This book can form the basis of a second course in algebraic geometry. As motivation, it
takes concrete questions from enumerative geometry and intersection theory, and provides …

Equivariant cohomology has become an indispensable tool in algebraic geometry and in
related areas including representation theory, combinatorial and enumerative geometry, and …

A new approach is described to the Schubert calculus on complete flag varieties, using the
volume polynomial associated with Gelfand-Zetlin polytopes. This approach makes it …

This book gives an introduction to the very active field of combinatorics of affine Schubert
calculus, explains the current state of the art, and states the current open problems. Affine …

We prove a root system uniform, concise combinatorial rule for Schubert calculus of
minuscule and cominuscule flag manifolds G/P (the latter are also known as compact …

We prove a conjecture of Knutson asserting that the Schubert structure constants of the
cohomology ring of a two-step flag variety are equal to the number of puzzles with specified …

We apply crystal theory to affine Schubert calculus, Gromov–Witten invariants for the
complete flag manifold, and the positroid stratification of the positive Grassmannian. We …

Richardson varieties play an important role in intersection theory and in the geometric
interpretation of the Littlewood–Richardson Rule for flag varieties. We discuss three natural …

We introduce a mutation algorithm for puzzles that is a three-direction analogue of the
classical jeu de taquin algorithm for semistandard tableaux. We apply this algorithm to prove …

We prove a Chevalley formula for the equivariant quantum multiplication of two Schubert
classes in the homogeneous variety X= G/P. Using this formula, we give an effective …