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274
Equivariant Cohomology, Koszul Duality, and the Localization Theorem
 Invent. Math
, 1998
"... This paper concerns three aspects of the action of a compact group K on a space ..."
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Cited by 265 (4 self)
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This paper concerns three aspects of the action of a compact group K on a space
Abelian varieties
, 2008
"... These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in ver ..."
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Cited by 158 (7 self)
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These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form.
Morava Ktheories and localisation
 MEM. AMER. MATH. SOC
, 1999
"... We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpotent spectra ..."
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Cited by 101 (19 self)
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We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpotent spectra. We give a number of useful extensions to the theory of vn self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible K(n)local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the BrownComenetz dual of MnS lies in the Picard group. We give a detailed analysis of some examples when n =1 or 2, and a list of open problems.
Gröbner geometry of Schubert polynomials
 Ann. Math
"... Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix S ..."
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Cited by 99 (15 self)
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Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are therefore positive sums of monomials, each monomial representing the torusequivariant cohomology class of a component (a schemetheoretically reduced coordinate subspace) in the limit of the resulting Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant Ktheory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in Ktheory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces is provided by the proof of the Gröbner basis property, bypassing what has come to be known as Kohnert’s conjecture [Mac91]. The coordinate subspaces, which are
Quantum Schubert Polynomials
 J. AMER. MATH. SOC
, 1997
"... We compute GromovWitten invariants of the flag manifold using a new combinatorial construction for its quantum cohomology ring. Our construction provides quantum analogues of the BernsteinGelfandGelfand results on the cohomology of the flag manifold, and the LascouxSchutzenberger theory of S ..."
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Cited by 88 (7 self)
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We compute GromovWitten invariants of the flag manifold using a new combinatorial construction for its quantum cohomology ring. Our construction provides quantum analogues of the BernsteinGelfandGelfand results on the cohomology of the flag manifold, and the LascouxSchutzenberger theory of Schubert polynomials. We also derive the quantum Monk's formula.
Equivariant cohomology and equivariant intersection theory
, 2008
"... This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures given at Montréal is Summer 1997. Our main aim is to obtain ..."
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Cited by 68 (4 self)
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This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures given at Montréal is Summer 1997. Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certain manifolds with group actions which arise in representation theory, e.g. homogeneous spaces and their compactifications. As another appplication of equivariant intersection theory, we obtain simple versions of criteria for smoothness or rational smoothness of Schubert varieties (due to Kumar [40], CarrellPeterson [16] and Arabia [4]) whose statements and proofs become quite transparent in this framework. We now describe in more detail the contents of these notes; the prerequisites are notions of algebraic topology, compact Lie groups and linear algebraic groups. Sections 1 and 2 are concerned with actions of compact Lie groups on topological spaces, especially on symplectic manifolds. The
A categorification of quantum sl(2)
 ADV. MATH
, 2008
"... We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs betwee ..."
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Cited by 67 (9 self)
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We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs between 1morphisms are graded lifts of a semilinear form defined on ˙U. Graded lifts of various homomorphisms and antihomomorphisms of U ˙ arise naturally in the context of our graphical calculus. For each positive integer N a representation of U˙ is constructed using iterated flag varieties that categorifies the irreducible (N + 1)dimensional representation of ˙ U.
Geometry of hyperKähler connections with torsion
 Comm. Math. Phys
"... Abstract The internal space of a N=4 supersymmetric model with WessZumino term has a connection with totally skewsymmetric torsion and holonomy in Sp(n). We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry, construct homogen ..."
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Cited by 66 (9 self)
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Abstract The internal space of a N=4 supersymmetric model with WessZumino term has a connection with totally skewsymmetric torsion and holonomy in Sp(n). We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.
Quadratic algebras, Dunkl elements, and Schubert calculus
 in Advances in Geometry (J.L. Brylinski
, 1997
"... We suggest a new combinatorial construction for the cohomology ring of the flag manifold. ..."
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Cited by 65 (10 self)
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We suggest a new combinatorial construction for the cohomology ring of the flag manifold.
The geometry of flag manifolds
 Proceedings of the London Mathematical Society
, 1959
"... A FLAG in projective space Sn is a 'nest ' of subspaces, one of each dimension from 0 to n — 1. The aggregate of all such flags is in unexceptional birational correspondence with the points of an algebraic variety, the flag manifold of the space, denoted by F(n\*\). The manifold can be re ..."
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Cited by 63 (0 self)
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A FLAG in projective space Sn is a 'nest ' of subspaces, one of each dimension from 0 to n — 1. The aggregate of all such flags is in unexceptional birational correspondence with the points of an algebraic variety, the flag manifold of the space, denoted by F(n\*\). The manifold can be represented as a