Floer Homotopy Theory and Twisted Algebraic K-Theory

The main purpose of this project is to apply methods from stable homotopy theory to solve questions in low dimensional topology and symplectic geometry.

In particularwe want to apply the theory of twisted objects such as twisted generating functions and twisted parametrized equivariant spectra in these settings (which are both related to twisted algebraic K-theory).

Our recent result with Abouzaid, Courte and Guillermou creating twisted generating functions serves as a first example of how this can be done.The main aims of the project areA) To further our understanding of exact Lagrangian submanifolds using our twisted generating functions.B) To construct a twisted parametrized S1 equivariant spectrum representing Seiberg-Witten Floer homology for 3-manifolds.

Use these to gain new insight into 3- and 4-manifolds.C) To define and compute twisted stable homotopy type invariants for Lagrangian intersection theory and Hamiltonian systems.D) To generalize to other settings such as: non-exact Lagrangians, Legendrians in general contact manifolds and more general gauge theories.E) Understanding how these structures relates to factorization homology, and finding deeper relations to physics.