In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as occur in the immersion problem, isometric immersion problem, and other areas.
The theory was started by works of Yakov Eliashberg, Mikhail Gromov and Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. These started with the Whitney–Graustein theorem, and followed the work of Stephen Smale and Morris W. Hirsch on immersions; also work by Nicolaas Kuiper and John Forbes Nash.
Read more about Homotopy Principle: Rough Idea, Ways To Prove The H-principle, Some Paradoxes
Famous quotes containing the word principle:
“To invent without scruple a new principle to every new phenomenon, instead of adapting it to the old; to overload our hypothesis with a variety of this kind, are certain proofs that none of these principles is the just one, and that we only desire, by a number of falsehoods, to cover our ignorance of the truth.”
—David Hume (1711–1776)