For many decades great mathematical interest has focused on problems associated with linear operators and the extension of the well-known results of linear algebra to an infinite-dimensional context. This interest has been crowned with deep insights, and the substantial theory that has been developed has had a profound influence throughout the mathematical sciences. However when one drops the assumption of linearity, the associated operator theory and the many concrete problems associated with such a theory represent a frontier of mathematical research. Nonetheless, the fundamental results so far obtained in this direction already form a deep and beautiful extension of this linear theory. Just as in the linear case, these results were inspired by and are highly relevant to concrete problems in mathematical analysis. The object of the lectures represented here is a systematic description of these fundamental nonlinear results and their applicability to a variety of concrete problems taken from various fields of mathematical analysis.

Here I use the term “mathematical analysis” in the broadest possible sense. This usage is in accord with the ideas of Henri PoincarC (one of the great pioneers of our subject). Indeed, by carefully scrutinizing the specific nonlinear problems that arise naturally in the study of the differential geometry of real and complex manifolds, classical and modern mathematical physics, and the calculus of variations, one is able to discern recurring patterns that inevitably lead to deep mathematical results. From an abstract point of view there are basically two approaches to the subject at hand. The first, as mentioned above, consists of extending specific results of linear functional analysis associated with the names of Fredholm, Hilbert, Riesz, Banach, and von Neumann to a more general nonlinear context. The second approach consists of viewing the subject matter as an infinite-dimensional version of the differential geometry of manifolds and mappings between them. Obviously, these approaches are closely related, and when used in conjunction with modern topology, they form a mode of mathematical thought of great power.