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Abstract: In his 1998 ICM talk, Dubrovin conjectured that a Fano variety X has semisimple quantum cohomology if and only if the derived category D(X) possesses a full exceptional collection. Using Gamma structures, this conjecture has been revisited, refined, and generalized to nonsemisimple cases in the last decade. 

In this talk, we investigate the quantum cohomolohy for projective bundles, blow-ups, and standard flips and its connection to derived categories. By restricting the quantum cohomology to a fiber curve direction, we obtain a decomposition of QH(X) into asymptotic Gamma classes, indexed by the spectrum of quantum multiplication of the first Chern class of X. Using a sequence of reductions and asymptotic expansions of Meijer G-functions, we show this decomposition is compatible with the semi-orthogonal decompositions of projective bundles, blow-ups, and standard flips studied by Orlov and Belmans-Fu-Raedschelders. 

The talk is based on work joint with Mark Shoemaker. If time permits, I will also talk about our recent progress on weighted flips.