# Dynamics and Related Topics

### Overview

The study of dynamical systems has had a long and distinguished history in mathematics. This study has ranged from applications involving differential equations and information theory, to more theoretical work focusing on systems with topological or algebraic structure. In the past few decades this field has grown dramatically, and completely new directions have opened up. One of the most significant recent developments has been the intense study of the joint action of several commuting transformations, or Zd-action. Although a comprehensive theory is currently out of reach, one class of such actions has been analyzed in great detail by Klaus Schmidt and his co-workers. These are the algebraic Zd-actions, which offer a rich and beautiful connection with commutative algebra and algebraic geometry, which are now being mined.

Another aspect of actions of higher-rank groups is the theory of aperiodic tilings and quasicrystals. The subject began with the work of Raphael Robinson and Roger Penrose, who showed that there were polygonal shapes that would tile Euclidean space, but only in aperiodic ways. The resulting tilings, however, are far from being completely random, and show a great deal of regularity. The dynamical point of view, where the transformations are shifts of tilings, has been particularly successful in analyzing these new structures. A singular physical discovery, namely crystal-like structures called quasicrystals which violated the traditional crystallographic rules, has led to very fruitful interactions with physical scientists and mathematicians.

During the last century, deep and fruitful connections were developed between dynamics and operator algebras. In recent years, interactions between topological dynamics and C*-algebras has grown with the development of tools in Connes's program for non-commutative geometry. This has been particularly evident in the theory of aperiodic order, largely through Bellissard's work connecting tilings, C*-algebras, K-theory, and solid state physics and the recent solution of the so-called gap-labeling problem.

There are quite a number of researchers dynamical systems and related areas at PIMS sites. Their interests are diverse, but they share expertise and create connections. The University of Washington has built up one of the strongest groups in dynamical systems in the world. It includes Christopher Hoffman, Douglas Lind, Steffen Rohde, Boris Solomyak, and Selim Tuncel. In addition, Manfred Einsiedler (a former Ph.D. student of Klaus Schmidt) will be starting a three-year appointment beginning the fall of 2002. Also Klaus Schmidt is planning to spend a sabbatical year in Seattle during 2002-03, and also possibly one or both of the adjacent summers. A number of other visitors in dynamics are expected that year as well, including Mike Boyle (Maryland), Christopher Denninger (Muenster), William Parry (Warwick), and Daniel Rudolph (Maryland).

At the University of Victoria, Ian Putnam works specifically in the area of interactions between operator algebras and dynamics. John Phillips and Marcelo Laca both work in areas of operator algebras that have strong ties with dynamics. Chris Bose works in ergodic theory. Rod Edwards works in the dynamics of biological systems and may have some interaction.

At the University of Alberta, Bob Moody works on aperiodic order and, to some extent, Al Weiss, also. Anthony Lau and Volker Runde work in operator algebras. At the University of Calgary, Berndt Brenken, Igor Nikolaev and Michael Lamoureux work in areas of operator algebras very closely linked with dynamics.

### Faculty

- U. Alberta: Robert Moody, Anthony Lau, Volker Runde, Al Weiss
- U. Calgary: Michael Lamoureux, Berndt Brenken, Igor Nikolaev
- U. Washington: Douglas Lind, Christopher Hoffman, Douglas Lind, Steffen Rohde, Boris Solomyak, Selim Tuncel, Manfred Einsiedler.
- U. Victoria: Ian Putnam, John Phillips, Marcelo Laca, Chris Bose, Rod Edwards
- Visitors and other contributors: Klaus Schmidt (Vienna), Mike Boyle (Maryland), Christopher Denninger (Muenster), William Parry (Warwick), Daniel Rudolph (Maryland).