TAILIEUCHUNG - Đề tài " Gromov-Witten theory, Hurwitz theory, and completed cycles "

There are two enumerative theories of maps from curves to curves. Our goal here is to study their relationship. All curves in the paper will be projective over C. The first theory, introduced in the 19th century by Hurwitz, concerns the enumeration of degree d covers, π : C → X, of nonsingular curves X with specified ramification data. In 1902, Hurwitz published a closed formula for the number of covers, | Annals of Mathematics Gromov-Witten theory Hurwitz theory and completed cycles By A. Okounkov and R. Pandharipande Annals of Mathematics 163 2006 517 560 Gromov-Witten theory Hurwitz theory and completed cycles By A. Okounkov and R. Pandharipande Contents 0. Introduction . Overview . Gromov-Witten theory . Hurwitz theory . Completed cycles . The GW H correspondence . Plan of the paper . Acknowledgements 1. The geometry of descendents . Motivation nondegenerate maps . Relative Gromov-Witten theory . Degeneration . The abstract GW H correspondence . The leading term . The full GW H correspondence . Completion coefficients 2. The operator formalism . The finite wedge . Operators E 3. The Gromov-Witten theory of P1 . The operator formula . The 1-point series . The n-point series 4. The Toda equation . The T-function . The string equation . The Toda hierarchy 5. The Gromov-Witten theory of an elliptic curve 518 A. OKOUNKOV AND R. PANDHARIPANDE 0. Introduction . Overview. . There are two enumerative theories of maps from curves to curves. Our goal here is to study their relationship. All curves in the paper will be projective over C. The first theory introduced in the 19th century by Hurwitz concerns the enumeration of degree d covers n C - X of nonsingular curves X with specified ramification data. In 1902 Hurwitz published a closed formula for the number of covers n P1 P1 with specified simple ramification over A1 c P1 and arbitrary ramification over TO see 17 and also 10 36 . Cover enumeration is easily expressed in the class algebra of the symmetric group S d . The formulas involve the characters of S d . Though great strides have been taken in the past century the characters of S d remain objects of substantial combinatorial complexity. While any particular Hurwitz number may be calculated very few explicit formulas are available. The second theory the Gromov-Witten theory of target curves X .

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