March 24, 2008
Michael Shapiro
Title: On Basic Ideas of Hypercomplex Analysis
"Hypercomplex analysis" is a generic name for those generalizations of one-dimensional complex analysis which involve hypercomplex numbers. Quaternionic analysis is the oldest and the most known version of it. In the talk, it will be discussed, first of all, in which sense quaternionic analysis is a "proper" or a "closest" version in low dimensions which includes as particular cases, or sub-theories, such classic theories as vector analysis and holomorphic mappings in two complex variables, as well as some systems of partial differential equations. This allows one, by developing quaternionic analysis, to obtain new results for the above classic theories and to refine known ones; some applications of this approach will be presented. Some comments on Clifford analysis and its applications will be also made.