Professor Emeritus, Norman Johnson, has published a new book titled Geometry of Derivation with Applications. Here is what he has to say about the book:
After about a ten-year hiatus from mathematics, I started rereading some of my previous work. I basically started over with a new perspective. So, I spent the last three or four years learning enough non-commutative algebra to connect with geometry in order to create this new book.
I came to the University of Iowa in 1969, after meeting Robert Oehmke at a meeting at Chicago Circle that connected various aspects of geometry. Oehmke worked in non-associative algebras and I was interested in connecting the finite geometry that I had learned from T.G. Ostrom, my advisor at Washington State University. I never did get around to that—until this new book.
I was hired for a one year visiting position after Oehmke, Erwin Kleinfeld, Ostrom and a few others stood on a street corner near Chicago Circle and basically discussed whether I was any good. The word from one of the participants got back to me, when the fellow said that he would have been happy with the assessment.
So certain algebras can act as coordinate structures for affine and projective planes. In 1964 Ostrom discovered “derivation” which transformed an affine plane admitting a certain class of subplanes to a “new” plane, a startling fact back then. The subplanes are contained in a so-called “derivable” net. All of this demanded a finite setting.
In the new book, I was able to give a classification of derivable nets by embedding the structure into 3-dimensional projective space—and the work is independent of cardinality. Given any skewfield, there is a derivable net that may use the skewfield as a fundamental coordinate system that provides an understanding of what Ostrom was actually doing more than 60 years ago.
The most fundamental type of skewfield (division ring) is the quaternion division ring. It turns out that the affine plane coordinatized by any quaternion division ring is derivable and creates some interesting affine planes in a non-commutative setting.
I would like to add that I did not even have a computer when I started this project. I am grateful to Brian Bacher who found and loaned me three different laptops that I seem to have burned through. I also thank Weimin Han for additional assistance with the book.
Many thanks to the University of Iowa Mathematics Department!