Many of the classical partial differential equations of mathematical physics (electromagnetics, fluids, acoustics, heat flow, etc.) can be reformulated in an equivalent integral equation form. The rapid numerical solution of the resulting integral equations require several tools: quadrature for singular functions, robust methods for handling complicated geometry, and fast algorithms for dense linear algebra (e.g. butterfly schemes and fast multipole methods). My research focuses on developing numerical schemes to address problems in all of these areas, as well as formulating novel integral equations which remedy analytical drawbacks of existing ones. Recently, I have also been applying many of these fast algorithms and analytical techniques to problems in computational statistics, namely Gaussian processes.