LAS LOOKING GLASS

The latest textile design pattern from New York’s Fashion Week or the Computer Model of a Recursive Math Equation?

Laura DeMarco, assistant professor of mathematics, explains how that what looks like pop art from the 1960s, or perhaps the latest textile design from New York’s Fashion Week, is actually the modeling of a mathematical equation "perturbed" to its limit.

"A ‘dynamical system’ is any configuration which evolves in time, for example, planetary motion, fluid flow, or the iteration of a simple recursive equation. Even for a simple dynamical system with a simple formula, there can be wildly chaotic behavior that we do not understand. The complication increases dramatically when we allow the formula itself to change in time. There is a general mathematical problem to understand or characterize conditions for "stability" in dynamical systems. That is, given one system, where in space can we predict behavior forever, and where do our predictions fail? Then, if we perturb the system, do the predictions still hold?

"This picture illustrates the "chaotic region" (in blue) for the iteration of one rational function. In other words, any pixel which isn't colored blue has a predictable path of motion. This example is a small perturbation of an even simpler formula which has a very different chaotic region; its blue region is only the outermost fractal ring of this picture. The internal rings suddenly appear with the perturbation, illustrating the instability of the system.

"This particular example is related to my work in complex dynamical systems. I concentrate on rational functions and look specifically at what happens when you perturb and keep perturbing the system all the way to the boundary of allowable systems (like the edge of a universe). In this example, the frontier system is a polynomial of smaller degree, where the chaotic behavior is simplified (the blue rings disappear), but at the expense of a significant loss of "entropy".

"Research in this direction: This picture, called a Julia set, was generated by Elizabeth Russell, a graduate student at Boston University. A group of professors and students there has studied related examples for a number of years now. The motivation is this: if we want to thoroughly analyze a dynamical system in the real world, we create simplified models to test our limits of understanding. These particular examples model complicated real-life ‘large perturbations’ of systems. But these model systems turn out to have their own universe of fascinating possibilities, with links to other areas of mathematics, like geometry or topology or even algebra. That's what drives our research and keeps it exciting for years."

Laura DeMarco

Laura DeMarco (PhD, Harvard University) is an assistant professor in the Department of Mathematics, Statistics and Computer Science with research interests in dynamical systems and complex analysis. DeMarco and her collaborators use many methods to analyze chaotic dynamical methods. Most of the methods are analytic, which means that their roots lie in the subject of calculus and its generalizations. But when the system itself is algebraic, such as those producing the Julia set pictures, DeMarco and her colleagues also try to use algebraic methods. She is the recipient of a 2008 Alfred J. Sloan Fellowship and a $550,000 NSF CAREER Award, which she plans to use "to understand what is special about these algebraic systems and to study some surprising connections between this field and others where the algebra is more apparent." The CAREER project includes a significant educational component, with funding for graduate student workshops, an undergraduate summer project, a new undergraduate course, research seminars and visiting professors.