Geometry of Derivation with Applications

Acknowledgements

Preface

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 K^a-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. a^a-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

 

4. Galois division rings & applications of multiple replacement
5. p-Adic numbers and Hensel's lemma
6. Field extensions of Qp
7. 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

Bibliography

Index

Biography

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.