Riemann Legacy

Riemannian Ideas in Mathematics and Physics

Häftad, Engelska, 2010

AvKrzysztof Maurin

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The study of the rise and fall of great mathematical ideas is undoubtedly one of the most fascinating branches of the history of science. It enables one to come into contact with and to participate in the world of ideas. Nowhere can we see more concretely the enormous spiritual energy which, initially still lacking clear contours, begs to be moulded and developed by mathematicians, than in Riemann (1826-1866). He perceived mathematics and physics as one discipline and thought of himself as both mathematician and physicist. His ideas as well as their contemporary descendants are the theme of this book. Audience: This volume will be useful to those interested in such diverse fields as the mathematics of physics, algebra and number theory, topology and geometry, analysis, and the history of science.

Produktinformation

Utgivningsdatum2010-12-06
Mått155 x 235 x 40 mm
Vikt1 107 g
FormatHäftad
SpråkEngelska
SerieMathematics and Its Applications
Antal sidor719
FörlagSpringer
ISBN9789048148769

Tillhör följande kategorier

I Riemannian Ideas in Mathematics and Physics.- 1 Gauss Inner Curvature of Surfaces.- 2 Sectional Curvature. Spaces of Constant Curvature. Weyl Hypothesis.- 3 Cohomology of Riemann spaces. Theorems of de Rham, Hodge, Kodaira.- 4 Chern—Gauss—Bonnet theorem.- 5 Curvature and Topology or Characteristic Forms of Chern, Pontriagin, and Euler.- 6 Kähler Spaces. Bergman Metrics. Harish-Chandra-Cartan Theorem. Siegel Space (once again!).- II General Structures of Mathematics.- 1 Differentiable Structures. Tangent Spaces. Vector Fields.- 2 Projective (Inverse) Limits of Topological Spaces.- 3 Inductive Limits. Presheaves. Covering Defined by Presheaf.- 4 Algebras. Groups, Tensors, Clifford, Grassmann, and Lie Algebras. Theorems of Bott—Milnor, Wedderburn, and Hurwitz.- 5 Fields and their Extensions.- 6 Galois Theory. Solvable Groups.- 7 Ruler and Compass Constructions. Cyclotomic Fields. Kronecker—Weber Theorem.- 8 Algebraic and Transcendental Elements.- 9 Weyl principle.- 10 Topology of Compact Lie Groups.- 11 Representations of Compact Lie Groups.- 12 Nilpotent, Semimple, and Solvable Lie Algebras.- 13 Reflections, Roots, and Weights. Coxeter and Weyl groups.- 14 Covariant Differentiation. Parallel Transport. Connections.- 15 Remarks on Rich Mathematical Structures of Simple Notions of Physics Based on Example of Analytical Mechanics.- 16 Tangent Bundle TM. Vector, Fiber, Tensor and Tensor Densities, and Associate Bundles.- 17 G-spaces. Group Representations.- 18 Principal and Associated Bundles.- 19 Induced Representations and Associated Bundles.- 20 Vector Bundles and Locally Free Sheaves.- 21 Axiom of Covering Homotopy.- 22 Serre Fibering. General Theory of Connection. Corollaries.- 23 Homology. Cohomology. de Rham Cohomology.- 24 Cohomology of Sheaves. Abstract deRham Theorem.- 25 Homotopy Group ?k(X, x0). Hopf Fibering. Serre Theorem on Exact Sequence of Homotopy Groups of a Fibering.- 26 Various Benefits of Characteristic Classes (Orientability, Spin Structures). Clifford Groups, Spin Group.- 27 Divisors and Line Bundles. Algebraic and Abelian Varieties.- 28 General Abelian Varieties and Theta Function.- 29 Theorems on Algebraic Dependence.- III The Idea of the Riemann Surface.- IV Riemann and Calculus of Variations.- 1 Introduction.- 2 The Plateau Problem.- 3 Teichmüller Theory. Riemann Moduli Problem.- 4 Riemannian Approach to Teichmiiller Theory. Harmonic Maps and Teichmüller Space.- 5 Teichmüller theory and Plateau—Douglas problem.- 6 Rescuing Riemann’s Dirichlet Principle. Potential Theory.- 7 The Royal Road to Calculus of Variations (Constantin Carathéodory).- 8 Symplectic and Contact Geometries. Conservation Laws.- 9 Direct Methods in Calculus of Variations for Manifolds with Isometries. Equivariant Sobolev Theorems. Yamabe Problem and its Relation to General Relativity.- V Riemann and Complex Geometry.- 1 Introduction.- 2 On Complex Analysis in Several Variables.- 3 Ellipticity, Runge Property, and Runge Type Theorems.- 4 Hörmander Method in Complex Analysis.- 5 Wirtinger Theorems. Metric Theory of Analytic Sets.- 6 The Problem of Poincaré and the Cousin Problems.- 7 Ringed Spaces and General Complex Spaces.- 8 Construction of Complex Spaces by Gluing and by Taking Quotient.- 9 Differential Geometry of Holomorphic Vector Bundles over Compact Riemann Surfaces and Kähler manifolds. Stable Vector Bundles, Hermite-Einstein Connections, and their Moduli Spaces.- VI Riemann and Number Theory.- 1 Introduction.- 2 The Riemann ? function.- 3 Hecke Theory.- 4 Dedekind ?K function for number field K and Selberg ?function.- Concluding Remarks.- Suggestions for Further Reading.