Symplectic Topology: A Fundamental Interface Between Pure Mathematics and Theoretical Physics
In our efforts to understand nature, physics and mathematics are allied disciplines. Einstein's theory of relativity offers a remarkable example of this mutual fertilization of mathematics and physics. In particular, it demonstrates the crucial role played by the mathematical structures that specialists call "topological spaces" and define as curved spaces of arbitrary dimensions which generate mathematical phenomena. And it continues today: as physicists work on developing a unified theory of the fundamental forces of physical reality, they necessarily turn again to mathematics, particularly to symplectic geometry and topology, which constitute one of the more remarkable breakthroughs in recent mathematical research. While these branches of mathematics are more than two hundred years old, it is only since the mid-1980s that they have received concerted attention from the international research community.
As Canada Research Chair in Differential Geometry and Topology, François Lalonde is tackling fundamental problems in symplectic topology, including the classification of symplectic spaces and their mathematical structures, the study of their transformations and behaviour under deformation as well as their connections to the quantum domain. He also takes an interest in possible applications of this theory to other, more traditional, branches of mathematics as well as to theoretical physics and hydrodynamics.
Prof. Lalonde has published a number of articles essential to this field and has been invited to speak at conferences all over the world. His work, which has helped to define the mathematical structure of symplectic spaces and transformations, commands as much interest among pure mathematicians as among mathematical physicists, who regard it as one of the ways to understand activity at the subatomic level. Lalonde carries out his research with a well-established network of collaborators in Europe, the United States and Asia, and with the team of students and postdoctoral researchers who work with him in Montréal.