Dr. Geddes applies the tools and techniques of modern dynamical systems theory to a variety of problems in science and engineering, especially those in mathematical biology.
An active area of interest is that of blood flow in the microvascular system. Blood often displays regular and irregular oscillations in flow velocity and red blood cell concentration as it propagates through a microvascular network. While there are active biological control elements at work, such as changes in the diameter of a vessel, passive properties of the network, such as hydraulic resistance, may be enough to account for these oscillations. Dr. Geddes and his colleagues and students at Olin College and the University of New Hampshire are currently designing fluid networks in order to test the predictions of the mathematical model.
A complimentary area of active interest is that of structural adaptation of microvascular networks. These networks are not static - they evolve over time in response to hemodynamic and biochemical stimuli. Dr. Geddes and his colleagues and students at Olin College are modeling the effect of endothelial dysfunction on microvascular remodeling in obesity, diabetes, and hypertension.
An earlier (and now inactive) area of interest is pulse dynamics in mode-locked lasers. Mode-locking is a technique used by laser physicists to produce very short pulses, sometimes on the order of a couple of femto-seconds. Mode-locked lasers can, however, exhibit complicated and seemingly turbulent dynamics. Dr. Geddes’ work in this area, conducted in collaboration with colleagues at Union College and Strathclyde University in Scotland, has focused on a detailed examination of the dynamics of pulse formation in a detuned, modulated laser.
Finally, as a graduate student at the University of Arizona, Dr. Geddes wrote his dissertation on pattern formation in nonlinear optical systems. Spontaneous pattern formation is ubiquitous in nature. Examples include the stripes on a zebra, wind-driven sand ripples, and hexagonal patterns in Rayleigh-Benard convection. While pattern formation in fluids is a well-established field, it was only in the late 1980's that seminal experiments on pattern formation in laser physics and nonlinear optics were being conducted. These experiments suggested that roll and hexagonal pattern formation could be observed in optics, but on time-scales measured in nano-seconds as opposed to tens of seconds typical of fluids experiments.