Memoirs of the American Mathematical Society 2014; 74 pp; softcover Volume: 232 ISBN10: 0821898450 ISBN13: 9780821898451 List Price: US$65 Individual Members: US$39 Institutional Members: US$52 Order Code: MEMO/232/1089
Not yet published. Expected publication date is October 24, 2014.
 Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all \(m\times n\) matrices over a division ring \(\mathbb{D}\) which preserve adjacency in both directions. Motivated by several applications the author studies a long standing open problem of possible improvements. There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes? A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings the author solves all three problems simultaneously, thus obtaining the optimal version of Hua's theorem. In the case of general division rings he gets such an optimal result only for square matrices and gives examples showing that it cannot be extended to the nonsquare case. Table of Contents  Introduction
 Notation and basic definitions
 Examples
 Statement of main results
 Proofs
 Bibliography
