Dynamical Systems: Connecting Theory to Applications

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Dynamical systems, a branch of mathematics concerned with the analysis of time evolutions of processes, is widely recognized as a basic tool for science. The research here is aimed at connecting dynamical systems theory to its potential applications. It will lead to significant advances within the field, greatly expanding its scope. It will promote the cross-fertilization between mathematics and other scientific disciplines including the biological sciences, focusing in this research on the mathematics of epidemiology and neuroscience. The research projects will offer training opportunities for students and junior scientists, and the principal investigator, a female mathematician, will serve as role model and contribute to diversity within the scientific work force. The research seeks to connect dynamical systems theory to applications in two different ways. One is to improve the applicability of existing theory by incorporating ideas of relevance (such as random noise and finite-time effects) and by introducing spatial and temporal scales to classical concepts. The principal investigator will work toward increasing the pool of concrete examples to promote the usefulness of the field. The other is to use dynamical systems tools to tackle problems that arise in the biological sciences. There are two specific topics in this research. One concerns the timeliness of the response to epidemic outbreaks; the problem is cast in the framework of infinite dimensional dynamical systems defined by delay differential equations. Another project analyzes, as a paradigm of the competition ubiquitous in biology, the dynamical interaction between excitatory and inhibitory neurons in local circuits of the cerebral cortex.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.