Despite their critical importance in the modern world, most prior research into the properties of complex networks has been done in an abstract setting, with characterizing quantities developed from topology limited entirely to generic networks. The major impediment to a deeper understanding of real biological distribution and functional networks has been that the promotion of simpler-to-understand two-dimensional networks, as in plant leaves, to fully realized three-dimensional complex systems, as in human vasculature, is fundamentally non-trivial. Consequently there have been no appropriate quantitative tools to characterize vasculature in a physically relevant mathematically compact way; this characterization is a necessary prerequisite to the development of predictive models for the effects of vascular network architecture and topology.
In order to fill this void, we have developed a classification algorithm that generalizes functional two-dimensional network characterization to a fully three-dimensional setting, applicable to many problems across all kinds of real distribution networks. This new algorithm can distinguish network architectures and capacity arrangements that originated in subtly different ways and has the potential to teach us a great deal about many critical and widely varied complex networks, from the function and vascular-associated dysfunction of mammalian neurovasculature to the complexly interwoven bile canaliculi in the liver to stress and load-bearing network structures in biofilms and more.
We have also been at the forefront of recent advances in shape programmable soft materials, playing a central role in the invention and development of thin sheet nematic solid systems that can be “blueprinted” with pre-determined nematic director fields that give rise to externally switchable, reversible shape change, potentially at the sub-micron scale. These systems lend themselves to a panoply of device design opportunities relevant to systems biology, biophysics, and medical applications, including soft robotics, drug delivery and encapsulation methods, peristaltic pumps and gateway switching for lab-on-a-chip and soft microfluidics. Furthermore, the theory and methods underlying these shape programmable materials may serve as a powerful model for complex shape determination in developing biological systems as well.
Our group seeks to leverage principles and methods of applied topology and geometry in order to better understand complex biological phenomena, with a particular focus on the role of network complexity in these systems. Some future projects include: