Steven Frankel, assistant professor of mathematics and statistics in Arts & Sciences at Washington University in St. Louis, received a CAREER Award from the National Science Foundation for his project "Universal Circles Between Dynamics and Geometry."
Frankel studies the relationships between the shape of a space, its symmetries and the kinds of dynamical systems it supports. His previous research showed that certain kinds of flows must always have orbits that close up on themselves to form loops.
"This project builds on these ideas, while asking a complementary question: what can this sea of orbits tell you about the shape of your space?" Frankel said.
"At first, talking about flows sounds like it's very specific, perhaps only useful in understanding the way a physical fluid might behave," he said. "In fact, virtually any phenomenon or system that changes with time can be thought of as a flow on a space. You just need to expand your concept of a 'space' from the physical notion of something inhabitable to something more abstract: a state space whose points correspond to possible instantaneous configurations of the underlying phenomenon or system."
"Virtually any phenomenon or system that changes with time can be thought of as a flow on a space."
Steven Frankel
"Systems of particles, predator-prey interactions and neural networks can all be thought of in this way," Frankel said.
For the educational component of his NSF CAREER effort, Frankel proposed an interdisciplinary project that he calls the State Space Project, drawing together faculty from Arts & Sciences and the McKelvey School of Engineering.
"The idea of the State Space Project is to use these unifying ideas from mathematical dynamical systems as an organizing principle to help people working in all sorts of fields - chemistry, neuroscience, physics, etc. - share ideas, and also to take advantage of some of the tools and perspective of mathematics."
