Title: Quantum Cohomology under Birational Maps and Transitions,
Abstract: 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.