Floer Homology and Khovanov Homology Reading Group

Consider $T^*P^1$ as the B-model of a mirror equivalence. It turns out that the A-model mirror depends on choices and I will describe two of these mirrors: one with underlying symplectic manifold the complement of a conic in $T^*S^2$, and the other with underlying symplectic manifold ${C^*}^2$. I will explain how these two mirrors can be related to each other without appealing to mirror symmetry, but that the correspondence between their Floer theory is still quite mysterious. At the end, I will speculate about the connection with Khovanov homology. The content of the talk is a combination of joint work with Auroux and Katzarkov, and an ongoing project with Smith.