Symplectic geometry is very useful for formulating clearly and concisely problems in classical physics and also for understanding the link between classical problems and their quantum counterparts. It is thus a subject from different viewpoints. Therefore, the issue addressed in this project is extending results from symplectic i.e. specific even-dimensional systems to contact i.e. specific odd-dimensional systems.
In more detail: The introduction of symplectic geometry comes from the investigation of light rays through a medium, this was extended to every mechanical system. One thing remained constant - each ray would be described according to two pieces of information - the position the ray is incident with the medium and the angle at which the ray hits the medium. According to Hamilton's infamous paper of 1828 `Caustics', the optical system used to study the geometry of rays of light, utilises four variables to locally specify the rays that enter a system from the left and exit from the right (the light rays to the left are straight line segments); two of the variables specify the point of intersection of the line with the plane perpendicular to the optical axis and two additional angular variables giving the angle of the line to this plane. Relating the incoming line segments on the left to the outgoing line segments on the right, is a transformation from the incoming coordinates to the outgoing ones, called symplectic diffeomorphisms.
Later Hamilton discovered that this same method applies without modification to mechanics. Replacing the optical axis by the time axis, light rays for trajectories of the system and the four incoming and four outgoing variables by the 2n incoming and outgoing variables of the phase space of the mechanical system. And therefore symplectic geometry was birthed and the associated momentum map and its image accordingly.
Contact systems exist on 2n-1 or odd dimensional manifolds. Contact geometry has been used to describe so many physical phenomena and is related to a lot of other mathematical structures. Sophus Lie was the first to introduce them through his work on partial differential equations. Then Gibbs reintroduced them through thermodynamics, Huygen's later through geometric optics and Hamiltonian dynamics. More recently contact structures have been found to describe heat transport through a medium. But this was only after previous work relating contact structures to Riemannian geometry, low dimensional topology and subelleptic operators.
Focussing on cooriented contact structures i.e. hyperplane fields on oriented smooth 2n-1 dimensional manifolds which are given by its kernels. We establish the momentum polytope for such structures.
Momentum theory is devoted to the interaction between group theory and symplectic geometry for the solution of various mechanical systems, as stated before, the various theories of light lead to the mathematical and physical ideas involved in symplectic geometry.
During my Marie-Curie fellowship, I have focussed on contact manifolds. Contact systems are the natural framework in thermodynamics, geometrical optics, geometric quantization, classical mechanics and dissipative Hamiltonian systems. Up till my work prominent mathematicians in the field have said that the momentum polytope for a contact system is not necessarily convex. Yael Karshon who has been described as "one of Canada's leading experts in symplectic geometry", Eugene Lerman who is one of the most respected and revered living mathematician in geometric mechanics, who is one of the pioneers of momentum polytope theory, have in their papers on the momentum polytope of the contact system focussed on symplectising the structure first before developing its polytope theory. This is a problem that has clearly gripped attention for decades.
To put the reason for interest into polytope theory in context, we will look at one of the many convex polytopes I produced during my PhD (see Images attached to the Summary for publication).
This simple rectangle represents three objects moving in relation to each other in eight dimensional space. Particularly, it tells us when these objects may be moving in line with each other, moving at right angles to each other, and when the reduced space of the system is a sphere, or another shape. We can also read off this rectangle when they are moving in a very stable way or a slightly more chaotic way. So you see, this is a minimal way to contain a lot of information. And this is how we can keep track of very complicated systems.
Therefore, if we can produce momentum polytopes for a system, we are more likely to comprehend very complicated versions of that system. The convexity theorem for the momentum polytope was widely generalised by Fields medal and Abel prize winner Michael Atiyah, Leroy P. Steele prize winner Victor Guillemin, and Savillion Professor of Geometry at Oxford, Frances Kirwan.
One of the reasons to which we can attribute this concentration of talent is that describing the image of a momentum map defined on a symplectic manifold has generated a large amount of research and remains, to this day, one of the most active areas in symplectic geometry and its applications to Hamiltonian dynamics, especially bifurcation theory.
