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Home > Mathematics and Science Textbooks > Mathematics > Geometry > Points and Lines: Characterizing the Classical Geometries(Universitext)
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Points and Lines: Characterizing the Classical Geometries(Universitext)

Points and Lines: Characterizing the Classical Geometries(Universitext)


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About the 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 Several chapters have new heretofore unpublished research results.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.

Table of Contents:
I.Basics.- 1 Basics about Graphs.- 2 .Geometries: Basic Concepts.- 3 .Point-line Geometries.-4.Hyperplanes, Embeddings and Teirlinck's Eheory.- II.The Classical Geometries.- 5 .Projective Planes.-6.Projective Spaces.- 7.Polar Spaces.- 8.Near Polygons.- III.Methodology.- 9.Chamber Systems and Buildings.- 10.2-Covers of Chamber Systems.- 11.Locally Truncated Diagram Geometries.-12.Separated Systems of Singular Spaces.- 13 Cooperstein's Theory of Symplecta and Parapolar Spaces.- IV.Applications to Other Lie Incidence Geometries.- 15.Characterizing the Classical Strong Parapolar Spaces: The Cohen-Cooperstein Theory Revisited.- 16.Characterizing Strong Parapolar Spaces by the Relation between Points and Certain Maximal Singular Subspaces.- 17.Point-line Characterizations of the “Long Root Geometries”.- 18.The Peculiar Pentagon Property.

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.

Review :
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 Grundhofer, Mathematical Reviews, Issue 2011 m)


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Product Details
  • ISBN-13: 9783642156267
  • Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
  • Publisher Imprint: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Height: 235 mm
  • No of Pages: 676
  • Returnable: Y
  • Sub Title: Characterizing the Classical Geometries
  • ISBN-10: 3642156266
  • Publisher Date: 20 Dec 2010
  • Binding: Paperback
  • Language: English
  • Returnable: Y
  • Series Title: Universitext
  • Width: 155 mm


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