Hamiltonian elements in algebraic K-theory

I will explain how to associate elements in algebraic K-theory of a commutative ring, and some categorified analogues of the latter, using Hamiltonian fiber bundles and Floer theory. This is analogous to the classical story of complex K-theory elements induced by finite dimensional complex vector bundles. In particular, this produces non-trivial elements in categorified algebraic K-theory (Bertrand Toen's secondary K-theory). The story gives a path to generalizing homological mirror symmetry to algebraic K-theory context, using Langlands dual group formalism.