The philosophy of deformations: introductory remarks and a guide to this volume

Abstract

One of the more prominent, specifically modern, and pervasive trends in mathematics has to do with perturbations and deformations. Instead of studying one particular model, e.g. one differential equation, or one particular algebra of operators, one is as least as interested in families of these things, and the question of how various properties change as the object under consideration is varied. One reason of this is no doubt the modern emphasis on the tenuous relation (logically speaking) between a mathematical model and the phenomena it is designed to deal with. Thus, to paraphrase Arnol’d, when dealing with models intended to apply to the real world, the question soon arises of choosing those properties of the model which are not very sensitive to small changes in the model and which thus have a chance of representing some properties of the real process.