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

Atiyah-Floer Conjecture: a Formulation, a Strategy to Prove and Generalizations