Hessian Polyhedra, Invariant Theory and Appell Hypergeometric Functions

图书简介

Our book gives the complex counterpart of Klein?s classic book on the icosahedron. We show that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equation of Picard modular functions (arithmetic) are in fact dominated by the structure of a single object, the Hessian group ???216. It provides another beautiful example on the fundamental unity of mathematics. Key Features: o Our book is the first book (in fact, the unique book up to the present), whose contents can compare with the contents in Klein?s icosahedral book (such as hypergeometric functions, invariant theory, the resolution of algebraic equations and modular equations) o Our book emphasizes the fundamental unity of mathematics (i.e., geometry, algebra, analysis and arithmetic are interacted with each other by the Hessian group) o Our book gives a higher dimensional analogue of Jacobi?s celebrated work "Fundamenta nova theoriae functionum" on the transformation property of elliptic integrals. Our book also provides a higher dimensional analogue of Dedekind?s celebrated ?-functions and Schwarz?s celebrated triangular s-functions

Geometry and Arithmetic Associated with Appell Hypergeometric Functions: Introduction; Four Derivatives and Their Properties; Nonlinear Partial Differential Equations and Modular Functions Associated to U(2, 1); ?-Functions Associated to U(2, 1) and Picard Curves; Transform Problems and Modular Equations Associated to Algebraic Surfaces; Triangular S-Functions Associated to U(2, 1) and Picard Curves; Four Derivatives and Nonlinear Evolution Equations; Hessian Polyhedra, Invariant Theory and Appell Hypergeometric Functions: Introduction; The Reciprocity Law About the Appell Hypergeometric Functions; The Algebraic Solutions of Appell Hypergeometric Partial Differential Equations; The Hessian Polyhedral Equations; Invariant Theory for the System of Algebraic Equations; Ternary Cubic Forms Associated to Hessian Polyhedra; Some Rational Invariants on ??2; Galois Representations Arising from Twenty-Seven Lines on a Cubic Surface and The Arithmetic Associated with Hessian Polyhedra: Introduction; Hessian Polyhedra and Cubic Forms Associated to G25, 920; Hessian Polyhedra and Picard Modular Forms; Hessian Polyhedra and Galois Representations Associated with Cubic Surfaces; Hessian Polyhedra and the Arithmetic of Rigid Calabi-Yau Threefolds;

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Hessian Polyhedra, Invariant Theory and Appell Hypergeometric Functions Hessian Appell 数学分析

Hessian Polyhedra, Invariant Theory and Appell Hypergeometric Functions

Hessian Appell
数学分析