Hilgert J. Mathematical Structures. From Linear Algebra...Rings to Geometry 2024 torrent

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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