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Points and Lines

Characterizing the Classical Geometries

  • Textbook
  • © 2011

Overview

Authors:
  1. Ernest Shult

    1. Kansas State University, Manhattan, USA

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  • Contains theorems characterizing the classical projective spaces, polar spaces and geometries of exceptional Lie groups, which are combined with proofs in one source
  • Contains an entirely new treatment of buildings
  • Students can follow all the proofs from elementary first principles without any appeal to external publications
  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xxii
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  2. Basics

    1. Front Matter

      Pages 1-1
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    2. Basics About Graphs

      • Ernest E. Shult
      Pages 3-41

    3. Geometries: Basic Concepts

      • Ernest E. Shult
      Pages 43-58

    4. Point-Line Geometries

      • Ernest E. Shult
      Pages 59-77

    5. Hyperplanes, Embeddings, and Teirlinck’s Theory

      • Ernest E. Shult
      Pages 79-101

  3. The Classical Geometries

    1. Front Matter

      Pages 103-103
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  4. The classical geometries.

    1. Projective Planes

      • Ernest E. Shult
      Pages 105-133

    2. Projective Spaces

      • Ernest E. Shult
      Pages 135-165

    3. Polar Spaces

      • Ernest E. Shult
      Pages 167-249

    4. Near Polygons

      • Ernest E. Shult
      Pages 251-287

  5. Methodology

    1. Front Matter

      Pages 289-289
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    2. Chamber Systems and Buildings

      • Ernest E. Shult
      Pages 291-397

    3. 2-Covers of Chamber Systems

      • Ernest E. Shult
      Pages 399-413

    4. Locally Truncated Diagram Geometries

      • Ernest E. Shult
      Pages 415-440

    5. Separated Systems of Singular Spaces

      • Ernest E. Shult
      Pages 441-453

    6. Cooperstein’s Theory of Symplecta and Parapolar Spaces

      • Ernest E. Shult
      Pages 455-494

  6. Applications to Other Lie Incidence Geometries

    1. Front Matter

      Pages 495-495
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  7. Applications to other Lie incidence geometries.

    1. Characterizations of the Classical Grassmann Spaces

      • Ernest E. Shult
      Pages 497-526

    2. Characterizing the Classical Strong Parapolar Spaces: The Cohen–Cooperstein Theory Revisited

      • Ernest E. Shult
      Pages 527-552
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Keywords

About this book

The classical geometries of points and lines include not only the projective and polar spaces, but similar truncations of geometries naturally arising from the groups of Lie type. Virtually all of these geometries (or homomorphic images of them) are characterized in this book by simple local axioms on points and lines. Simple point-line characterizations of Lie incidence geometries allow one to recognize Lie incidence geometries and their automorphism groups. These tools could be useful in shortening the enormously lengthy classification of finite simple groups. Similarly, recognizing ruled manifolds by axioms on light trajectories offers a way for a physicist to recognize the action of a Lie group in a context where it is not clear what Hamiltonians or Casimir operators are involved. The presentation is self-contained in the sense that proofs proceed step-by-step from elementary first principals without further appeal to outside results. Several chapters have new heretofore unpublished research results. On the other hand, certain groups of chapters would make good graduate courses. All but one chapter provide exercises for either use in such a course, or to elicit new research directions.

Reviews

From the reviews:

“In this expansive volume, Shult (Kansas State Univ.) attempts to provide a thorough, self-contained characterization of geometries of Lie type by means of local axioms on points and lines. … There is enough material here for several semester-long graduate courses; alternatively, the book could be used as a reference work. Shult makes every effort to remain chatty … and intuitive while ensuring precision and detail. Summing Up: Recommended. Graduate students and researchers.” (S. J. Colley, Choice, Vol. 48 (11), July, 2011)

“This book presents characterizations of the classical geometries of Lie type by axioms on points and lines; it is a teaching book with detailed proofs, written for beginning graduate students. … Most chapters end with a set of exercises … . The whole book shows the author’s love of incidence geometry and of teaching.” (Theo Grundhöfer, Mathematical Reviews, Issue 2011 m)

“Shult has designed the book as a self-contained resource for a graduate student who plans to pursue research in this area. … the book gives detailed proofs and offers exercises at the end of each chapter, organized by topic. … geometrically inclined readers will wish to illustrate the text, in addition to working through the official exercises. … the intensity of detail and sparsity of illustrations make it more suitable as a handbook for experts … .” (Ursula Whitcher, The Mathematical Association of America, July, 2012)

Authors and Affiliations

  • Kansas State University, Manhattan, USA

    Ernest Shult

About the author

Ernest Shult studied finite groups with Michio Suzuki, and held visiting fellowships at the University of Chicago and the Princeton Institute for Advanced Study in the 1960’s. He continued to contribute to finite groups until he got interested in incidence geometry. In 1987-8, he received a US Scientist Award from the Alexander von Humboldt Foundation in Freiburg Germany.

Bibliographic Information

  • Book Title: Points and Lines

  • Book Subtitle: Characterizing the Classical Geometries

  • Authors: Ernest Shult

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-3-642-15627-4

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2011

  • Softcover ISBN: 978-3-642-15626-7Published: 20 December 2010

  • eBook ISBN: 978-3-642-15627-4Published: 13 December 2010

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: XXII, 676

  • Number of Illustrations: 88 b/w illustrations

  • Topics: Geometry, Topological Groups, Lie Groups

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