Intersection Cohomology

Intersection Cohomology presents an introduction to the mathematical theory developed by Mark Goresky and Robert MacPherson in the 1970s. This text is based on lectures given by Borel at a University of Texas research seminar. The book progresses from foundational concepts through increasingly complex applications in algebraic geometry and topology. The material covers intersection homology groups, characteristic classes, and the decomposition theorem, with detailed proofs and explanations throughout. Dense mathematical notation and formal proofs form the core of this work, making it suitable for graduate students and researchers in topology and related fields. The text assumes familiarity with algebraic topology, sheaf theory, and derived categories. This work stands as a systematic treatment of a mathematical breakthrough that unified elements of topology, geometry, and algebra. The theory presented continues to influence modern developments in representation theory and mathematical physics.

This academic text has very limited online reviews available, making it difficult to gauge broad reader sentiment. Readers note it provides a mathematical introduction to intersection cohomology, though several mention the material requires significant background knowledge in algebraic topology and sheaf theory. A mathematics graduate student on Mathematics Stack Exchange indicated it served as a "helpful companion text" to understanding perverse sheaves. The main criticism cited is that the book assumes too much prerequisite knowledge without adequate explanations of foundational concepts. One reader on ResearchGate mentioned struggling with the "terse presentation style." No ratings or reviews are available on Goodreads or Amazon. The book appears to be primarily used as a reference text in graduate-level mathematics courses rather than receiving broader readership reviews. [Note: Due to this book being a specialized academic mathematics text from 1984, there are very few public reader reviews available to analyze]

Perverse Sheaves and Applications to Representation Theory by Dan Ciubotaru and Peter E. Trapa The text develops perverse sheaves from foundations through applications to geometric representation theory, intersecting with many themes from Borel's work.

D-Modules, Perverse Sheaves, and Representation Theory by Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki This comprehensive treatment connects D-module theory with intersection cohomology and representation theory of Lie algebras.

Introduction to Hodge Theory by José Bertin, Jean-Pierre Demailly, Luc Illusie, and Chris Peters The book builds from classical Hodge theory to mixed Hodge structures and intersection cohomology, providing mathematical context for Borel's work.

An Introduction to Intersection Homology Theory by Frances Kirwan and Jonathan Woolf This text presents the fundamentals of intersection homology theory and its applications to singular spaces.

The Geometry and Topology of Coxeter Groups by Michael W. Davis The book explores geometric group theory and singular spaces through the lens of Coxeter groups, connecting with intersection cohomology methods.

✦ Armand Borel was a Swiss mathematician who worked at the Institute for Advanced Study in Princeton for over 50 years, influencing generations of mathematicians through his research and mentorship. ✦ Intersection cohomology, introduced in the 1970s by Mark Goresky and Robert MacPherson, revolutionized algebraic geometry by providing a way to study singular spaces that preserved important mathematical properties. ✦ The theory presented in this book helped resolve several long-standing conjectures, including the Kazhdan-Lusztig conjecture about representations of Weyl groups. ✦ The book grew out of lectures given by Borel at the Indian Institute of Science in Bangalore in 1984, making advanced mathematical concepts accessible to graduate students. ✦ Intersection cohomology has found applications beyond pure mathematics, including in theoretical physics, particularly in string theory and mirror symmetry.