersymmetry, and topics in physical mathematics, offering a panoramic view of the subject.

Key Highlights

• Comprehensive Second Edition: Updated with a new chapter on derived motivic spectra and an extended introduction to infinity categories.
• Self-Contained Approach: Builds from basic schemes and simplicial sets to advanced topics like spectral algebraic geometry and higher categories.
• Interdisciplinary Scope: Connects derived algebraic geometry with supersymmetry, Goodwillie calculus, and physical mathematics.
• Rigorous Yet Accessible: Written for graduate students and researchers with a solid background in algebra and topology.
• Premium Hardbound Edition: Durable and long-lasting, ideal for library or personal reference.

Inside the Book

The book is structured to guide the reader from classical foundations to modern developments. Early chapters cover schemes, simplicial sets, and model categories, laying the groundwork for higher category theory. Later chapters explore derived algebraic geometry, spectral algebraic geometry, stacks, and motives. The new chapter on derived motivic spectra introduces cutting-edge material, while the revised chapter on stacks clarifies key concepts. Each chapter includes detailed proofs, examples, and exercises to reinforce understanding.

Key Topics

• Schemes and Simplicial Sets: Foundational tools for algebraic geometry and homotopy theory.
• Higher Categories and Infinity Categories: Extended introduction to these essential structures.
• Model Categories: Framework for homotopy theory in derived contexts.
• Derived Algebraic Geometry: Core theory and applications.
• Spectral Algebraic Geometry: Modern approach using spectra.
• Stacks and Derived Motivic Spectra: Revised and new chapters for deeper insight.
• Motives, Goodwillie Calculus, and Higher Galois Theory: Advanced topics linking geometry, topology, and number theory.
• Supersymmetry and Physical Mathematics: Connections to theoretical physics.

Reader Benefits

• Deep Conceptual Understanding: Gain a thorough grasp of derived algebraic geometry from the ground up.
• Research-Ready Knowledge: Equip yourself with the latest tools and theories for original research.
• Cross-Disciplinary Skills: Apply concepts from homotopy theory, category theory, and physics to algebraic geometry.
• Updated Content: Benefit from the author’s revisions and new chapters that reflect recent advances.
• Indian Context: Ideal for students and faculty at Indian universities pursuing advanced mathematics and mathematical physics.

Learning Outcomes

By the end of this book, readers will be able to navigate the landscape of derived algebraic geometry with confidence. They will understand the role of infinity categories and model categories in modern geometry, work with spectral algebraic structures, and explore motives and Goodwillie calculus. The book also prepares readers to engage with current research in supersymmetry and physical mathematics, making it a valuable stepping stone for doctoral work and beyond.

Who Should Read

This book is intended for graduate students, postdoctoral researchers, and professional mathematicians in algebraic geometry, topology, and category theory. It is also highly relevant for theoretical physicists working in supersymmetry, string theory, or quantum field theory who seek a rigorous mathematical foundation. Indian readers pursuing advanced degrees at institutions like IISc, TIFR, IITs, and central universities will find this text an indispensable companion.

About the Author

Renaud Gauthier is a mathematician known for his contributions to derived algebraic geometry and its interfaces with theoretical physics. His research spans higher categories, motives, and supersymmetry, and he brings a clear, pedagogical style to complex material. This second edition reflects his ongoing engagement with the field and his commitment to making advanced topics accessible.

About the Publisher

de Gruyter is a renowned academic publisher with a legacy spanning over 270 years. Based in Berlin, it is celebrated for publishing high-quality research in mathematics, science, and the humanities. This hardcover edition upholds de Gruyter’s tradition of excellence, ensuring that Indian students and researchers receive a reliable and authoritative text.

Conclusion

Derived Algebraic Geometry: An Elemental Approach, Second Edition, is a masterful synthesis of foundational and cutting-edge material. Whether you are a student embarking on advanced study or a researcher seeking a comprehensive reference, this book offers clarity, depth, and inspiration. Order your copy from Bookshops.in today and add this essential volume to your mathematical library.