Rigid and Flexible Symplectic Topology

Stanford University, Stanford CA

Investigators

Abstract

From the very first steps of symplectic topology in 1980s flexible and rigid methods played important complementary roles, though with introduction by M. Gromov of the method of holomorphic curves, rigid methods played the dominating role in the development of the subject. However, it became recently feasible, with a number of important recent advances, to push the results on the flexible side almost to the borderline with the rigid side. The research under the current grant pursues the development of rigid and flexible methods in symplectic topology. Most important tasks on the rigid side include: completion of the algebraic foundations of symplectic field theory (SFT), including the relative theory concerning invariants of Legendrian submanifolds, connections with the theory of integrable systems, and further development and extension of Legendrian surgery technique to all SFT invariants with applications to Hamiltonian dynamics, especially to the theory of quasi-states and quasi-morphisms. On the flexible side, the research will be directed towards further development of the theory of flexible Weinstein manifolds, a new class of symplectic manifolds previously discovered by K. Cieliebak, E. Murphy and the PI, as well as the theory of Lagrangian immersions with minimal number of intersection points. Another new subject which should benefit from both, flexible and rigid methods is symplectic pseudo-isotopy theory, a necessary ingredient towards understanding topology of groups of symplectic and contact transformations. Symplectic topology is at the crossroads of several mathematical disciplines such as low-dimensional, algebraic and geometric topology, Hamiltonian dynamics, algebraic geometry, mathematical theory of mirror symmetry and theory of integrable systems. Its development, and in particular the development of symplectic field theory, significantly changed the face of these areas. The research under the current grant should bring both, the further development of symplectic topological machinery, and in particular of the algebraic formalism of symplectic field theory, as well as new horizons for its applications to dynamics, low-dimensional topology and theory of quantum integrable systems.