Manin stressed that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing cycles which I have read are projective, I am curious about the behaviour of quantum cohomology under monodromy.
Edit: A coming seminar: "Quantum motives: realizations, detection, applications", incl. a lecture "Quantum motives: review (of) the classical idea of how to linearize algebraic geometry with an eye to utilizing it in the quantum setup." and a minicourse "Geometric Langlands and quantum motives: a link".
Edit: The slides on Manin's talk on the concept of classical motives and it's relation to quantum cohomology are here.
Edit: Manin's and Smirnov's interesting computations and thoughts on e.g. a "membrane quantum cohomology"cohomology" (+ russian videos of a talk in june 2011 on the program "... elucidation of this "self-referentiality" of quantum cohomology has just begun. The talk tries to outline the contours of this huge program and the first steps of it.": part 1, part 2). BTW, has anyone the text by Kapranov which is mentioned in the paper above and in Hacking's introduction?