Curve neighborhoods and gromov–witten invariants of Pieri type

In this thesis we present two independent results related to the Quantum Cohomology and the Quantum K-theory rings of flag varieties.In the first part, we prove an explicit description of a two-pointed curve neighborhood Γd (Xw0 w , X u ) as a Schubert variety, where w ∈ W comin and X is a flag variety of type A. We also prove a reduction result that works in all types, that the description holds in if it holds for a flag variety defined by a maximal parabolic. In the second part, we consider a maximal orthogonal grassmannian X = OG(n; 2n), and compute its K-theoretic Gromov–Witten invariants Id (XP , X Q , X p ) where XP, XQ are Schubert and opposite Schubert varieties, and X p is an opposite Schubert variety of Pieri type. Along the way we prove a parametrization conjecture for Richardson varieties in submaximal orthogonal grassmannians Y = OG(m; 2n) stated by Ravikumar.The first result confirms a conjecture of Buch–Chaput–Perrin in type A, while the second result extends the results of Buch–Chaput–Perrin and Ravikumar in type D.