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出版社:高等教育
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ISBN:9787040474190
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作者:季理真//(荷)路易安嘎
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頁數:221
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出版日期:2017-04-01
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印刷日期:2017-04-01
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包裝:精裝
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開本:16開
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版次:1
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印次:1
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字數:240千字
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由季理真、路易安嘎著的《黎曼曲面和熱帶曲線 的模空間導引(英文版)(精)》黎曼曲面及其模空間的 概念由黎曼分別在其博士畢業論文和一篇著名的文章 中定義。由於同數學與物理的許多學科聯繫廣泛,黎 曼曲面及其模空間得到了深入的研究,並將繼續吸引 人們的關注。近期熱帶曲線的研究迅速崛起。熱帶代 數曲線是經典復數域上代數曲線以及黎曼曲面在熱帶 半環上的一種模擬。本書深入淺出地介紹了以上幾個 重要學科,並且重點強調如代數幾何、復幾何、雙曲 幾何、拓撲、幾何群理論和數學物理等不同學科之間 的關聯。
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Part I Moduli Spaces of Riemann Surfaces 1 Mapping class groups and Dehn twists Mapping class groups Dehn twists Fundamental groups and mapping class groups 2 Conformal structures and a rough classification Conformal structures Spherical cases Flat cases Hyperbolic cases 3 Geometry of the upper half plane 4 Hyperbolic surfaces Closed geodesics on a hyperbolic surface Geodesic shear Geodesic representation Disjunction of geodesics Pair of pants Pants decompositions Fenchel-Nielsen coordinates Hyperbolic surfaces with cusps Pants decomposition in the presence of cusps 5 Quadratic differentials Local normal form Structure defined by a holomorphic quadratic differential Quadratic differentials and the Teichmtiller flow 6 Ribbon graphs and ideal triangulation of Teichmuller space Jenkins-Strebel differentials Combinatorial description of a graph Ribbon graphs The dual ribbon graph and the arc complex Metrized ribbon graphs Ideal triangulation of thickened Teichmtuler space 7 The homotopy type of the moduli spaces □ The homotopy type of □ The moduli space of curves as a virtual classifying space 8 A brief review of Dolbault cohomology and Serre duality for Riemann surfaces Dolbault cohomology of a coherent sheaf Riemann-Roch and Serre duality for a Riemann surface 9 Deformation theory of Riemann surfaces Orbifolds Towards the complex tangent bundle of Teichmiiller space Deformation theory The Kodaira Spencer map The Weil-Petersson metric The quasi-projective structure on the universal family 10 Harvey bordification and Deligne-Mumford compactification The curve complex Harvey's bordification Stable pointed curves Deformations of nodes Deformations of nodal curves The Deligne-Mumford compactification The idea behind Geometric Invariant Theory Projectivity of the moduli space of stable pointed curves Getting Harvey's bordification from the D-M compactification... 11 Cohomological properties of □ Harer's stability theorem Hopf algebra structure on stable cohomology 12 Tautological algebras Duality on orbifolds Review of the Gysin map The moduli space of stable pointed curves as a category Vector bundles on moduli spaces of curves Weighted graphs The notion of a tautological algebra A theorem of Kontsevich Faber's conjectures Primitivity of the kappa classes References Part II Introduction to Tropical Curves, Tropical Moduli and Teichmiiller Spaces 1 Introduction 1.1 A general overview 1.2 A specific motivation from geometric group theory 1.3 Interaction between moduli spaces of tropical curves and outer space of metric graphs 2 Definitions and basic facts on Out(Fn) and the outer space Xn 2.1 Combinatorial group theory and motivations from mapping class groups 2.2 Geometric group theory, metric graphs and outer space Xn 2.3 Action of Out(Fn) on outer space Xn and applications 2.4 A natural approach to construct complete geodesic metrics on Xn 3 Tropical semifields and tropical polynomials 3.1 The tropical semifield T 3.2 Tropical polynomials 4 Tropical curves 4.1 Plane tropical curves гp 4.2 Newton polygons and tropical curves 4.3 Balancing condition at vertices 4.4 Abstract smooth tropical curves 4.5 Identification between smooth tropical curves and metric graphs 5 Definitions of tropical varieties and motivations 5.1 Tropical hypersurfaces and higher codimension tropical subvarieties of Rn 5.2 Tropicalization of polynomials and tropical varieties as non-Archimedean amoebas 5.3 Tropical varieties as logarithmic limit sets 5.4 Tropical varieties as limits of Archimedean amoebas 5.5 Affine manifolds and tropical affine manifolds 6 Some alternative descriptions of tropical varieties and related spaces 6.1 Fans and tropical fans 6.2 Stacky fans 6.3 Tropical hypersurfaces as super currents 6.4 Tropical varieties versus Berkovich spaces 7 Applications of tropical geometry 7.1 Logarithmic compactification of algebraic varieties 7.2 Topology of real algebraic curves 7.3 Geometric and combinatorial group theory 7.4 Three-dimensional topology 7.5 Boundary of Teichmuller space □ 7.6 Enumerative algebraic geometry 7.7 Mirror symmetry of Calabi-Yau manifolds 7.8 Arithmetic algebraic geometry 8 Moduli spaces of tropical curves and tropical Teichmuller spaces... 8.1 Moduli space □ of tropical curves and a general philosophy on moduli spaces 8.2 Gromov-Hausdorffdistance and topology on □ 8.3 The moduli space □ and tropical Teichmuller space □ as stacky fans 8.4 The tropical Teichmiiller space □ as a tropical space 8.5 The moduli space □ as a tropical orbifold 8.6 Compactifications of the moduli space □ 9 Jacobian variety of a Riemann surface and the Siegel upper-half space 9.1 Polarized algebraic varieties 9.2 Abelian varieties 9.3 Jacobian varieties of Riemann surfaces 9.4 Moduli space of principally polarized abelian varieties □ 10 Tropical abelian varieties, their moduli spaces, and Jacobians of tropical curves 10.1 Jacobian variety of tropical curves and metric graphs 10.2 Tropical abelian varieties 10.3 Principally polarized tropical abelian varieties and their moduli space □ 11 The Torelli Theorem for tropical curves 11.1 Period of tropical curves and contractions of graphs 11.2 Tropical Torelli theorem and its failure 12 Complete invariant metrics on outer space Xn 12.1 Tropical Jacobian map and invariant complete geodesic metrics on Xn 12.2 Tropical Jacobian map and a complete pesudo-Riemannian metric on Xn 12.3 Complete pseudo-Riemannian metric on Xn via lengths of pinching loops 12.4 Finite Riemannian volume of the quotient Out(Fn)\Xn References Index
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