## ABSTRACT

* 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.

Acknowledgements. Preface. Part 1. Classical theory of derivation. Chapter 1. Coordinate methods. Chapter 2. Embedding theory of derivable nets. Part 2. Classifying derivable nets over skewfields. Chapter 3. Fundamentals & background. Chapter 4. Classification theory over skewfields. Part 3. Types *i* of derivable nets. Chapter 5. The types. Part 4. Flocks of *a*-cones. Chapter 6. Klein quadric and generalization. Part 5. Flock geometries. Chapter 7. Related geometries. Part 6. Twisted hyerbolic flocks. Chapter 8. Hyperbolic flocks and generalizations. Part 7. Lifting. Chapter 9. Chains & surjectivity of degree 1/*k*. Lifting skewfields. Chapter 10. General theory. Part 9. Bilinearity. Chapter 11. General bilinear geometries. Part 10. Multiple replacement theorem. Chapter 12. The general theorem. Part 11. Classification of subplane covered nets. Chapter 13. Suspect subplane covered nets. Part 12. Extensions of skewfields. Chapter 14. Quaternion division ring extensions. Chapter 15. General ideas on Klein extensions. Bibliography. Index.