Title: Extremal quantum cohomology and Gamma structure for standard flips

Abstract: In his 1998 ICM talk, Dubrovin conjectured that for a Fano variety X, the derived category of X possesses a full exceptional collection if and only if the quantum cohomology of X is generically semi-simple, suggesting a deep connection between the derived category of a variety and its Gromov--Witten theory. The general relationship between quantum cohomology and the derived category has been clarified in recent years through a number of conjectures. In this talk I will describe some of these conjectures, focusing on the quantum cohomology at special values of the quantum parameters. In the case of projective bundles, blow-ups, and standard flips, I will show how known semi-orthogonal decompositions of the derived category are closely connected to the quantum cohomology at special values. This talk is based on joint work with Yefeng Shen.