CCMS Colloquium: Dynamical Systems and the Period 3 Implies Chaos Theorem (Michelle Manes, AIM)

Speaker: Michelle Manes, AIM

Title: Dynamical Systems and the Period 3 Implies Chaos Theorem

Abstract: Sharkovskii’s theorem, sometimes called “period 3 implies chaos,” concerns a one-dimensional real dynamical system: a function from the real line to itself that you iterate and study the orbits of points. This theorem is fascinating because of its very simple hypotheses, its surprisingly strong conclusion, and its clever and elementary proof that uses not much more than the intermediate value theorem and some careful bookkeeping. Like Sharkovskii’s theorem, my mathematics research lives in the world of iterated dynamical systems, and I’ll end by telling you a bit about some interesting number theoretic questions one can ask (and sometimes answer) in this context.

Bio: Michelle Manes received her AB in mathematics from UC Berkeley, an MEd in Deaf Education from Boston University, and an ScM and PhD in mathematics from Brown University. She was a professor in the mathematics department at the University of Hawaii for 15 years before leaving to join the American Institute of Mathematics (in Pasadena) as Deputy Director. She has also worked as a program officer at the National Science Foundation. She has taught mathematics at every level from 3rd grade through graduate school and has co-authored textbooks for middle grades mathematics, high school geometry, linear algebra, and preservice elementary teachers.

Dr. Manes does research in number theory, primarily in the field of arithmetic dynamics. She has authored over 30 articles, most of them in collaboration with some of her best friends. She believes deeply in collaboration, mentoring, and professional service. She has served on committees for the Association for Women in Mathematics (AWM), the Mathematical Association of America (MAA), and the American Mathematical Society (AMS). She currently serves as the Associate Secretary for the Western Section of the AMS and as an editor-in-chief for La Matematica, the flagship journal of the AWM. She has received numerous accolades including the Regents Medal for Excellence in Teaching at the University of Hawaii and the Haimo Award for Distinguished College or University Teaching of Mathematics from the MAA Golden Section and an AWM Service Award. She has been named a Fellow of the AWM and of the AMS.