My primary interests and areas of research are
- Topological Hamiltonian and contact dynamics
- Symplectic and contact geometry and topology
- Topological dynamics
and the deeper topological structure underlying symplectic and contact topology. Topological Hamiltonian dynamics and topological contact dynamics are natural and genuine extensions of smooth Hamiltonian and contact dynamics to topological dynamics, and of symplectic and contact diffeomorphisms to homeomorphisms as transformations preserving the additional geometric structure. These new theories have numerous applications to their smooth counterparts, as well as to other areas of mathematics, such as topological dynamics, in particular in low dimensions, and to geodesic flows on Riemannian manifolds. Applications are also expected to billiard dynamics. I also have an interest in action selectors in Hamiltonian Floer theory and related subjects.
A topological Hamiltonian or contact dynamical system consists of a topological Hamiltonian or contact isotopy (a continuous isotopy of homeomorphisms), together with a possibly non-smooth topological Hamiltonian function on the underlying symplectic or contact manifold, and a topological conformal factor (a continuous function) in the contact case. By definition, this Hamiltonian function is the limit of a sequence of smooth Hamiltonian functions with respect to the usual Hofer metric. Moreover, the corresponding sequence of smooth Hamiltonian or contact isotopies converges uniformly to the above continuous isotopy, and their smooth conformal factors converge uniformly to the above continuous function associated to the limit isotopy. In other words, we consider limits of smooth Hamiltonian and contact dynamical systems with respect to a metric that combines topological and dynamical information. Similarly, topological automorphisms of a symplectic or contact structure or form are C^0-limits of symplectic or contact diffeomorphisms (together with their conformal factors in the contact case).
A topological Hamiltonian or contact isotopy is not generated by a vector field, and may not even be Lipschitz continuous; nonetheless, it is uniquely determined by its associated topological Hamiltonian function, and in turn, in the contact case it determines uniquely its topological conformal factor. Composition and inversion can therefore be defined as in the smooth case, and the usual transformation law continues to hold. Conversely, every topological Hamiltonian or contact isotopy possesses a unique topological Hamiltonian function. The topological automorphism groups of a contact structure and a contact form exhibit surprising rigidity properties analogous to the well-known Eliashberg-Gromov rigidity in the case of a symplectic structure.
The uniqueness theorems for topological Hamiltonian and contact dynamical systems, rigidity, the group structures, and the transformation law, provide ample evidence that Hamiltonian and contact dynamics are in fact topological theories, and a priori smooth invariants can be extended to topological Hamiltonian and contact dynamical systems. The study of these topological theories is to a large extend motivated by physical considerations and by various continuity phenomena already present in their smooth counterparts, and in turn, they have applications to smooth Hamiltonian and contact dynamics as well as to topological dynamics and low-dimensional topology. For example, one can answer important questions of V. I. Arnold regarding the continuity of the helicity of a volume preserving isotopy, and its behavior under conjugation by volume preserving homeomorphisms, provided the isotopy and the transformation can be described as lifts from a manifold one dimension lower. The relation between topological Hamiltonian dynamics and topological contact dynamics is … [More]