  This is a mixture of survey article and research anouncement. We discuss
Instanton Floer homology for 3 manifolds with boundary. We also discuss a
categorification of the Lagrangian Floer theory using the unobstructed immersed
Lagrangian correspondence as a morphism in the category of symplectic
manifolds. During the year 1998-2012, those problems have been studied
emphasising the ideas from analysis such as degeneration and adiabatic limit
(Instanton Floer homology) and strip shrinking (Lagrangian correspondence).
Recently we found that replacing those analytic approach by a combination of
cobordism type argument and homological algebra, we can resolve various
difficulties in the analytic approach. It thus solves various problems and also
simplify many of the proofs.
