Geometry of Derivation with Applications  book cover
1st Edition

Geometry of Derivation with Applications

  • Available for pre-order on May 16, 2023. Item will ship after June 6, 2023
ISBN 9781032349169
June 6, 2023 Forthcoming by Chapman & Hall
404 Pages

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USD $120.00

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Book Description

Geometry of Derivation with Applications is the fifth work in a longstanding series of books on combinatorial geometry (Subplane Covered Nets, Foundations of Translation Planes, Handbook of Finite Translation Planes, and Combinatorics of Spreads and Parallelisms). Like its predecessors, this book will primarily deal with connections to the theory of derivable nets and translation planes in both the finite and infinite cases. Translation planes over non-commutative skewfields have not traditionally had a significant representation in incidence geometry, and derivable nets over skewfields have only been marginally understood. Both are deeply examined in this volume, while ideas of non-commutative algebra are also described in detail, with all the necessary background given a geometric treatment.

The book builds upon over twenty years of work concerning combinatorial geometry, charted across four previous books and is suitable as a reference text for graduate students and researchers. It contains a variety of new ideas and generalizations of established work in finite affine geometry and is replete with examples and applications.

Table of Contents




Part 1. Classical theory of derivation

Chapter 1. Coordinate methods

    1. Translation planes and quasifibrations
    2. Quasifields
    3. Left quasifields
    4. T -extension

Chapter 2. Embedding theory of derivable nets

    1. Co-dimension 2 construction
    2. Structure theory and contraction of embedded nets
    3. Embedding of subplane covered nets
    4. Transversals to derivable nets

Part 2. Classifying derivable nets over skewfields

Chapter 3. Fundamentals & background

    1. Uniform representation for quaternion division rings
    2. Quaternion division ring planes
    3. Matrices and determinants over skewfields
    4. Classifying derivable nets

Chapter 4. Classification theory over skewfields

    1. Notation
    2. Extension of skewfields theorem/Skewfield bimodules
    3. Preliminary types 1, 2, 3
    4. Standard framework
    5. Generalized quaternions over skewfields
    6. Matrix skewfields are generalized quaternion
    7. Generalized (a, b)F contains (a, b)Z(F )
    8. Brauer groups
    9. Extending skewfields

Part 3. Types i of derivable nets

Chapter 5. The types


    1. Type 0
    2. Double regulus type 0 derivable nets
    3. The ambient space
    4. Derivable nets of type 3
    5. Order in type 3 derivable nets
    6. Derivable nets of type 2
    7. Fake type 2 derivable nets
    8. Open form derivable nets of type 2
    9. Order in type 2 derivable nets
    10. Derivable nets of type 1
    11. Examples of type 1 derivable nets
    12. Carrier nets
    13. Derivable nets in translation planes

Part 4. Flocks of a-cones

Chapter 6. Klein quadric and generalization

    1. a-Klein quadric
    2. Construction of general flocks
    3. The field case
    4. Algebraic construction for a-cones
    5. Elation groups and flokki planes
    6. Maximal partial spreads and a-flokki
    7. The second cone
    8. Baer groups for flokki Planes
    9. q-Flokki and lifting
    10. Collineations and isomorphisms of a-flokki planes

Part 5. Flock geometries

Chapter 7. Related geometries

    1. Kantor's coset technique
    2. Quasi-BLT-sets
    3. s-Inversion & s-square
    4. A census
    5. Quasi-flock derivations
    6. Herds of Hyperovals
    7. Hyperbolic fibrations
    8. The correspondence theorem
    9. Flocks to cyclic planes.

Part 6. Twisted hyperbolic flocks

Chapter 8. Hyperbolic flocks and generalizations.

    1. Algebraic theory of twisted hyperbolic flocks
    2. Simultaneous a-Flocks & twisted hyperbolic spreads


  1. Flocks of D-cones
  2. j planes and twisted hyperbolic flocks
  3. Joint theory of a-flocks
  4. The Ka-Klein quadric
  5. Baer theory
  6. Quasi-flocks
  7. The Baer forms
  8. Algebraic and a-Klein methods
  9. Infinite flocks of hyperbolic quadrics

Part 7. Lifting

Chapter 9. Chains & surjectivity of degree 1

1. Restricted surjectivity

2. Hughes-Kleinfield look-alikes

3. The remaining quasifibrations of dimension 2

4. Large dimension quasifibrations

5. T -copies of generalized twisted field planes

Part8. Lifting skewfields

Chapter 10. General theory

1. Matrix forms and replacement

2. The general skewfield spread

3. Generalized quaternion division rings

4. Retraction

Part 9.Bilinearity

Chapter 11. General bilinear geometries

    1. Star flocks and rigidity
    2. Bilinear a-flocks
    3. Bilinear flocks of quadratic cones
    4. Translation planes admitting SL(2, K)
    5. Double covers
    6. nm-Linear flocks of quadratic cones
    7. Nests of reguli
    8. Group replaceable translation planes
    9. Circle geometry over K(✓--)
    10. aa-1 -nest planes
    11. Flocks of elliptic quadrics
    12. Klein quadric and Pappian spreads
    13. n-Linear elliptic flocks
    14. Tangential packings of ovoids
    15. Part 10. Multiple replacement theorem


      Chapter 11. General bilinear geometries

    16. Star flocks and rigidity
    17. Bilinear a-flocks
    18. Bilinear flocks of quadratic cones
    19. Translation planes admitting SL(2, K)
    20. Double covers
    21. nm-Linear flocks of quadratic cones
    22. Nests of reguli
    23. Group replaceable translation planes
    24. Circle geometry over K(✓--)
    25. aa-1 -nest planes
    26. Flocks of elliptic quadrics
    27. Klein quadric and Pappian spreads
    28. n-Linear elliptic flocks
    29. Tangential packings of ovoids

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Norman L. Johnson is an Emeritus Professor (2011) at the University of Iowa where he has had ten Ph.D. students. He received his BA from Portland State University, MA from Washington State University and Ph.D. also at Washington State University as a student of T.G. Ostrom. He has written 580 research items including articles, books, and chapters available on  Additionally, he has worked with approximately 40 coauthors and is a previous Editor for International Journal of Pure and Applied Mathematics and Note di Matematica.  Dr. Johnson plays ragtime piano, enjoys studying languages and 8-ball pool.