Nonlinear Fourier Analysis

In this talk, I will give an overview of spectral theory as a nonlinear analog of Fourier analysis. As an illustration, I will discuss a system of exponentially interacting particles known as the Toda lattice whose solution is based on the spectral theory of Jacobi matrices. I will discuss some classical and more recent results in spectral theory of Jacobi matrices and, in particular, present nonlinear analogs of Riemann-Lebesgue lemma, Parseval's identity, and Paley-Wiener theorem.