by Cumrun Vafa, Eric Zaslow
Publisher: American Mathematical Society 2003
Number of pages: 950
The aim of the book is to provide a pedagogical introduction to the field of mirror symmetry from both a mathematical and physical perspective. After covering the relevant background material, the main part of the monograph is devoted to the proof of mirror symmetry from various viewpoints. More advanced topics are also discussed. In particular, topological strings at higher genera and the notion of holomorphic anomaly.
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by J. S. Milne
This is an introduction to the arithmetic theory of modular functions and modular forms, with an emphasis on the geometry. Prerequisites are the algebra and complex analysis usually covered in advanced undergraduate or first-year graduate courses.
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From the table of contents: Affine Varieties; Ideals and varieties. Hilbert's Basis Theorem. Regular functions and regular mappings. Projective and Abstract Varieties; Dimension Theory; Regular and singular points; Intersection theory.
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The monograph gives a detailed exposition of the algorithmic real algebraic geometry. It is well written and will be useful both for beginners and for advanced readers, who work in real algebraic geometry or apply its methods in other fields.
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This book discusses two subjects of quite different nature: Construction methods for quotients of quasi-projective schemes by group actions or by equivalence relations and properties of direct images of certain sheaves under smooth morphisms.