Methods of Geometry in the Theory of Partial Differential Equations

Mathematical models are used to describe the essence of the real world, and their analysis induces new predictions filled with unexpected phenomena.



In spite of a huge number of insights derived from a variety of scientific fields in these five hundred years of the theory of differential equations, and its

extensive developments in these one hundred years, several principles that ensure these successes are discovered very recently.



This monograph focuses on one of them: cancellation of singularities derived from interactions of multiple species, which is described by the language of geometry, in particular, that of global analysis.



Five objects of inquiry, scattered across different disciplines, are selected in this monograph: evolution of geometric quantities, models of multi-species in biology, interface vanishing of d - δ systems, the fundamental equation of electro-magnetic theory, and free boundaries arising in engineering.



The relaxation of internal tensions in these systems, however, is described commonly by differential forms, and the reader will be convinced of further applications of this principle to other areas.