My research is in dynamical systems (sometimes popularly termed "Chaos Theory"), which seeks to understand how systems change with time and how this evolution can be understood, classified and predicted. This area of research is an exciting interdisciplinary field which relates to, and uses ideas from pure and applied mathematics, physics and computer science. My current work focuses on two broad areas: the study of integrable systems (ordered dynamics based on rotations) and the study of arithmetic dynamics (a hybrid of dynamical systems with number theory). I work with my students, research associates and national and international collaborators.
My research centres around two main (sometimes overlapping) topics in dynamical systems:
Integrable dynamics means “special”, loosely-speaking, and may refer to one or more “special” properties that the dynamics possesses, for example: ability to solve the dynamics; possession of enough constants of the motion or conservations laws; possession of so-called Lax pairs; slow growth of observables under the iterated dynamics; dynamics satisfying a consistency relation or having a lot of symmetry. These properties are non-generic in general dynamical systems. So systems that present with them, integrable systems, have been found to be rich sources of mathematical and physical beauty.
Integrable ordinary differential equations have been studied intensively since the time of Kepler and Newton, integrable partial differential equations since the discovery of solitons in the Korteweg-de Vries equation by Kruskal and collaborators. The question thus arises how to generalize this theory and its applications from the continuous case (integrable differential equations) to the discrete case (integrable difference equations), i.e. when all independent variables are discrete. Discrete integrable systems are currently the focus of many international research efforts, including my own.
Arithmetic dynamics is an exciting breakthrough science, merging various areas of mathematics (dynamics, number theory, algebra, geometry). This research area is rapidly expanding, has enormous potential for cross-fertilization, and is rich in important applications (cellular automata, pseudo-randomness and cryptography). Whereas the dynamics of maps over the continuum (of the real or complex numbers) has received enormous attention, the study of iterated maps over finite sets is relatively new and offers distinct challenges because some strictures from the continuum do not apply. Together with international collaborators, my research has used orbit statistics as a key to characterising and understanding the dynamics of birational maps. We have found universal distributions in the orbit statistics, related them to dynamical structure of the map, and then built probabilistic and combinatorial models to reproduce these distributions.
I am very happy to supervise Honours projects in the broad area of dynamical systems. Recent projects include:
Likewise, I would love to talk to (good!) prospective PhD students interested in discrete integrable systems and/or arithmetic dynamics. Thesis titles of my recent students include:
Dynamical systems is a wonderful area to work in. It affords an opportunity to do computer experimentation, look for patterns and test conjectures and, ultimately, to prove theorems that explain what you observe. Systems evolving with time are ubiquitous, from celestial mechanics [the historical origin of dynamical systems] to the stock market to mathematical biology to cryptosystems. Research in mathematics is becoming increasingly a cross-pollination of ideas and techniques from various mathematical subdisciplines, and nowhere is this more evident than in dynamical systems! Please contact me if you wish to know more.
MATH3201 Dynamical Systems and Chaos
Current: Member, EDI Committee, Australian Mathematical Society
Former: Deputy Head of School (2012-2020); School Research Director (2007-2020); Vice-President, Australian Mathematical Society (2016-2017); Chief Investigator of the ARC Centre of Excellence MASCOS
2012 Foreign Sabbatical Fellowship, Spanish Ministry of Science (Universitat Autonoma Barcelona)
2009 Isaac Newton Institute Visiting Fellow (Cambridge University)
2006 IAS Visiting Fellow (La Trobe University)
2006 AMSI/MASCOS Research Fellow
2004 UNSW John Yu Fellow