Speaker: Stephen Casey (American University)

Title: Sampling and Tomography in Euclidean and non-Euclidean Spaces

Abstract: We discuss harmonic analysis in the settings of both Euclidean and non-Euclidean spaces, and then focus on two specific problems using this analysis - ampling theory and network tomography. These show both the importance of non-Euclidean spaces and some of the challenges one encounters when working in non-Euclidean geometry. Starting with an overview of surfaces, we demonstrate the importance of hyperbolic space in general surface theory,and then develop harmonic analysis in general settings, looking at the Fourier-Helgason transform and its inversion. We then focus on sampling and tomography.

Sampling theory is a fundamental area of study in harmonic analysis and signal and image processing. We connect sampling theory with the geometry of the signal and its domain. It is relatively easy to demonstrate this connection in Euclidean spaces, but one quickly gets into open problems when the underlying space is not Euclidean. We discuss how to extend this connection to hyperbolic geometry and general surfaces, outlining an Erlangen-type program for sampling theory.

The second problem we discuss is network tomography. We demonstrate a way to create a system that will detect viruses as early as possible and work simply on the geometry or structure of the network itself. Our analysis looks at weighted graphs and how the weights change due to an increase in traffic. The analysis is developed by applying the tools of harmonic analysis in hyperbolic space.