Mini-workshop in Symmetries and Stability

University of Canberra, 26-29 June 2007

Overview

The Australian Mathematical Sciences Institute in collaboration with the Australian Defence Force Academy and the University of Canberra will be hosting the Mini-workshop Symmetries and Stability from June 26 to June 29, 2007 in Canberra, Australia. The theme of the meeting will be the interaction of symmetry and stability in the study of differential equations.

Symmetry is a central idea throughout mathematics and physics. Galois’ seminal idea to study the group that permutes the roots (solutions) of a polynomial equation was extended to the solutions of differential equations by Sophus Lie in the late ninteenth century and has been profoundly developed since that time. Nowadays, the notion of a Lie group is ubiquitous in many quantitative fields including dynamical systems in the form of equivariant bifurcation theory, theoretical chemistry in the form of quasi-solvable potentials in Schrodinger’s equation, and in innumerable areas of physics. For many years and particularly since the 1960s, mathematicians have been developing and using Lie group methods to study problems of importance to soil physics, continuum mechanics, integrable systems, pattern formation and differential geometry to name but a few.

This Mini-workshop will seek to bring together a variety of mathematicians working in these and related fields to discuss the latest advances, particularly in bifurcation theory and in symmetry reduction, broadly understood. A fundamental fact, discovered by Lie, is that the solutions of a differential equation that are invariant under its Lie symmetry group can be constructed by solving ordinary differential equations. However, Lie did not touch on the important notion of the stability of such solutions. Stability of solutions of differential equations will be a major concern of the meeting.

Program & Invited Speakers

Invited Speakers included:

Prof. George Bluman, University of British Columbia, Canada
Dr. Georg Gottwald, University of Sydney
Prof. Bjorn Sandstede, University of Surrey, United Kingdom
Dr. Claudia Wulff, University of Surrey, United Kingdom

Topics

The workshop included topics in the following general areas:

1) Qualitative analysis of differential equations, for example

(a) pattern formation and nonlinear waves: existence, stability and bifurcation of special solutions of nonlinear dynamical systems; dynamics and stability of patterns
(b) dynamical systems with symmetry and the relationship between symmetry and stability

2) The theory and application of Lie symmetry methods for differential equations, for example

(a) numerical approximations that preserve symmetry and/or conservation laws
(b) various types of symmetries of differential equations: potential, classical, nonclassical, etc.
(c) symmetries and integrable systems

Presentations

Nonlocal Extensions of Similarity Methods, Prof. George Bluman, University of British Columbia, Canada
Bifurcation analysis of a normal form for one-dimensional excitable media, Dr. Georg Gottwald, University of Sydney
Bifurcations of spiral waves I, Prof. Bjorn Sandstede, University of Surrey, United Kingdom
An undular bore solution for the higher-order Korteweg-de Vries equation, Assoc. Prof. Tim Marchant, University of Wollongong
Stability of Hamiltonian relative equilibria with isotropy, Dr. Claudia Wulff, University of Surrey, United Kingdom

Participants

	Ian Benn	Sebastian Hermann	Jude Offor	Claudia Wulf
	George Bluman	Omar Hijazi	Richard Pennifold
	Philip Broadbridge	Jonathan Kress	Bjorn Sandstede
	Emmanuel Brown	Ian Lisle	Jason Sharples
	Robert Dewar	Malachy Luke Onyeama	Harvinder Sidhu
	Georg Gottwald	Tim Marchant	Peter Vassiliou
	Celso Grebogi	Geoff Mercer	Rodney Weber

Venue

University of Canberra

More information

Email Graham Keen, AMSI at

Organising Committee

Phil Broadbridge, AMSI
Ian Lisle, University of Canberra
Peter Vassiliou, University of Canberra
Rodney Weber, UNSW at ADFA

Other sponsors

ANZIAM (throught the ACT Branch)