The objective of the cooperative research project is to apply categorical methods of modern representation theory to classification problems of linear algebra. Professors V.V. Sergeichuk (Ukraine) and R.A. Horn (Utah) will develop a universal method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings, and will apply this new method to known canonical form problems. They will also give miniversal deformations of the canonical matrices that they obtain; that is, they will find a simplest possible normal form, to which not only a given matrix, but also an arbitrary family of matrices close to it, can be reduced by means of an admissible transformation that depends smoothly on the entries of the matrix entries.
Elaboration of general methods for solving classification problems of linear algebra is of great practical importance since matrix problems underlie most methods of modern computational mathematics. Students in science and engineering first meet matrix problems when they study systems of linear equations. An investigation of miniversal deformations of matrices is important for applications in which one has matrices that arise from physical measurements, which means that their entries are known only approximately. Thus, one is compelled to study the structure, not only of a given matrix, but also of all matrices close to it.
