Soit G un groupe algbrique complexe quasi-simple et G/P une varit de drapeaux partiels. La projection sur G/P des varits de Richardson (de la varit des drapeaux complets) forment une stratification de G/P. Nous montrons que les relations dadhrence des varits de Richardson projetes correspondent celles dun certain sous-ensemble de varits de Schubert sur la varit de drapeaux affine de G. Nous comparons aussi les classes de cohomologie quivariante et de K-thorie de ces deux stratifications. Notre travail gnralise celui de Knutson, Lam et Speyer pour la grassmannienne de type A.

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I will discuss results of three different types in geometry and topology.(1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras.(2) The computations of intersection numbers of the moduli spaces of flat connections on a Riemann surface by using heat kernels.(3) The mirror principle about counting curves in Calabi-Yau and general projective manifolds by using hypergeometric series.

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.