Four Applications of Nonlinear Analysis to Physics and Engineering

Abstract

My goal is to describe, in as accessible terms as possible, four separate applications of nonlinear analysis to relativity, elasticity, chaotic dynamics and control theory that I have recently been involved with. The descriptions are in some sense superficial since many interesting technical points are glossed over. However, this is necessary to efficiently convey the flavor of the methods. Most applications of mathematics to "real-life" problems of immediate need do not involve deep methods and ideas. For example, the force exerted on an aircraft frame by the landing gear when the vehicle lands is best computed, at least at first, by using undergraduate mathematics, engineering and experience. However, applied mathematics in the broad sense ranges from such problems of urgency to "practical" problems involving deeper mathematics (compute the lift and flutter characteristics for a design modification of the 747) through to fundamental physical problems involving interactions with the frontier of mathematics that need not be of any immediate "need" (is turbulence predictable from the Navier Stokes equations alone?). The applications I shall speak about are of the fundamental kind involving current research in mathematics and basic questions in physics and engineering that are normally not considered "practical". Most, if not all, of the other lectures I have heard at this conference fall into the same category