Speaker: Dr. Oscar Bruno, California Institute of Technology
Title: Vibrations are everything: a visit to the world of eigenfunctions
Abstract: Just as is the case for a stretched guitar string, for which any deformation can be viewed as a combination of sines and cosines of a multitude of amplitudes and frequencies (that is, it can be expressed as a combination of "pure oscillations", or "pure vibrations", or "eigenfunctions"), physics seems determined that the same should happen with absolutely everything: from sound and light, to matter and time, and including laws that govern them, our understanding of the physical world is based, at a fundamental level, on consideration of eigenfunctions. With this motivation in mind we explore some areas in the mathematics of the world of vibrations, and we consider, in particular, a certain special combination of eigenfunctions: the Green function. After reviewing basic ideas concerning eigenfunctions, we will mention recent numerical methods in which the eigenfunctions are computed precisely using the aforementioned Green function. Based on Green's functions, we will build, study and use eigenfunctions related to various physical problems, including, for example, oscillations describing disturbances on the surface of wine in a glass (which are given by Steklov eigenfunctions), the interaction of light with nano-optical devices and the design of such devices (Maxwell eigenfunctions), quantum mechanics and sound propagation (Schrodinger and Helmholtz eigenfunctions), problems related to probability theory (eigenfunctions of Laplace and fractional Laplacian operators), and, notably, the temporal sinusoidal vibrations themselves. We will also briefly consider certain geometric characteristics of the eigenfunctions, such as the distribution of the corresponding eigenvalues (oscillation frequencies) and their "nodal curves"--that is, the points in space where the eigenfunctions vanish--including a discussion, with some details, of the intriguing nodal lines of the Steklov eigenfunctions related to the aforementioned wine-filled glass.