The subject of Points and Lines is incidence geometry; the classical geometries referred to in the subtitle are Lie incidence geometries. Shult has designed the book as a self-contained resource for a graduate student who plans to pursue research in this area.
Shult’s background is in finite group theory, and his treatment is highly algebraic and axiomatic. This approach allows a student to approach open research questions with a minimum of prerequisites. On the other hand, because Schult does not assume his reader has any background in topology or differential and algebraic geometry, historical and topological motivations for constructions can be obscured. For example, Schult does not assume his readers have encountered universal covering spaces, and the explanation he gives for “why universal covering spaces are important” is confined to graph theory and incidence geometries.
In keeping with its role as a self-contained resource, the book gives detailed proofs and offers exercises at the end of each chapter, organized by topic. Diagrams are rather sparse, so geometrically inclined readers will wish to illustrate the text, in addition to working through the official exercises. Though the book’s tone is friendly and even chatty, the intensity of detail and sparsity of illustrations make it more suitable as a handbook for experts and would-be experts than as a first introduction to incidence geometry.
Ursula Whitcher is an assistant professor at the University of Wisconsin-Eau Claire.
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.