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Positivity determines the quantum cohomology of GrassmanniansMay 14 2019We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with non-negative structure ... More

Applying Parabolic Peterson: Affine Algebras and the Quantum Cohomology of the GrassmannianJul 07 2017Aug 13 2018The Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson's map is only a surjection, and one needs to quotient by a suitable ideal on the affine side ... More

Transitive factorizations of permutations and geometryJul 28 2014We give an account of our work on transitive factorizations of permutations. The work has had impact upon other areas of mathematics such as the enumeration of graph embeddings, random matrices, branched covers, and the moduli spaces of curves. Aspects ... More

Equivariant Quantum Cohomology of the Grassmannian via the Rim Hook RuleMar 25 2014May 18 2018A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton ... More

Equivariant Quantum Cohomology of the Grassmannian via the Rim Hook RuleMar 25 2014Apr 15 2016A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton ... More

Mutations of puzzles and equivariant cohomology of two-step flag varietiesJan 14 2014Oct 29 2014We 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 our conjectured puzzle formula for the equivariant Schubert structure ... More

Positivity and Kleiman transversality in equivariant K-theory of homogeneous spacesAug 20 2008Feb 17 2012We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological ... More

Positivity and Kleiman transversality in equivariant K-theory of homogeneous spacesAug 20 2008Mar 13 2017We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological ... More

Gromov-Witten invariants and quantum cohomology of GrassmanniansJun 29 2003This is the written version of my five lectures at the Banach Center mini-school on "Schubert Varieties", in Warsaw, May 18-22, 2003.

Gromov-Witten invariants on GrassmanniansJun 27 2003We prove that any three-point genus zero Gromov-Witten invariant on a type A Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these cases the two-step ... More

Quantum cohomology of partial flag manifoldsMar 19 2003We give elementary geometric proofs of the main theorems about the (small) quantum cohomology of partial flag varieties SL(n)/P, including the quantum Pieri and quantum Giambelli formulas and the presentation.

Quantum cohomology of GrassmanniansJun 29 2001Jul 01 2001We give elementary proofs of the main theorems about (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, Siebert and Tian's presentation, and a recent theorem of Fulton and Woodward ... More