A Panorama Of Pure Mathematics By J Dieudonne Pdf

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Transcript of Dieudonne, Panorama of Pure Mathematics

• CoverTitle pageDate-lineContentsIntroductionA I. Algebraic and differential topology1. Techniques. Homotopy. Homotopy groups. Homotopy and cohomology. Homology and cohomology. Cohomology and homology rings. Fibrations2. Results. The different sorts of "manifolds." The Poincard conjecture. Cobordism. Immersions, embeddings, and knot theory. Fixed points; spaces with group action3. Connections with the natural sciences4. The originatorsReferencesA II. Differential manifolds. Differential geometry1. The general theory. Singularities of differentiable mappings. Vector fields on differential manifolds2. G-structures. Riemannian manifolds3. The topology of differential manifolds4. Infinite-dimensional differential manifolds5. Connections with the natural sciences6. The originatorsReferencesA III. Ordinary differential equations1. The algebraic theory2. Ordinary differential equations in the complex domain3. The qualitative study of ordinary differential equations4. The classification problem5. Boundary-value problems6. Connections with the natural sciences7. The originatorsReferencesA IV. Ergodic theory1. The ergodic theorem2. Classification problems3. Connections with the natural sciences4. The originatorsReferencesA V. Partial differential equations1. The local study of differential systems2. Completely integrable systems and foliations3. Linear partial differential equations: general theory. Problems. Techniques. Results4. Equations with constant coefficients. Invariant operators on homogeneous spaces5. Boundary-value problems for linear equations: I. General theory6. Boundary-value problems for linear equations: II. Spectral theory of elliptic operators. Second-order elliptic operators and potential theory7. Boundary-value problems for linear equations: III. Equations of evolution. Strictly hyperbolic equations. Parabolic equations8. Pseudodifferential operators on compact manifolds9. Nonlinear partial differential equations10. Connections with the natural sciences11. The originatorsReferencesA VI. Noncommutative harmonic analysis1. Elementary cases: compact groups and abelian groups2. The fundamental problems3. Harmonic analysis on real reductive Lie groups4. Harmonic analysis on reductive $p$-adic groups5. Harmonic analysis on nilpotent and solvable Lie groups6. Representations of group extensions7. Connections with the natural sciences8. The originatorsReferencesA VII. Automorpbic forms and modular forms1. The analytic aspect2. The intervention of Lie groups3. The intervention of adele groups4. Applications to number theory, (a) Extensions of abelian class-field theory, (b) Elliptic curves and modular forms, (c) The Ramanujan-Petersson conjecture, (d) Congruences and modular forms5. Automorphic forms, abelian varieties, and class fields6. Relations with the arithmetic theory of qnadratic forms7. Connections with the natural sciences8. The originatorsReferencesA VIII. Analytic geometry1. Functions of several complex variables and analytic spaces. Domains of holomorphy and Stein spaces. Analytic subspaces and coherent sheaves. Globalization problems. Continuation problems. Properties of morphisms and automorphisms. Singularities of analytic spaces. Singularities of analytic functions; residues2. Compact analytic spaces; Kahler manifolds. Classification problems3. Variations of complex structures and infinite-dimensional manifolds4. Real and $p$-adic analytic spaces5. Connections with the natural sciences6. The originatorsReferencesA IX. Algebraic geometry1. The modern framework of algebraic geometry2. The fundamental notions of the theory of schemes, (a) Local properties and global properties, (b) Quasi-coherent Modules and subschemes. (c) Relativization and base change, (d) The various types of morphisms. (e) Techniques of construction and representable functors3. The study of singularities4. The "transcendental" theory of algebraic varieties. Monodromy. Topology of subvarieties. Algebraic cycles. Divisors and abelian varieties. Divisors and vector bundles. Ample divisors and projective embeddings5. Cohomology of schemes. The various cohomologies. Intersection multiplicities and homology. The fundamental group and monodromy6. Classification problems. The classification of surfaces. "Moduli" problems7. Algebraic groups. Abelian varieties. Linear algebraic groups. Invariant theory8. Formal schemes and formal groups9. Connections with the natural sciences10. The originatorsReferencesA X. Theory of numbers1. The modern viewpoint in number theory. Local fields, adeles and ideles. Zeta functions and L-functions. Local fields and global fields2. Class-field theory. Particular class-fields. Galois extensions of local and global fields3. Diophantine approximations and transcendental numbers4. Diophantine geometry. Diophantine geometry over a finite field. Abelian varieties denned over local and global fields. Diophantine geometry over a ring of algebraic integers5. Arithmetic linear groups. The arithmetic theory of quadratic forms6. Connections with the natural sciences7. The originatorsReferencesB I. Homological algebra1. Derived functors in abelian categories. Examples of derived functors2. Cohomology of groups. Variants. Galois cohomology3. Cohomology of associative algebras4. Cohomology of Lie algebras5. Simplicial structures6. K-theory7. Connections with the natural sciences8. The originatorsReferencesB II. Lie groups1. Structnre theorems2. Lie groups and transformation groups3. Topology of Lie groups and homogeneons spaces4. Connections with the natural sciences5. The originatorsReferencesB III. Abstract groups1. Generators and relations2. Chevalley groups and Tits systems3. Linear representations and characters. The classical theory. The modular theory. Characters of particular groups4. The search for finite simple groups5. Connections with the natural sciences6. The originatorsReferencesB IV. Commutative harmonic analysis1. Convergence problems2. Normed algebras in harmonic analysis. Homomorphisms and idempotent measures. Sets of uniqueness and pseudofunctions. The algebras A(E) and harmonic synthesis. Functions acting on algebras3. Symmetric perfect sets in harmonic analysis: connections with number theory4. Almost-periodic functions and mean-periodic functions5. Applications of commutative harmonic analysis6. Connections with the natural sciences7. The originatorsReferencesB V. Von Neumann algebras1. Tomita theory and the Connes invariants2. Applications to $C^3. Connections with the natural sciences4. The originatorsReferencesB VI. Mathematical logic1. Noncontradiction and undecidability2. Uniform effective procedures and recursive relations3. The technique of ultraproducts4. Connections with the natural sciences5. The originatorsReferencesB VII. Probability theory1. Fluctuations in sequences of independent random variables2. Inequalities for martingales3. Trajectories of processes4. Generalized processes5. Random variables with values in locally compact groups6. Connections with the natural sciences7. The originatorsReferencesC I. Categories and sheaves1. Categories and functors. Opposite categories; contravariant functors. Functorial morphisms2. Representable functors. Examples: final objects, products, kernels, inverse limits. Dual notions. Adjoint functors. Algebraic structures on categories3. Abelian categories4. Sheaves and ringed spaces. Direct and inverse images of sheaves. Morphisms of ringed spaces5. Sites and topoi6. Connections with the natural sciences7. The originatorsReferencesC II. Commutative algebra1. The principal notions. Localization and globalization. Finiteness conditions. Linear algebra over rings. Graduations and nitrations. Topologies and completions. Dimension. Integral closure. Excellent rings. Henselian rings. Valuations and absolute values. Structure of complete Noetherian local rings2. Problems of field theory. Quasi-algebraically closed fields. Subextensions of a pure transcendental extension. Hilbert's 14th problem3. Connections with the natural sciences4. The originatorsReferencesC III. Spectral theory of operators1. Riesz-Frednolm theory. Refinements and generalizations2. Banach algebras3. Hllbert-von Neumann spectral theory4. Connections with the natural sciences5. The originatorsReferencesBibliographyIndex

A Panorama Of Pure Mathematics By J Dieudonne Pdf

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