  Around 1988, Floer introduced two important theories: instanton Floer
homology as invariants of 3-manifolds and Lagrangian Floer homology as
invariants of pairs of Lagrangians in symplectic manifolds. Soon after that,
Atiyah conjectured that the two theories should be related to each other and
Lagrangian Floer homology of certain Lagrangians in the moduli space of flat
connections on Riemann surfaces should recover instanton Floer homology.
However, the space of flat connections on a Riemann surface is singular and the
first step to address this conjecture is to make sense of Lagrangian Floer
homology on this space. In this note, we formulate a possible approach to
resolve this issue. A strategy to construct the desired isomorphism in the
Atiyah-Floer conjecture is also sketched. We also use the language of A
infty-categories to state generalizations of the Atiyah-Floer conjecture.
