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349
Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 529 (3 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
Noncommutative manifolds, the instanton algebra and isospectral deformations
 Comm. Math. Phys
"... We give new examples of noncommutative manifolds that are less standard than the NCtorus or Moyal deformations of R n. They arise naturally from basic considerations of noncommutative differential topology and have nontrivial global features. The new examples include the instanton algebra and the ..."
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Cited by 172 (29 self)
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obtained from this equation as a noncommutative Grassmanian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC3sphere intimately related to quantum group deformations SUq(2) of SU(2) but for unusual values (complex values of modulus one
Formal (non)commutative symplectic geometry
 THE GELFAND MATHEMATICAL SEMINARS, 1990–1992”, BIRKHÄUSER
, 1993
"... ..."
Combinatorial Hopf Algebras and KHomology of Grassmanians
 MATH. RES. NOT., ARTICLE ID RNM125
, 2007
"... Motivated by work of Buch on setvalued tableaux in relation to the Ktheory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as Ktheoretic analogues of the by now classical “square” of Hopf algebras consisting of symmetric functions, quasisymmetr ..."
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Cited by 16 (3 self)
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, quasisymmetric functions, noncommutative symmetric functions and the Malvenuto–Reutenauer Hopf algebra of permutations. In addition, we develop a theory of setvalued Ppartitions and study three new families of symmetric functions which are weight generating functions of reverse plane partitions, weak set
NONCOMMUTATIVE GEOMETRY YEAR 2000
, 2000
"... Our geometric concepts evolved first through the discovery of NonEuclidean geometry. The discovery of quantum mechanics in the form of the noncommuting coordinates on the phase space of atomic systems entails an equally drastic evolution. We describe a basic construction which extends the familiar d ..."
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Cited by 4 (0 self)
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Our geometric concepts evolved first through the discovery of NonEuclidean geometry. The discovery of quantum mechanics in the form of the noncommuting coordinates on the phase space of atomic systems entails an equally drastic evolution. We describe a basic construction which extends the familiar
Manifest supersymmetry in noncommutative geometry
 JHEP 0002 (2000) 022 [hepth/9912153
"... We consider the open superstring ending on a Dbrane in the presence of a constant NSNS B field, using the GreenSchwarz formalism. Quantizing in the lightcone gauge, we find that the anticommutation relations for the fermionic variables of superspace remain unmodified. We also derive the unbroke ..."
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Cited by 28 (2 self)
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the unbroken supersymmetry algebra living on the Dbrane. This establishes how the Moyal product is extended in a superspace formulation of noncommutative field theories. The superfield formulation of noncommutative During the last years, there has been remarkable progress in understanding the dynamics
Results 1  10
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