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While studying physics at the Technische Universität Berlin, I was awarded a Fulbright Scholarship, which led to me obtaining an M.A. in mathematics from the University of Maryland in College Park. I have been at Tufts University since completing my doctorate at the California Institute of Technology in 1989. During research leaves I have held visiting appointments at the Institut des Hautes Etudes Scientifiques near Paris, the Institut de Recherche Mathématique Avancée and the Collège Doctoral Européen in Strasbourg, the Eidgenössische Technische Hochschule in Zürich, and the Graduate School of Mathematical Sciences of the University of Tokyo. I have also held the Jean Morlet Chair at the Centre International de Rencontres Mathématiques in Marseille, and I have served within the governance of the American Mathematical Society, the American Statistical Association and the Mathematical Association of America. I serve as founding editor of the Journal of Modern Dynamics, Editor in Chief of Electronic Research Announcements in Mathematical Sciences, and on the Steering Committee of the Boston Higher Education Innovation Council. In the Faculty of Arts, Sciences and Engineering at Tufts I have chaired numerous standing faculty committees as well as committees created for accreditation and for strategic planning. My administrative roles at Tufts University have been chair of the Department of Mathematics and Associate Provost.
Dynamical systems is the mathematical theory of systems, such as in classical dynamics, that evolve in time. The motivations for this field come from classical and celestial mechanics as well as, more recently, from population dynamics, meteorology, economics, physiology, neuroscience, medicine, genomics, i.e. all across the natural and social sciences. In the modern theory of these systems the central aim is often to answer qualitative questions about long-term behavior directly from a study of the governing laws rather than through explicit expressions of the evolution itself. The grandest of these questions is whether the solar system is stable (or whether instead, the earth might, without external influences, leave the solar system for good at some point). Just over 100 years ago this question started the modern theory of dynamical systems and led to the first glimpse of "chaos", and it led to another quantum leap, the KAM-theory of "order", in the 1950s. "Chaos theory" is a popular name for the study of dynamics in which the cumulative effects of the tiniest discrepancies grow exponentially over time and give rise to behavior that looks random and unpredictable. (If two people with different calculators start with \(x=0.3\), compute \(4 \cdot x \cdot (1-x)\), and repeatedly apply the same formula to the output of the previous step, they will likely quite soon find wild mismatches between their results due to accumulated deviations from differences in rounding.) Hyperbolic dynamics is the mathematical study of these systems, and much of my work has been in uniformly hyperbolic dynamics, which represents this sensitivity to initial conditions in the purest and strongest way and has been called the crown jewel of dynamical systems.
For a pure mathematician this is a beautiful subject to study because in these systems the truly messy long-term behavior of any particular time evolution coexists with smooth and orderly global structures in the space of possible states whose study on one hand provides a way of understanding the possible long-term behaviors and on the other hand can provide insights into the origins of the system, such as, whether it arises from an algebraic system. It is an exciting field for me because in addition to intrinsic beauty it provides interactions with other fields of mathematics (mostly differential geometry, but also number theory and coding theory, for example). Moreover, the study of hyperbolic dynamics blends into the study of real-world systems with complex behavior, providing methods and concepts to the sciences for describing, classifying, studying and understanding chaotic behavior, such as the intrinsic difficulty of weather prediction or the diagnosis of an impending heart attack from the decreasing (!) complexity of a patient's heart beat, but also the exquisitely subtle design of chaotic trajectories for recent spacecraft missions.Research topics >
Doctoral dissertations supervised >
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