Yuri Manin (Max-Planck-Institute for Mathematics)
Quantum cohomology, motives, derived categories I & II
Quantum Cohomology - and Mirror Symmetry - started in 1991, with the discovery made by a group of physicists of a remarkable identity for the generating series for the number of rational curves of various degrees on a quintic threefold.
When algebraic geometers started studying this problem it turned out that even the correct definition of curve count is a highly non-trivial task. After several years of arduous efforts, it turned out that such a count is a by-product of a very vast new structure involving ALL smooth projective varieties. Namely the motive, in Grothendieck's sense, of any such variety is acted upon by the motives of Deligne-Mumford stacks of stable curves with marked points, and this action is operadic.
In my brief minicourse, I will give a review of basic structures involved in this picture. I will discuss the problem: how can we describe the action of the operad of moduli spaces upon its own members, e.g., what is quantum cohomology of $\bar{M}_{0,n}$? Finally, I will touch upon the recent generalization of motives, in which the motive of a manifold is represented by its coherent derived category, or more sophisticated enhanced dg category. How can we extend Quantum Cohomology to this universe?