Complex Singularities in Numerical Analysis and Nonlinear Dynamics

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This project will investigate the precise analytic form of the complex singularities ofthe Lorenz system, the figure-eight solution of the three-body problem, and signals obtainedfrom the Navier-Stokes boundary layer. Both analytic and numerical techniqueswill be used. Research on the Lorenz system has the aim of understanding the analyticcontinuation of the Lorenz solutions to the entire complex plane. With regard to thefigure-eight solution, the project will show that the complex singularities of this systemhave an appealingly simple structure. The Navier-Stokes boundary layer has beenstudied primarily using techniques from spectral analysis. We will locate and elucidatesingularities of the analytic continuation of the signal in the complex plane, and andobtain a measurement as well as an understanding of the time scales imposed by theouter flow.This project endeavors to benefit the public in two respects. Firstly, we aim to obtainnew insights into the Navier-Stokes boundary layer which is of immense importance inengineering and meteorology. As an example of its importance, we mention that amajor part of the energy intake of automobiles is dissipated in the boundary layer.Secondly, we will write a new book that puts computer architecture at the heart ofscientific computing. This book will introduce a style of scientific computing that isdeeply informed by recent progress in computer architecture to a wider audience.