IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

The theory of higher Reidemeister torsion yields characteristic classes of (stable) fiber bundles of smooth manifolds. We use this theory to define a new family of invariants for Legendrians in 1-jet spaces which we collectively call Legendrian higher torsion. Any version of Legendrian higher torsion yields a Legendrian isotopy invariant consisting of a collection of real cohomology classes of the base manifold. For the class of tube bundles in the sense of Waldhausen we call the invariant tube torsion and we show that it consists of a union of cosets of a suitably normalized version of the Pontryagin character. For a nearby Lagrangian (with stably trivial Gauss map) we moreover show that there is a distinguished coset which is a Hamiltonian isotopy invariant and which we call nearby Lagrangian torsion. We do not know whether nearby Lagrangians must have trivial higher torsion, as would follow from the nearby Lagrangian conjecture. Although we work with generating functions, the story has a Floer-theoretic counterpart and I will state some concrete conjectures about the expected behavior of this theory. Joint work with K. Igusa and M. Sullivan.