Dynamical Systems

The theory of dynamical systems is a branch of mathematics that has been created in order to describe, explain, and predict the behavior of mathematical models of various phenomena from the experimental sciences. Those models are systems that evolve in time. This evolution is determined by the state of the system at a given moment, and the laws for the evolution, specific for the system. Those laws may be given by a system of differential equations, or just by one transformation that is iterated.

In order to discover and examine possible interesting features of various systems, mathematicians have introduced many systems that are not models of any concrete real phenomena. One can study them by purely theoretical methods or investigate them on computers, or, very often, both. This approach has turned out to be successful. For instance, it has led to the discovery of chaotic phenomena. Those phenomena were subsequently found in most of the experimental sciences and this has had a tremendous impact on the way scientists think. It is now widely recognized that the deterministic nature of a system does not imply long-term validity (the best example is weather prediction), except in a statistical sense.

William Geller

Prof. Geller's research primarily relates to low dimensional, topological, and symbolic dynamics, with increasing connections to asymptotic geometry/topology and geometric group theory. Some current projects include: an investigation into the effect of the asymptotic geometry of large random and nonrandom finite graphs on the existence of critical behavior for threshold dynamics on the graphs (with computational assistance by B. Ramsey); an attempt to extend work of Ceccherini-Silberstein, Machi, Scarabotti, and of Gromov on cellular automata on the Cayley graph of an amenable group to address a conjecture of Furstenberg on the nonamenable case (with T. Sinclair); and a program to complete the classification of the lamplighter groups up to quasi-isometry, with possible broader dynamical implications

Bruce Kitchens

Prof. Kitchens is interested mainly in symbolic dynamics. Given any discrete system, one can divide the phase space into pieces, assign a symbol to each of them, and trace the orbits of the system via sequences of those symbols. This leads to the investigation of the symbolic systems. Those systems are also extremely important because of their applications in building actual communication and information storage devices. While employed at IBM, he worked on the design of communication channels for magnetic disks using ideas developed for symbolic dynamics. Bruce investigates symbolic systems from both topological and measure (ergodic theory) points of view. Currently he works on algebraic actions of the group Zd; that is, systems for which the phase space is a compact topological group and the "time" is d-dimensional. The goal of his work is to construct a "dictionary" between the dynamics of the Zd-action and the algebraic properties of the module of characters

Michal Misiurewicz

Prof. Misiurewicz is interested mainly in low-dimensional discrete systems. Those systems are simple enough to be examined more thoroughly than more complicated ones, yet on the other hand, complex enough to display all general types of behavior, including chaos (in fact, it was the study of those systems where the term "chaos" started to be used in the modern sense). He also works with higher dimensional systems and applications. Recently he started with Alexander Blokh a big project on the investigation of two-dimensional systems with branching points, similar to one-dimensional piecewise expanding maps. Those systems exhibit a surprisingly wide variety of behaviors. Misiurewicz collaborates with William Geller, and with mathematicians from other universities.