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E-grāmata: Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

  • Formāts: PDF+DRM
  • Sērija : Theoretical and Mathematical Physics
  • Izdošanas datums: 11-Feb-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540711759
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  • Formāts: PDF+DRM
  • Sērija : Theoretical and Mathematical Physics
  • Izdošanas datums: 11-Feb-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783540711759
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This book gives an introduction to modern geometry. Starting from an elementary level, the author develops deep geometrical concepts that play an important role in contemporary theoretical physics, presenting various techniques and viewpoints along the way. This second edition contains two additional, more advanced geometric techniques: the modern language and modern view of Algebraic Geometry and Mirror Symmetry.



This book gives an introduction to modern geometry. Starting from an elementary level, the author develops deep geometrical concepts that play an important role in contemporary theoretical physics, presenting various techniques and viewpoints along the way. This second edition contains two additional, more advanced geometric techniques: the modern language and modern view of Algebraic Geometry and Mirror Symmetry.

Recenzijas

From the reviews of the second edition:









"As the title suggests, this book is an introduction to Riemann surfaces, with the target audience being students of string theory. an excellent book to use to become familiar with these concepts, and as a result the book is able to touch on a wide variety of concepts which are not broached by more traditional treatments of the subject. I would certainly recommend the book for anyone who wants an enjoyable conceptual introduction to what can be a highly technical subject." (Mark Gross, Mathematical Reviews, Issue 2008 k)



"This book is an introduction to the language of modern algebraic geometry, designed mainly for Physics students who are interested in string theory. Overall the book is very readable and it fulfills its goal remarkably well that are useful to physicists interested in string theory, with all the necessary references for further reading. This book will be an excellent addition to the bookshelf of any physics student or researcher who wants to learn about the mathematical aspects of string theory." (Valentino Tosatti, Zentrablatt MATH, Vol. 1153, 2009)

Introduction from a Physicist's Viewpoint 1(6)
Manifolds
7(10)
Generalities
7(2)
Complex Manifolds
9(4)
The Classification Problem
13(4)
Hints for Further Reading
14(3)
Topology of Riemann Surfaces
17(14)
Fundamental Group
17(4)
Simplicial Homology
21(7)
Universal Covering Space
28(3)
Hints for Further Reading
29(2)
Analytic Structure
31(12)
Holomorphic and Meromorphic Functions
31(4)
Divisors and the Theorem of Riemann--Roch
35(3)
Meromorphic Functions on the Torus
38(5)
Hints for Further Reading
41(2)
Differentials and Integration
43(10)
Tangent Space and Differentials
43(5)
Differential Forms of Second Order
48(2)
Integration
50(3)
Hints for Further Reading
52(1)
Tori and Jacobians
53(8)
Higher Dimensional Tori
53(2)
Jacobians
55(6)
Hints for Further Reading
59(2)
Projective Varieties
61(10)
Generalities
61(4)
Embedding of One-Dimensional Tori
65(2)
Theta Functions
67(4)
Hints for Further Reading
69(2)
Moduli Spaces of Curves
71(16)
The Definition
71(3)
Methods of Construction
74(4)
The Geometry of the Moduli Space and Its Compactification
78(9)
Hints for Further Reading
85(2)
Vector Bundles, Sheaves and Cohomology
87(16)
Vector Bundles
87(4)
Sheaves
91(4)
Cohomology
95(8)
Hints for Further Reading
100(3)
The Theorem of Riemann--Roch for Line Bundles
103(16)
Divisors and Line Bundles
103(6)
An Application: The Krichever--Novikov Algebra
109(10)
Hints for Further Reading
117(2)
The Mumford Isomorphism on the Moduli Space
119(14)
The Mumford Isomorphism
119(6)
The Grothendieck--Riemann--Roch Theorem
125(8)
Hints for Further Reading
131(2)
Modern Algebraic Geometry
133(22)
Varieties
133(6)
The Spectrum of a Ring
139(7)
Homomorphisms
146(3)
Noncommutative Spaces
149(6)
Hints for Further Reading
153(2)
Schemes
155(14)
Affine Schemes
155(4)
General Schemes
159(3)
The Structure Sheaf OR
162(2)
Examples of Schemes
164(5)
Hints for Further Reading
167(2)
Hodge Decomposition and Kahler Manifold
169(14)
Some Introductory Remarks on Mirror Symmetry
169(2)
Compact Complex Manifolds and Hodge Decomposition
171(6)
Kahler Manifolds
177(4)
Hodge Numbers of the Projective Space
181(2)
Hints for Further Reading
182(1)
Calabi-Yau Manifolds and Mirror Symmetry
183(20)
Calabi-Yau Manifolds
183(4)
K3 Surfaces, Hypersurfaces and Complete Intersections
187(5)
Geometric Mirror Symmetry
192(4)
Example of a Calabi-Yau Three-fold and Its Mirror: Results of Givental
196(7)
Hints for Further Reading
200(3)
Appendix p-adic Numbers 203(10)
Index 213
Martin Schlichenmaier is full professor for mathematics at the University of  Luxemburg. He has held several teaching and research positions in the mathematics department of the University of Mannheim.