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MATH 613: Topics in Analysis (Spring 2007)Laplacian: the triangle at the center of mathematics Instructor: Robert Strichartz Prerequisites: MATH 611 No text. One of the central concepts in mathematics is the Laplacian, which appears in many different guises. The purpose of this course is to introduce you to some of them and to see some connections among them. We will try to cover a lot of material, and we will try not to get bogged down in technical details. So this will be something of a "survey course." We will look at Laplacians on Riemannian manifolds, graphs, and fractals, and also the Hodge-deRham Laplacian on differential forms. We will begin with the theory of spherical harmonies, which relates to the Laplacian on the round sphere. We will emphasize fundamental ideas such as energy, harmonic functions, and the spectrum (eigenvalues and eigenfunctions). We will look at estimates for the bottom of the spectrum, as well as the asymptotic distribution of eigenvalues (Weyl's law). We will look at solutions of differential equations involving the Laplacian, and this will lead to the theory of Sobolev spaces and pseudodifferential operators. We will also investigate differential equations related to the Laplacian: the heat equation, the wave equation, and Schrödinger's equation. Last modified:October 31, 2006 |
