Final Report Summary - GEODYCON (Geometry and dynamics via contact topology)
The project has been exploring the interactions between the geometry of a space, and the dynamical properties of a special type of evolutive systems on this space: the flow of a Reeb vector field. To give a concrete picture, one can imagine a space filled with water so that the water moves in a given way. Can one evaluate the number of circular trajectories one sees in this movement in terms of the shape of the ambient space?
The outcomes have been twofolds : when the space was complicated, we have obtained the existence of remarkable periodic orbits of our dynamical system, and a quantification of its complexity by a notion of entropy. For example we have shown that for most three dimensional spaces, the dynamics is chaotic and the number of periodic orbits grows exponentially with their length. We also obtained a new proof of the so called "Weinstein conjecture" which tells that on a closed three dimensional space there always are such periodic orbits. On the other side, out of dynamical properties of a system, we were able to propose the construction of new invariants describing the complexity of the ambient space. Amongst them, we get an interpretation of a famous invariant of a knot in the three dimensional sphere, called its Khovanov homology, in terms of a count of chords of a Reeb flow in a five dimensional space.
The outcomes have been twofolds : when the space was complicated, we have obtained the existence of remarkable periodic orbits of our dynamical system, and a quantification of its complexity by a notion of entropy. For example we have shown that for most three dimensional spaces, the dynamics is chaotic and the number of periodic orbits grows exponentially with their length. We also obtained a new proof of the so called "Weinstein conjecture" which tells that on a closed three dimensional space there always are such periodic orbits. On the other side, out of dynamical properties of a system, we were able to propose the construction of new invariants describing the complexity of the ambient space. Amongst them, we get an interpretation of a famous invariant of a knot in the three dimensional sphere, called its Khovanov homology, in terms of a count of chords of a Reeb flow in a five dimensional space.