Our project, lying at the crossroads of symplectic and contact topology and mathematical physics, aims to reshape our understanding of Hamiltonian dynamics and to push the subject beyond its traditional boundaries. It will investigate the mysterious algebraic and geometric structures of groups of symplectic and contact diffeomorphisms, which encode relationships among dynamical systems and unexpectedly combine features of Lie groups and hyperbolic groups. It will develop homological, homotopical, and categorical structures arising from dynamics and governed by moduli spaces of pseudoholomorphic curves. We will address long-standing open problems concerning periodic orbits, minimal invariant sets, Poincaré recurrence, and contact and Liouville dynamics, and develop applications to the topology of Liouville manifolds and thermodynamics. The grant will support graduate students, postdoctoral researchers, workshops, and research visits that foster collaboration, knowledge exchange, and the development of young researchers.  

We thank the Simons Foundation for supporting our targeted research group.