Abelian Groups and Fields
Definitions and examples of abelian groups and fields
Categories
- Math / Linear Algebra 20
- Math / Set Theory 26
- Math / Category Theory 9
- Math / Algebraic Structures 21
- Math / Group Theory 4
- Math / Ring Theory 3
- Math / Multilinear Algebra 14
- Math / Field Theory 10
- Math / Homological Algebra 8
- Math / Commutative Algebra 22
- Math / Representation Theory 2
- Math / Topology 19
- Math / Algebraic Topology 10
- Math / Manifold 16
- Math / Lie Theory 4
- Math / Riemannian Geometry 2
- Math / Algebraic Varieties 20
- Math / Scheme Theory 16
- Math / Toric Geometry 4
- Math / Symplectic Geometry 3
- Math / Mirror Symmetry 3
Math / Linear Algebra
Vector Spaces
The definition of vector spaces, simple properties, and examples
Subspaces
Subspaces of a vector space and linear combinations of vectors
Basis of a Vector Space
Basis of a vector space, linear combination
Dimension of Vector Spaces
Bases and Dimension of Vector Spaces
Linear Maps
The definition and examples of linear maps
Isomorphisms
Equivalent vector spaces
Matrices
Definition and operations of matrices
Space of Linear Maps
Hom and Dual Space
Fundamental Theorem of Linear Algebra
Fundamental Theorem of Linear Algebra
Gaussian Elimination
Gaussian elimination and inverse matrices
Determinant
The definition and geometric meaning of the determinant
Existence and Uniqueness of the Determinant
Existence and uniqueness proof of the determinant, and methods for computing it
Characteristic Polynomial
The characteristic polynomial of a matrix
Eigenspace Decomposition
Eigenspace decomposition of a vector space
Jordan Canonical Form
Construction of the Jordan canonical form through generalized eigenspace decomposition
Dual Space
Dual spaces, dual maps, and orthogonal complements
Bilinear Forms
Bilinear forms and dual spaces
Inner Product Spaces
Properties of inner products defined over the real numbers
Least Squares Method
Orthogonal Projection and Least Squares Method
Math / Set Theory
ZFC Axioms
The axioms of set theory
Ordered Pairs
Subset relations and the definition of ordered pairs
Binary Relations
The definition of binary relations
Operations on Binary Relations
Inverse and Composition of Binary Relations
Functions
Basic Definition of Functions
Operations on Functions
Inverse and composition of functions, surjective and injective functions
Retraction and Section
Properties of Surjective and Injective Functions
Union and Intersection
Union and intersection of sets
Sum of Sets
Sum (disjoint union) of a family of sets
Product of Sets
Product of Sets
Properties of Products
Partial products, associativity and distributivity
Equivalence Relations
The definition and properties of equivalence relations
Examples of Equivalence Relations
Examples of equivalence relations, saturation of equivalence relations, isomorphism theorems
Definition of Order Relations
Definitions and properties of order relations
Monotone Functions
Operations on ordered sets and monotone functions
Elements of Ordered Sets
Greatest, least, maximal, and minimal elements of ordered sets
Directed Sets
Directed sets and lattices
Filters, Ideals, and Galois Connections
Filter and ideal
Ordinals and Well-Ordered Sets
Definition of well-ordered sets, motivation for ordinals
Properties of Well-Ordered Sets
Definition of ordinals and properties of well-ordered sets
Axiom of Choice
The Axiom of Choice and its equivalents
Order Relations Between Ordinals
Order relations between ordinals and the rigorous definition of cardinals
Cardinals
Definition of Cardinal number
Operations on Cardinals
Operations on cardinal numbers
Natural Numbers and Infinite Sets
Definition of natural numbers and properties of infinite sets
Limits
Inverse limit and direct limit
Math / Category Theory
Categories
Definitions and basic concepts of category theory
Functors
The definition and examples of functors
Natural Transformations
Natural transformations and equivalence between categories
Representable Functors
Initial object, terminal object, representable functor
Limits
Limits and colimits
Monoidal Categories
The definition of monoidal categories and coherence conditions
Monoid Objects
Monoid objects in a monoidal category and their examples
Adjoint Functors
Definition of left and right adjoint functors
Abelian Categories
Abelian Categories
Math / Algebraic Structures
Algebraic Structures
Binary operations defined on sets
Semigroups, Monoids, and Groups
Definitions of semigroups, monoids, and groups
Grothendieck Groups
The Grothendieck group and the definition of integers
Group Homomorphisms
Definition and properties of group homomorphisms, kernels and images
Quotient Groups
Normal subgroups and quotient groups
Isomorphism Theorems
Isomorphism theorems
Direct Product of Groups
Direct product of groups
Restricted Sums
Restricted sums of groups
Free Products
Free product and universal property
Abelian Groups
Free abelian group, tensor product
Group Actions
Group action
Definition of a Ring
The definition of a ring and its basic properties
Quotient Rings and Ring Homomorphisms
Quotient ring and ring isomorphism theorems
Products, Coproducts, and Tensor Products of Rings
Definition of the product, coproduct, and tensor product of rings
Field of Fractions
Localization, ring of fractions, prime ideal
Graded Rings
Definition and basic properties of a graded ring indexed by a monoid
Modules
Definition of modules
Direct Products, Direct Sums, and Tensor Products of Modules
Product, coproduct, and tensor product in the module category
Change of Base Ring
Restriction and extension of scalars via a ring homomorphism
Graded Modules
The definition of graded modules over a graded ring
Algebras
The definition of an algebra over a commutative ring, and various kinds such as associative, unital, and commutative algebras
Math / Group Theory
Symmetric Groups
Cycle decomposition and sign of the symmetric group, and the alternating group
Group Extensions
Group extensions as short exact sequences, and semidirect products
Series of Groups
Commutators and normal, composition, and derived series; solvability
Sylow Theorems
p-subgroups of finite groups and the three Sylow theorems
Math / Ring Theory
Integral Domains
Definitions of Euclidean domains, PIDs, and UFDs, and their inclusion relationships
Chinese Remainder Theorem
Chinese remainder theorem for products of ideals and comaximal ideals
Polynomial Rings
Factorization of polynomial rings over commutative rings and Gauss’s lemma
Math / Multilinear Algebra
Exact Sequences
Exact sequences of modules, and short/long exact sequences
Projective, Injective, and Flat Modules
Definitions and equivalent conditions for projective, injective, and flat modules
Basis
Definition of free modules, basis, and the universal property
Dual Spaces
Hom of modules, dual modules, and the bidual map
Hom and the Tensor Product
Adjunction and exactness of the Hom functor and the tensor product
Matrices
Definition and multiplication of matrices over free modules over a general ring
Matrices and Linear Maps
Matrix representations of linear maps between free modules and coordinate systems
Change of Basis
Square matrices and invertible matrices, transformation of matrices under change of basis
Tensor Algebras
Tensor algebras, symmetric algebras, exterior algebras
Determinants
The determinant of an endomorphism of a free module and its basic properties
Norms and Traces
The norm and trace defined by elements of an algebra
Derivations
Differential module
Differential Modules
Differential modules with derivations on graded algebras
Symmetric Tensors
The action of the symmetric group, symmetric tensors, and symmetric powers
Math / Field Theory
Fields
The definition of a field, prime fields, and characteristic
Algebraic Extensions
The definition and degree of algebraic extensions of fields
Algebraic Closures
The existence of algebraically closed fields and algebraic closures
Radical Extensions
The definition of radical extensions and their role in Galois theory
Étale Algebras
The definition of étale algebras over a field and a characterization via diagonalizability
Separable Extensions
Characterization of separable extensions through étale algebras
Separable Degree
The decomposition of separable and inseparable degrees
Galois Extensions
The definition of a Galois extension satisfying both normal and separable
Properties of Galois Groups
The structure of infinite Galois groups with the Krull topology
The Fundamental Theorem of Galois Theory
The Galois correspondence between subgroups and intermediate fields
Math / Homological Algebra
Diagram chasing
Five lemma, snake lemma
Homology
Basic definitions
The Long Exact Sequence
The long exact sequence
Resolutions
Projective and injective resolutions in an abelian category
Derived Functors
Definition of right/left derived functors via δ-functors
Ext and Tor
Definitions and properties of Ext and Tor, the derived functors of Hom and tensor
Spectral Sequences
Spectral sequences that approximate the cohomology of a filtered complex page by page
Derived Categories
Construction of derived categories via chain complexes and quasi-isomorphisms
Math / Commutative Algebra
Basic Notions
Basic conventions and definitions for rings and algebras used in commutative algebra
Localization
The localization of rings and modules, and the construction of local rings
Properties of Localization
Compatibility of localization with Hom and tensor, and local properties
Localization of Graded Rings
Homogeneous localization of graded rings and graded modules
The Jordan-Hölder Theorem
Uniqueness of composition series and well-definedness of length
Associated Primes
The prime avoidance lemma and the definition and properties of associated primes
Primary Decomposition
Primary decomposition and uniqueness for modules over Noetherian rings
Integral Extensions
The Cayley-Hamilton theorem, integral elements, and integral extensions
Integral Extensions and Ideals
The lying over and going up theorems for integral extensions and prime ideals
Nullstellensatz
Proofs of Jacobson rings and Hilbert’s Nullstellensatz
Blowup Algebras
Rees algebras and associated graded rings constructed from an ideal
Flatness
Definition of flat modules, characterization via Tor, and basic properties
Flatness and Localization
A local criterion for flatness via checking at the maximal ideal
Completion
Completion of rings and modules defined by a filtration
Properties of Completion
Compatibility of completion with exact sequences, Artin-Rees lemma
Dimension
Krull dimension, defined by prime chains, and its basic properties
System of Parameters
The relationship between the system of parameters of a local ring and dimension
Regular Local Rings
Characterizations of regular systems of parameters and regular local rings
Fractional Ideals
Fractional ideals, invertible modules, and the Picard group
Divisors
Cartier divisors and class groups in Dedekind domains
Noether Normalization
The Noether normalization theorem for finitely generated algebras and its applications
Differentials
The algebraic definition of the Kähler differential module and its universal property
Math / Representation Theory
Representation Theory of Finite Groups
Definition of representations of finite groups and irreducible decomposition
Characters
Definition of character functions and orthogonality relations
Math / Topology
Open Sets
Topological space, open sets
Bases of a Topological Space
Bases, subbases, and local bases of a topological space
Interior, Closure, and Boundary
Basic concepts in topology
Other Definitions of Topological Spaces
Definitions of topology using closed sets, closure, and neighborhood filters
Continuous Functions
Properties of continuous functions
Initial and Final Topology
Initial/final topology and their examples
Subspaces
Properties of subspaces
Presheaves
The gluing lemma and the definition of a presheaf
Sheaves
Sheaves defined on a topological space
Quotient Spaces
Properties of Subspaces
Product Spaces
Properties of product spaces
Open Mappings and Closed Mappings
Definitions of open maps and closed maps, and their relationship to quotient maps
Hausdorff Spaces
Convergence of sequences and the Hausdorff axiom
Compact Spaces
Compact spaces, defined by the existence of a finite subcover for every open cover
Compactness and Filter Convergence
A characterization of compactness through filter convergence
Compactness
Tychonoff’s theorem, local compactness, and paracompactness
Proper Maps
The relationship between proper maps as universally closed maps and compactness
Connected Spaces
Connected spaces, path-connectedness, and connected components
Dimension
Definitions of covering dimension and Krull dimension for algebraic geometry
Math / Algebraic Topology
Topological Manifolds
The definition and properties of topological manifolds as locally Euclidean spaces
Homology
Definitions and properties of simplices
Homotopy
Classifying spaces via topological invariants and the fundamental group
Covering Spaces
Equivalent conditions for simply connected, covering spaces, and the Seifert–van Kampen theorem
Computation of Homology
Practical computation of homology via relative homology and Mayer–Vietoris
Cohomology
The definition of cohomology and the universal coefficient theorem
Acyclic models theorem
The acyclic models theorem for categories with models and its applications
Cup Product
The exterior product in cohomology, the definition of cup product, and the resulting ring structure
Poincaré Duality
Duality between homology and cohomology via orientation sheaves and fundamental classes
Characteristic Classes
The definition of characteristic classes of fiber bundles and their interpretation via classifying spaces
Math / Manifold
Smooth Manifolds
Definition of smooth manifolds
Examples of Differentiable Manifolds
Various examples of differentiable manifolds
Tangent Space
Tangent Vectors and Tangent Space
Cotangent Space
Tangent Vectors and Tangent Space
Differentials
The differential between two tangent spaces
Examples of Differentials
Examples of smooth functions and differentials
Submanifolds and the Inverse Function Theorem
Substructures of differentiable manifolds
Uniqueness of Submanifolds
The topological structure of immersed submanifolds and the factorization of smooth functions
Implicit Function Theorem
The implicit function theorem on differentiable manifolds and its consequences
Tangent and Cotangent Bundles
The definition of vector bundles and the tangent and cotangent bundles
Vector Fields
Vector fields
Differential Forms
Differential form
Distribution
The definition of a distribution and the Frobenius theorem
Lie Derivative
Lie derivative and Lie bracket
Differential Ideal
Differential ideals and Frobenius’s theorem
Orientation
Orientation on a manifold
Math / Lie Theory
Lie Groups
Definition and properties of Lie groups
Torus Actions
Actions of a torus and weight space decomposition
Root Systems
Root systems obtained from the weight decomposition of the adjoint representation
Borel Subgroups and Flag Varieties
Dynkin diagrams, ADE classification, and flag varieties
Math / Riemannian Geometry
Riemannian Metric
The Riemannian metric as a positive-definite symmetric 2-tensor on the tangent bundle
Connection
Differentiation on a vector bundle
Math / Algebraic Varieties
Affine Varieties
Affine varieties and their basic properties
Projective Varieties
Projective varieties and homogeneous coordinates
Quasi-Projective Varieties
Quasi-projective varieties and regular maps
Rational Maps
Rational maps and birational equivalence
Dimension
Dimension of algebraic varieties
Tangent Spaces and Smoothness
Tangent spaces and smoothness of algebraic varieties
Grassmannians
Grassmannians as parameter spaces of linear subspaces
Divisors
Weil divisors, Cartier divisors, and divisor class groups
Line Bundles and Vector Bundles
Line bundles, invertible sheaves, and the Picard group
Linear Systems
Complete linear systems, base loci, and ampleness
Canonical Bundle
Canonical Bundle and Canonical Divisor
Sheaf Cohomology
Sheaf cohomology and its applications
Cohomology of Projective Space
Bott’s formula and the cohomology of line bundles on projective space
Serre Duality
Serre duality theorem and its applications
The Riemann–Roch Theorem for Curves
The Riemann–Roch theorem for curves
The Riemann–Roch Theorem for Surfaces
Intersection theory on surfaces and its applications
Kodaira Vanishing Theorem
The Kodaira vanishing theorem and its applications
Chow Groups
Chow groups and the cycle class map
Intersection Product
The intersection product on Chow groups
Bézout’s Theorem
Bézout’s theorem and its applications
Math / Scheme Theory
Spectra
The prime spectrum of a commutative ring and the Zariski topology
Affine Schemes
Affine schemes defined by the structure sheaf on the spectrum of a ring
Schemes
The definition of a scheme as a locally affine locally ringed space
Topological Structure of Schemes
Generic points, Zariski topology, and irreducible components
Algebra of Schemes
Definitions and properties of reduced schemes and integral schemes
Morphisms of Schemes
Four perspectives on scheme morphisms as morphisms of locally ringed spaces
Properties of Scheme Morphisms
Basic properties of scheme morphisms such as affine, finite, and finite type
Valuation Rings
Valuative criteria for separatedness and properness
Flat Morphisms
Flat morphisms in algebraic geometry
Fiber Products
Definition and existence of fiber products in the category of S-schemes
Closed Subschemes
Closed subschemes and vanishing schemes defined by ideal sheaves
Projective Schemes
The Proj construction from a graded ring and projective schemes
Closed Subschemes of Projective Space
The correspondence between closed subschemes of projective space and homogeneous ideals
Complete Intersections
Codimension of vanishing schemes and complete intersections
Dimension
The definition of dimension for schemes and its relation to the Krull dimension of local rings
Algebraic Groups
Algebraic group action
Math / Toric Geometry
Affine Toric Varieties
Construction of affine toric varieties from strongly convex rational polyhedral cones
Definition of Toric Varieties
General toric varieties obtained by gluing affine toric varieties from a fan
Torus-Invariant Divisors and Line Bundles
Torus-invariant divisors and line bundles arising from the rays of a fan
Fano Varieties
Reflexive polytopes and the corresponding Gorenstein Fano toric varieties
Math / Symplectic Geometry
Classical Mechanics
Classical mechanics and phase space
Symplectic Vector Spaces
Definition of symplectic forms
Symplectic Manifolds
Definitions and properties of symplectic manifolds
Math / Mirror Symmetry
Overview of Mirror Symmetry
Historical background and the Hori-Vafa mirror
Frobenius Manifolds
Frobenius manifolds and the WDVV equation
Dubrovin Connection
A flat connection on a Frobenius manifold with a spectral parameter