Chapter 4 - The Method of Stationary Phase and Applications to Geometry and Analysis on Lie Groups

Abstract

Many problems arise in optics and electromagnetic theory, number theory, probability theory, and quantum physics leads to questions involving integrals of rapidly oscillating functions, especially the asymptotic behavior of these integrals as the frequency parameter becomes large. The methods used in these are invariably what are known as the method of stationary phase that goes back to Stokes and Kelvin. It is equally the case, although less widely known, that oscillatory integrals and their asymptotics play an important role in many more modern parts of mathematics, such as topology of manifolds, geometry and analysis on homogeneous and locally homogeneous spaces, differential equations on complex manifolds, and so on. This chapter provides a brief introduction to oscillatory integrals and the method of stationary phase, and explain how it relates to the various areas of mathematics.