Quantum Cohomology under Birational Maps and Transitions

This is an expanded version of the third author’s lecture in String-Math 2015 at Sanya. It summarizes some of our works in quantum cohomology. After reviewing the quantum Lefschetz and quantum Leray–Hirsch, we discuss their applications to the functoriality properties under special smooth flops, flips and blow-ups. Finally, for conifold transitions of Calabi–Yau 3-folds, formulations for small resolutions (blow-ups along Weil divisors) are sketched.