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SOURCE: Hilgert J. Mathematical Structures. From Linear Algebra...Rings to Geometry 2024
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MEDIAINFO
Textbook in PDF format This textbook is intended to be accessible to any second-year undergraduate in mathematics who has attended courses on basic real analysis and linear algebra. It is meant to help students to appreciate the diverse specialized mathematics courses offered at their universities. Special emphasis is on similarities between mathematical fields and ways to compare them. The organizing principle is the concept of a mathematical structure which plays an important role in all areas of mathematics. The mathematical content used to explain the structural ideas covers in particular material that is typically taught in algebra and geometry courses. The discussion of ways to compare mathematical fields also provides introductions to categories and sheaves, whose ever-increasing role in modern mathematics suggests a more prominent role in teaching. The book is the English translation of the second edition of “Mathematische Strukturen” (Springer, 2024) written in German. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content. Preface Algebraic Structures Rings Elementary Definitions and Examples Some Structure Theory for Rings Special Classes of Rings Modules Structure Theory of Modules Elementary Definitions and Examples Homomorphisms, Submodules, and Quotient Modules Generated and Free Modules Excursion: Direct Sums and Products Applications to Linear Mappings Decomposition of Finitely Generated Modules over Euclidean Rings Minimal Polynomial and Rational Normal Form Characteristic Polynomial and Jordan Normal Form Multilinear Algebra Tensor Products Tensor Products of Two Modules over R Tensor Products of Modules over Commutative Rings Tensor Algebras Universal Property and Construction Symmetric and Exterior Algebras The Symmetric Algebra over M The Exterior Algebra over M Pattern Recognition Universal Algebra Naive Category Theory Categories Functors Categorical Constructions: Limits Products and Sums Adjoint Functors Uniqueness of Adjoint Functors Equivalence of Categories Part II Local Structures Sheaves Presheaves and Sheaves Sheaves Sheafification of Presheaves Étalé Spaces Ringed Spaces Module Sheaves Manifolds Charts and Parametrizations Tangent Spaces and Derivatives Tangent Spaces Derivatives Tangent and Tensor Bundles Cotangent Bundle Vector Bundles Tensor Bundles Vector Fields Differential Forms The Exterior Derivative Integration on Real Manifolds Partition of Unity Integration of n-Forms and Stokes' Theorem Applications to Complex Differentiability Algebraic Varieties Algebraic Sets Hilbert's Nullstellensatz Construction of the Structure Sheaf Algebraic Varieties The Noether Property Prevarieties over Algebraically Closed Fields Varieties over Algebraically Closed Fields Schemes Geometric Interpretation of Affine Schemes Schemes Over (General) Fields Outlook Transfer of Arguments and Structures Technologies of Structural Comparison Homological Algebra Group Objects and Group Actions New Structures from Orbits and Quotients Specialization, Generalization and Unification of Structures Special Tensors Pseudo-Riemannian Metrics Symplectic Forms Kähler Manifolds Poisson Manifolds Connections and Fiber Bundles Covariant Derivatives Curvature Fiber Bundles Connections on Fiber Bundles Structured Analysis? Algebraic Analysis References Index
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