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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 523 (3 self)
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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
GromovWitten classes, quantum cohomology, and enumerative geometry
 Commun. Math. Phys
, 1994
"... The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological ..."
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Cited by 474 (3 self)
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Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given. Let V be a projective algebraic manifold. Methods of quantum field theory recently led to a
Finding rational solutions of rational systems of autonomous ODEs
"... In this paper we provide an algorithm to find explicitly rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its invariant algebraic curves. The method is based on the rational parametrization of the rational invariant algebraic curves and intensively us ..."
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In this paper we provide an algorithm to find explicitly rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its invariant algebraic curves. The method is based on the rational parametrization of the rational invariant algebraic curves and intensively
EXERCISES IN THE BIRATIONAL GEOMETRY OF ALGEBRAIC VARIETIES
, 2008
"... The book [KM98] gave an introduction to the birational geometry of algebraic varieties, as the subject stood in 1998. The developments of the last decade made the more advanced parts of Chapters 6 and 7 less important and the detailed treatment of surface singularities in Chapter 4 less necessary. H ..."
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Cited by 322 (1 self)
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. Although singular, this is the “best ” model to work with. There are lots of these but they appear less frequently outside algebraic geometry. There are also two types of “mongrels”: (Ruled): Birational to P 1 × (curve of genus ≥ 1). (Elliptic): These fiber over a curve with general fiber an elliptic curve
Vector bundles over an elliptic curve
 Proc. London Math. Soc
, 1957
"... THE primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve (defined over an algebraically closed field k). The interest of the elliptic curve lies in the fact that it provides the first nontrivial case, Grothendieck (6) having shown that for a rational curve e ..."
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Cited by 293 (0 self)
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THE primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve (defined over an algebraically closed field k). The interest of the elliptic curve lies in the fact that it provides the first nontrivial case, Grothendieck (6) having shown that for a rational curve
Symplectic reflection algebras, CalogeroMoser space, and deformed HarishChandra homomorphism
 Invent. Math
"... To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic ..."
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Cited by 280 (39 self)
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‘rational ’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group of g = gl n. We construct a 1parameter deformation of the HarishChandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n
Rational Points on Elliptic Curves
, 1992
"... Abstract. We give a quantitative bound for the number of Sintegral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the jinvariant, the degree [K: Q], and the number of primes in S. Let K be a number field of degree d and MK the set of ..."
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Cited by 125 (1 self)
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Abstract. We give a quantitative bound for the number of Sintegral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the jinvariant, the degree [K: Q], and the number of primes in S. Let K be a number field of degree d and MK the set
RELATIONSHIPS AMONG BIRATIONAL INVARIANTS OF ALGEBRAIC PLANE CURVES
"... Let S be a nonsingular rational surface S andD a nonsingular curve on S. Suppose that m ¸ a ¸ 1. Then Pm;a[D] = dim jmKS + aDj + 1. ..."
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Let S be a nonsingular rational surface S andD a nonsingular curve on S. Suppose that m ¸ a ¸ 1. Then Pm;a[D] = dim jmKS + aDj + 1.
Equivariant GromovWitten invariants
 INTERNAT. MATH. RES. NOTICES
, 1996
"... The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the GromovWitten (GW) theory, i.e., intersection theory on spaces of (pseudo) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a co ..."
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Cited by 127 (10 self)
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symplectic and complex geometry on a compact Kähler CalabiYau nfold and, respectively, complex and symplectic geometry on another CalabiYau nfold, called the mirror partner of the former one. The remarkable application [8]ofthe mirror conjecture to the enumeration of rational curves on CalabiYau 3folds
Algebraic and rational differential invariants
"... These notes start with an introduction to differential invariants. They continue with an algebraic treatment of the theory. The algebraic, differential algebraic and differential geometric tools that are necessary to the development of the theory are explained in detail. We expose the recent result ..."
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results on the topic of rational and algebraic differential invariants. Finally we give a new algebraic version of the finiteness theorem of Lie–Tresse for the case of finite dimensional algebraic groups.
Results 1  10
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