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Hyperplane Arrangements

An Introduction

  • Textbook
  • © 2017

Overview

Authors:
  1. Alexandru Dimca

    1. Université Côte d’Azur, Nice, France

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  • Provides a clear and rapid introduction to many hot topics in the field of hyperplane arrangements
  • Introduces basic definitions and key notions and takes the reader right up to open questions
  • Contains a wealth of exercises of varying levels of difficulty

Part of the book series: Universitext (UTX)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xii
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  2. Invitation to the Trip

    • Alexandru Dimca
    Pages 1-14

  3. Hyperplane Arrangements and Their Combinatorics

    • Alexandru Dimca
    Pages 15-43

  4. Orlik–Solomon Algebras and de Rham Cohomology

    • Alexandru Dimca
    Pages 45-60

  5. On the Topology of the Complement \(M({\mathscr {A}})\)

    • Alexandru Dimca
    Pages 61-84

  6. Milnor Fibers and Local Systems

    • Alexandru Dimca
    Pages 85-98

  7. Characteristic Varieties and Resonance Varieties

    • Alexandru Dimca
    Pages 99-123

  8. Logarithmic Connections and Mixed Hodge Structures

    • Alexandru Dimca
    Pages 125-149

  9. Free Arrangements and de Rham Cohomology of Milnor Fibers

    • Alexandru Dimca
    Pages 151-185

  10. Back Matter

    Pages 187-200
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Keywords

About this book

This textbook provides an accessible introduction to the rich and beautiful area of hyperplane arrangement theory, where discrete mathematics, in the form of combinatorics and arithmetic, meets continuous mathematics, in the form of the topology and Hodge theory of complex algebraic varieties.


The topics discussed in this book range from elementary combinatorics and discrete geometry to more advanced material on mixed Hodge structures, logarithmic connections and Milnor fibrations. The author covers a lot of ground in a relatively short amount of space, with a focus on defining concepts carefully and giving proofs of theorems in detail where needed. Including a number of surprising results and tantalizing open problems, this timely book also serves to acquaint the reader with the rapidly expanding literature on the subject.


Hyperplane Arrangements will be particularly useful to graduate students and researchers who are interested in algebraic geometry or algebraic topology. The book contains numerous exercises at the end of each chapter, making it suitable for courses as well as self-study.

Reviews

“In this book, Alexandru Dimca provides a readable and concise introduction to the theory of hyperplane arrangements. ... the textbook is well written and contains the necessary information for an introduction to the modern study of hyperplane arrangements. Those new to the area will benefit from gaining enough knowledge to proceed to research problems quickly, and those who have studied hyperplanes for longer will benefit from the carefully curated list of references to current research.” (Kristopher Williams, Mathematical Reviews, 2018)



“In the book under review, the author provides an interesting introduction to the theory of hyperplane arrangements. … It is worth pointing out that the text is accessible even for advanced undergraduate students (especially the first three chapters) and motivating since the book focuses on current research problems and provides some open problems. … this is a very nice introduction to the subject.” (Piotr Pokora, zbMATH 1362.14001, 2017)

Authors and Affiliations

  • Université Côte d’Azur, Nice, France

    Alexandru Dimca

About the author

Alexandru Dimca is a world-leading authority in Singularity Theory and Hyperplane Arrangements, with a strong track record of ground-breaking research. He is the author of four books and over 120 research papers, many of them devoted to the topics discussed in this book. He has an established reputation for his clear writing style, and his vast teaching experience helps him to convey the main ideas in an accessible and efficient way.

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