Ebook: Contemporary trends in algebraic geometry and algebraic topology

In the 25 years of its existence Soliton Theory has drastically expanded our understanding of "integrability" and contributed a lot to the reunification of Mathematics and Physics in the range from deep algebraic geometry and modern representation theory to quantum field theory and optical transmission lines. The book is a systematic introduction to the Soliton Theory with an emphasis on its background and algebraic aspects. It is the first one devoted to the general matrix soliton equations, which are of great importance for the foundations and the applications.Differential algebra (local conservation laws, Bäcklund-Darboux transforms), algebraic geometry (theta and Baker functions), and the inverse scattering method (Riemann-Hilbert problem) with well-grounded preliminaries are applied to various equations including principal chiral fields, Heisenberg magnets, Sin-Gordon, and Nonlinear Schrödinger equation Mathematics in the 20th Century (M Atiyah); The [PHI]4 of Minimal Gorenstein 3-Folds of General Type (M Chen); Morphisms of Curves and the Fundamental Group (M Cushman); Iterated Integrals and Algebraic Cycles: Examples and Prospects (R Hain); Chen's Interated Integrals and Algebraic Cycles (B Harris); On Algebraic Fiber Spaces (Y Kawamata); Local Holomorphic Isometric Embeddings Arising from Correspondences in the Rank-1 Case (N Mok); Multiple Polylogarithms: Analytic Continuation, Monodromy, and Variations of Mixed Hodge Structures (J-Q Zhao); Deformation Types of Real and Complex Manifolds (F M E Catanese); The Life and Work of Kuo-Tsai Chen (R Hain & P Tondeur)