Award: DMS 1105813, Principal Investigator: Lev Buhovski
Rigidity in functional spaces is a relatively recent discovery in symplectic geometry, and these projects will continue the principal investigator's study of rigidity of Poisson brackets and other multilinear differential operators on symplectic manifolds. An approximation problem will also be studied: Given two smooth functions on a symplectic manifold, is it possible to approximate them by another pair of functions whose Poisson bracket is small in the uniform norm? Another line of investigation concerns the group of Hamiltonian diffeomorphisms, in the Hofer metric. The behavior of geodesics under the Hofer metric and under perturbations of it is an example of the kinds of questions that remain open regarding these diffeomorphisms.
Symplectic geometry is the study of the background structure for the Hamiltonian version of classical mechanics, and the Poisson bracket referred to above is an operation on pairs of functions that is used in mechanics to compare integrals of motion for a mechanical system and in passing from a classical mechanical system to a counterpart system in quantum theory. The subject has been studied since the nineteenth century, but the introduction of new tools into symplectic geometry from the 1980s onward has opened up many surprising phenomena, including the rigidity property cited above, which shows that small perturbations of the definition of a Poisson bracket operation have only small effects on the way it acts on pairs of functions.
Hamiltonian dynamics has started as a language to describe classical mechanics. It also has numerous connections to string theory and other modern and fast developing directions in physics. In 1990 Helmut Hofer introduced a metric on the Hamiltonian group. Given a path, its length is given by integrating mechanical energy along the path. The distance between two points is the length of the shortest path which connects them. It is a deep result that this construction defines a metric. Later on Buhovski and Ostrover showed that this metric is natural in a sense that it is equivalent to any other nicely behaved metric. Introduction of the metric allowed to apply powerful geometric tools to study the dynamics as one can talk about distance estimates, shortest paths end etc. Hofer's metric has many nice properties but is extremely difficult to compute. There is very limited set of tools that provide estimates of distance and most of them use complicated machinery. Until now geometry of the Hamiltonian group is almost unexplored and very basic questions remain open. My results provide estimates of the metric for certain constructions which allowed to solve two open questions. One of them considers Hofer's energy needed to translate a circle in a two-dimensional surface and bring it back to the original location. In the case when the surface is a punctured disk, this energy gives rise to an interesting system of invariants.