Emmy Noether Lectures

2004 Lecturer: Svetlana Katok

Symbolic dynamics for geodesic flows

Abstract:
At the dawn of modern dynamics, in the 1920s, E. Artin and M. Morse discovered that geodesics on surfaces of constant negative curvature may be described by sequences of symbols via certain “coding” procedures. They found one of the first instances of what much later became widely known as “chaotic” behavior. Artin’s code of geodesics on the modular surface is closely related to continued fractions. Morse’s procedure is more geometric and more widely applicable. For 80 years these classical works provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. Major contributions were made by R. Bowen, C. Series, R. Adler and L. Flatto who interpreted and expanded the classical works in the modern lan guage of symbolic dynamics.
Quite surprisingly, there was room for new results in this well-developed area. Even more surprisingly, Gauss reduction theory leads to a variant of continued fractions that provides a particularly elegant coding of geodesics on the modular surface. This, in turn, brings about new connections with topological Markov chains that, mysteriously, are related to the five Platonic solids.
Brief Biography:
Svetlana Katok grew up in Moscow in an environment saturated by mathematics: family, mathematical circles at the university, special mathematical schools, mathematical olympiads. She was especially influenced by her father Boris Rosenfeld, a renowned geometer and one of the most distinguished historians of science in the world. At the early age of thirteen, she decided to become a professional mathematician. She earned an M.A. with honors from Moscow State University in 1969. Her first published paper, based on her master’s thesis, was reviewed by Jurgen Moser. However, due to the anti-Semitic and anti-intelligentsia policies of the time, she was denied admission to the university Ph.D. program and worked for several years in the area of early and secondary mathematical education.
After emigrating to the United States in 1978, she returned to research mathematics and entered the Ph.D. program at the University of Maryland. She changed her research area from dynamical systems to number theory and completed her degree under Don Zagier in 1983. She was awarded an NSF postdoc and was associated with Caltech and four campuses of the University of California before moving to Penn State in 1990, where she was promoted to full professor in 1993. Her mathematical interests center on the interaction between number theory, geometry and dynamical systems with the latter field, her first mathematical specialty, coming to the fore in the last decade.
She has a life-long interest in mathematical education, which has borne such diverse fruits as innovative programs for primary school students from the Soviet period, the popular graduate text Fuchsian Groups and the unique MASS program for undergraduates at Penn State that she created together with her husband Anatole Katok. The topics of her many invited talks are quite varied, ranging from her mathematical research to models for integrating research into the undergraduate experience. In 1995 she foundedERA-AMS, the first electronic-only AMS journal, and is managing editor of the journal. She has been a member of the Editorial Board of the Journal of the Institute of Mathematics of Jussieusince 2000. She has served on many AMS, NSF and NRC committees and panels and was a Member-at-Large of the AMS Council for 1993-1996. In 2001 she received the Eberly College of Science Alumni Society Distinguished Service Award.
She has three children whose careers range from operations research to software development and architecture to classical singing.