Speaker: Tyrus Berry (George Mason University)

Title: Frame theory and a global approach to the exterior calculus

Abstract: Multivariable calculus studies vector fields and associated operators such as the gradient and divergence in Euclidean space. This generalizes to smooth Riemannian manifolds as the exterior calculus. Recently there has been interest in defining discrete analogs of the exterior calculus on simplicial complexes. In this talk we go even further and present a generalization of the exterior calculus to graphs (without the extra structure of a complex). To achieve this, the exterior calculus on smooth manifolds is first reformulated entirely in terms of the eigenvalues and eigenfunctions of the Laplacian operator. The key to this global approach to manifolds is representing objects (functions, vector fields, operators, etc.) in a frame (dependent spanning set) instead of a basis. We call this reformulation the Spectral Exterior Calculus (SEC). The primary goal of the talk is to explain why frame theory is the natural setting for analysis on manifolds and then introduce the SEC. We then transfer this formulation to a graph using the eigenvalues and eigenvectors of the graph Laplacian. In numerical experiments we show that coarse-grained topological features of a graph are reflected in the SEC, in direct analogy to classical results in differential geometry.