can be thought of as a case of Lagrangian Floer homology in which the ambient manifold is M crossed with M and the Lagrangian submanifolds are the diagonal and the graph of the symplectomorphism. The construction of Heegaard Floer homology is based on a variant of Lagrangian Floer homology for totally real submanifolds defined using a Heegaard splitting of a three-manifold. Seidel–Smith and Manolescu constructed a link invariant as a certain case of Lagrangian Floer homology, which conjecturally agrees with Khovanov homology, a combinatorially-defined link invariant. Atiyah–Floer conjecture The Atiyah–Floer conjecture connects the instanton Floer homology with the Lagrangian intersection Floer homology. Consider a 3-manifold Y with a Heegaard splitting along a surface . Then the space of flat connections on modulo gauge equivalence is a symplectic manifold of dimension 6g − 6, where g is the genus of the surface . In the Heegaard splitting, bounds two different 3-manifolds; the space of flat connections modulo gauge equivalence on each 3-manifold with boundary embeds into as a Lagrangian submanifold. One can consider the Lagrangian intersection Floer homology. Alternately, we can consider the Instanton Floer homology of the 3-manifold Y. The Atiyah–Floer conjecture asserts that these two invariants are isomorphic. Salamon–Wehrheim and Daemi–Fukaya are working on their programs to prove this conjecture. Relations to mirror symmetry The homological mirror symmetry conjecture of Maxim Kontsevich predicts an equality between the Lagrangian Floer homology of Lagrangians in a Calabi–Yau manifold and the Ext groups of coherent sheaves on the mirror Calabi–Yau manifold. In this situation, one should not focus on the Floer homology groups but on the Floer chain groups. Similar to the pair-of-pants product, one can construct multi-compositions using pseudo-holomorphic n-gons. These compositions satisfy the -relations making the category of all (unobstructed) Lagrangian submanifolds in a symplectic manifold into an -category, called the Fukaya category. To be more precise, one must add additional data to the Lagrangian – a grading and a spin structure. A Lagrangian with a choice of these structures is often called a brane in homage to the underlying physics. The Homological Mirror Symmetry conjecture states there is a type of derived Morita equivalence between the Fukaya category of the Calabi–Yau and a dg category underlying the bounded derived category of coherent sheaves of the mirror, and vice versa. Symplectic field theory (SFT) This is an invariant of contact manifolds and symplectic cobordisms between them, originally due to Yakov