Quantum cohomology as a deformation of symplectic cohomology

Consider a positively monotone (Fano) closed symplectic manifold $M$ and a symplectic simple crossings divisor $D$ in it. Assume that the Poincare dual of the anti-canonical class is a positive rational linear combination of the classes $[D_i]$, where $D_i$ are the components of $D$ with their symplectic orientation. A choice of such coefficients, called the weights, (roughly speaking) equips $M-D$ with a Liouville structure. I will start by discussing results relating the symplectic cohomology of $M-D$ with quantum cohomology of $M$. These results are particularly sharp when the weights are all at most 1 (hypothesis A). Then, I will discuss certain rigidity results (inside $M$) for skeleton type subsets of $M-D$, which will also demonstrate the geometric meaning of hypothesis A in examples. The talk will be mainly based on joint work with Strom Borman and Nick Sheridan.