Topological methods in low-dimensional dynamics

Abstract

Dynamical systems on two-dimensional manifolds have been the focus of intense interest and activity for the entire history of modern dynamical systems theory. These systems can be of almost unimaginable complexity, yet there are many mathematical tools available which allow for a fairly complete understanding of their dynamical behavior. In addition, these systems are studied because of their physical importance in areas such as forced oscillations and fluid mixing. Foremost in the topological methods for dynamics on surfaces are the collections of methods and ideas surrounding Thurston's classification theorem of surface isotopy classes. Prof's Boyland and de Carvalho are both leading world experts in these and related methods. One main goal of this proposal is to provide an opportunity for them to combine their complementary areas of expertise during an extended visit of Boyland to USP. Plans include joint research projects and interaction with mathematics graduate students focused around a jointly taught graduate course in surface dynamics whose notes will form a book to be submitted to the next Brazilian Colloquium of Mathematics. Boyland has also made significant contributions to understanding the dynamics on the annulus and torus, especially to systems of physical interest. Another main purpose of his visit is to foster collaboration with USP Prof's Zanata and Tal who have recently made great progress in areas of Boyland's previous work. Finally, Boyland also worked in one-dimensional dynamics and classical and fluid mechanics. During his visit it is also anticipated that he will have significant interaction with the of USP mathematicians working in these areas especially Prof. Vargas. In addition to these specific projects, Boyland will give a research level mini-course “The pseudo-Anosov Spectrum and its Applications” and be a resource for the graduate students in Dynamical Systems. He will also visit other major mathematics institutions in Brazil with active Dynamics groups including IMPA and UFF where he will give lectures and explore potential collaborations. Prof. Boyland is an internationally recognized expert who has made significant contributions to many areas of low-dimensional dynamics including dynamics on hyperbolic surfaces, on the torus and annulus, and on the circle, as well as on dynamical systems coming from classical and fluid mechanics. All these topics are active areas of research at Instituto de Matemática e Estatística and so the research accomplishments and mathematics graduate education of USP would greatly benefit by his one semester visit. (AU)