1st Edition

Geometry of Derivation with Applications

By Norman L. Johnson Copyright 2023
    372 Pages
    by Chapman & Hall

    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.



    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 G(a, b)F contains (a, b)Z(F )
      8. Artin-Wedderburn theorem & 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
      3. Flocks of D-cones
      4. j—planes and twisted hyperbolic flocks
      5. Joint theory of a-flocks
      6. The Ka-Klein quadric
      7. Baer theory
      8. Quasi-flocks
      9. The Baer forms
      10. Algebraic and a-Klein methods
      11. Infinite flocks of hyperbolic quadrics

    Part 7. Lifting

    Chapter 9. Chains & surjectivity of degree 1/k

    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

    Part 8. 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

    Part 10. Multiple replacement theorem

    Chapter 12. The general theorem

      1. Skewfields of finite dimension/Fixo
      2. (a,b)K-inner automorphisms 
      3. Automorphisms of infiinite order
      4. (a,b)K/K^(o)xk^K(p)-outer automorphisms 
      5. Cyclic algebras and additive quasifibrations
      6. Cyclic division ring automorphisms

    Part 11. Classification of subplane covered nets

    Chapter 13. Suspect subplane covered nets



      1. Ambient theory for subplane covered nets
      2. Fundamentals of Kummer theory
      3. Galois theory for division rings

      5. Galois division rings & applications of multiple replacement
      6. p-Adic numbers and Hensel's lemma
      7. Field extensions of Qp
      8. The quaternions division rings (a,b)Qp

    Part 12. Extensions of skewfields

    Chapter 14. Quaternion division ring extensions

      1. Ore's method of localization of a ring
      2. Skew polynomial rings
      3. Generalized cyclic algebras
      4. Extending division rings using rational function fields
      5. Derivable nets over twisted formal Laurent series
      6. Lifted semifields in PG(3,L(p))
      7. A garden of division rings & multiple replacements

    Chapter 15. General ideas on Klein extensions

      1. Klein varieties
      2. Translation planes admitting twisted pseudo-regulus nets
      3. Skewfield a-flocks
      4. Skewfield 'cyclic' translation planes
      5. Spreads of large dimension
      6. Finite and infinite theories of flocks
      7. Division ring connnections
      8. Group theoretic differences
      9. Specific differences in the theory
      10. Hyperbolic flocks, finite and infinite
      11. Remarks on corrections to earlier work




    Norman L. Johnson is an Emeritus Professor (2011) at the University of Iowa where he has had ten PhD students. He received his BA from Portland State University, MA from Washington State University and PhD 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 Researchgate.net. 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 and enjoys studying languages and 8-ball pool.