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Notes On Stable Maps And Quantum Cohomology
, 1996
"... Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. GromovWitten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10. Varia ..."
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Cited by 87 (12 self)
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Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. GromovWitten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10. Variations 43 References 46 0. Introduction 0.1. Overview. The aim of these notes is to describe an exciting chapter in the recent development of quantum cohomology. Guided by ideas from physics (see [W]), a remarkable structure on the solutions of certain rational enumerative geometry problems has been found: the solutions are coefficients in the multiplication table of a quantum cohomology ring. Associativity of the ring yields nontrivial relations among the enumerative solutions. In many cases, these relations suffice to solve the enumerative problem. For example, let N d be the number of degree d, rational plane curves passing through 3d \Gamma 1 general points in P . Since there is a un
Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2 ..."
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Cited by 32 (2 self)
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
Semisimple quantum cohomology and blowups
 Int. Math. Res. Not
"... Abstract. Using results of Gathmann, we prove the following theorem: If a smooth projective variety X has generically semisimple (p,p)quantum cohomology, then the same is true for the blowup of X at any number of points. This a successful test for a modified version of Dubrovin’s conjecture from t ..."
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Cited by 7 (0 self)
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Abstract. Using results of Gathmann, we prove the following theorem: If a smooth projective variety X has generically semisimple (p,p)quantum cohomology, then the same is true for the blowup of X at any number of points. This a successful test for a modified version of Dubrovin’s conjecture from the ICM 1998. 1.
Generalized enrichment of categories
 Also Journal of Pure and Applied Algebra
, 1999
"... We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmultica ..."
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Cited by 3 (1 self)
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We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmulticategory extends the (more or less wellknown) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fcmulticategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper is elementary and selfcontained, we also explain why, from one point of view, fcmulticategories are the natural structures in which to enrich categories.
Counting Curves on Irrational Surfaces
 SURVEY IN DIFFERENTIAL GEOMETRY, EDITED BY S.T. YAU
, 1999
"... In this paper we survey recent results and conjectures concerning enumeration problems on irrational surfaces. ..."
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Cited by 3 (2 self)
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In this paper we survey recent results and conjectures concerning enumeration problems on irrational surfaces.
An introduction to Frobenius manifolds, moduli spaces of stable maps and quantum cohomology
, 1997
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Quantum generalized cohomology
 in Operads, Contemp. Math
, 1997
"... Abstract. We construct a ring structure on complex cobordism tensored with Q, which is related to the usual ring structure as quantum cohomology is related to ordinary cohomology. The resulting object defines a generalized twodimensional topological field theory taking values in a category of spect ..."
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Cited by 2 (2 self)
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Abstract. We construct a ring structure on complex cobordism tensored with Q, which is related to the usual ring structure as quantum cohomology is related to ordinary cohomology. The resulting object defines a generalized twodimensional topological field theory taking values in a category of spectra.
MODULI SPACES AND MULTIPLE POLYLOGARITHM MOTIVES
, 2006
"... Abstract. In this paper, we give a natural construction of mixed Tate motives whose periods are a class of iterated integrals which include the multiple polylogarithm functions. Given such an iterated integral, we construct two divisors A and B in the moduli spaces M0,n of npointed stable curves of ..."
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Abstract. In this paper, we give a natural construction of mixed Tate motives whose periods are a class of iterated integrals which include the multiple polylogarithm functions. Given such an iterated integral, we construct two divisors A and B in the moduli spaces M0,n of npointed stable curves of genus 0, and prove that the cohomology of the pair (M0,n −A, B −B ∩A) is a framed mixed Tate motive whose period is that integral. It generalizes the results of A. Goncharov and Yu. Manin for multiple ζvalues. Then we apply our construction to the dilogarithm and calculate the period matrix which turns out to be same with the canonical one of Deligne. 1.
Let V and W be smooth and projective varieties over the field k. In this
, 1997
"... We prove that the system of GromovWitten invariants of the product of two varieties is equal to the tensor product of the systems of ..."
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We prove that the system of GromovWitten invariants of the product of two varieties is equal to the tensor product of the systems of
A PRODUCT FORMULA FOR GROMOVWITTEN INVARIANTS.
, 904
"... Abstract. We establish a product formula for GromovWitten invariants for closed, connected, relatively semipositive Hamiltonian fibrations over any symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi)fibration map from the moduli space of pseudohol ..."
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Abstract. We establish a product formula for GromovWitten invariants for closed, connected, relatively semipositive Hamiltonian fibrations over any symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi)fibration map from the moduli space of pseudoholomorphic maps with marked points in the total space of the Hamiltonian fibration to the corresponding moduli space of pseudoholomorphic maps with marked points in the base. We use this induced map to recover the product formula by means of integration. Finally, we give applications to csplitting and symplectic uniruledness. 1.