Limits of Functions
Defining limits of functions via ε-δ and proving limit laws and the squeeze theorem
Categories
- Calculus 20
- Linear Algebra 27
- Set Theory 26
- Category Theory 9
- Algebraic Structures 22
- Group Theory 4
- Ring Theory 6
- Multilinear Algebra 15
- Field Theory 10
- Homological Algebra 8
- Commutative Algebra 22
- Representation Theory 2
- Topology 19
- Algebraic Topology 12
- Manifolds 15
- Lie Theory 6
- Riemannian Geometry 2
- Algebraic Varieties 20
- Scheme Theory 20
- Toric Geometry 4
- Symplectic Geometry 3
- Mirror Symmetry 4
Calculus
Continuous Functions
Definition of continuity and properties of continuous functions: extreme and intermediate value theorems
Limits of Sequences
Convergence, limit laws, standard limits, e, and monotone convergence
Infinite Series
Partial sums and convergence, geometric and p-series, convergence tests, absolute and conditional convergence
Power Series
Power series, radius of convergence, elementary function expansions, and analytic functions
Differentiation and Derivatives
Definition of derivative, differentiability and continuity, derivatives and higher-order derivatives
Differentiation
Termwise differentiation of power series, derivatives of elementary functions, product, quotient, and chain rules
Mean Value Theorem
Mean value theorem and its applications: monotonicity, extrema, convexity, L’Hospital’s rule, optimization
Taylor’s Theorem
Taylor polynomials, Lagrange remainder, Maclaurin series, approximation and limits
Integration
Antiderivatives, Riemann sums, definite integrals, properties, and the mean value theorem
The Fundamental Theorem of Calculus
Fundamental theorem, existence of primitives, Leibniz rule, termwise integration of power series
Improper Integrals
Infinite and singular integrals, comparison test, absolute convergence
Curves and Vector-Valued Functions
Vector-valued functions, parametric curves, velocity and tangent, arc length, curvature and acceleration decomposition
Functions of Several Variables and Partial Derivatives
Partial derivatives, gradient, differentiability, multivariable chain rule, extrema
Multiple Integrals
Multiple integrals, Fubini’s theorem, change of variables and the Jacobian determinant
Vector Fields
Vector fields, gradient and conservative fields, divergence and curl, differential identities
Line Integrals
Scalar and vector line integrals, work, fundamental theorem, and conservative fields
Green’s Theorem
Green’s theorem, area formulas, circulation and flux forms, simply connected domains and conservative fields
Surface Integrals and Flux
Parametric surfaces, normal vectors, surface area, scalar surface integrals, flux
The Divergence Theorem and Stokes’ Theorem
Divergence theorem, Stokes’ theorem, irrotational and conservative fields, unification of integral theorems
Linear Algebra
Abelian Groups and Fields
Definitions and examples of abelian groups and fields
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
Quotient Spaces
Quotient spaces of vector spaces by subspaces
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
Constructing Jordan form via generalized eigenspace decomposition
Minimal Polynomial
Cayley-Hamilton theorem and minimal polynomial
Dual Space
Dual spaces, dual maps, and orthogonal complements
Bilinear Forms
Bilinear forms and dual spaces
Inner Product Spaces
Properties of inner products over the real numbers
Complex Inner Product Spaces
Hermitian inner products over complex numbers
Least Squares Method
Orthogonal projections and least squares
Spectral Theorem
Orthogonal diagonalization of self-adjoint operators
Complex Spectral Theorem
unitary diagonalization of normal operators
Singular Value Decomposition
Orthogonal decomposition of arbitrary real matrices
Quadratic Forms and Sylvester’s Law of Inertia
Classification of real symmetric bilinear forms
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, and 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
Category Theory
Category
Definition and basic concepts of categories
Functor
Definition and examples of functors
Natural Transformations
Natural transformations and equivalence between categories
Representable Functors
Initial objects, terminal objects, and representable functors
Limits
Limits and colimits
Monoidal Categories
Definition and coherence conditions of monoidal categories
Monoid Objects
Monoid objects in monoidal categories and their examples
Adjoint Functors
Definitions of left and right adjoint functors
Abelian Categories
Abelian categories
Algebraic Structures
Algebraic Structures
Binary operations defined on a set
Semigroups, Monoids, and Groups
Definitions of semigroups, monoids, and groups
Grothendieck Groups
The Grothendieck group and the definition of integers
Group Homomorphisms
Definitions and properties of group homomorphisms, kernels and images
Quotient Groups
Normal subgroups and quotient groups
Group Homomorphisms
Homomorphism theorems
Direct Product of Groups
Categorical definition and universal property of group products
Restricted Direct Sum
Restricted sums of groups
Free Products
Free products and the universal property
Abelian Groups
Free abelian groups and tensor products
Group Actions
Group actions on sets
Definition of a Ring
Definition and basic properties of rings
Quotient Rings and Ring Homomorphisms
Quotient rings and ring isomorphism theorems
Products, Coproducts, and Tensor Products of Rings
Categorical definitions of ring products, coproducts, and tensor products
Field of Fractions
Localization, ring of fractions, prime ideal
Graded Rings
Definition and basic properties of monoid-indexed graded rings
Modules
Definition of a module
Direct Products, Direct Sums, and Tensor Products of Modules
Products, coproducts, and tensor products in the module category
Change of Scalars
Restriction and extension of scalars via ring homomorphism
Graded Modules
Definition of graded modules over a graded ring
Algebras
Definitions and types of algebras over commutative rings
Direct Product, Direct Sum, and Tensor Product of Algebras
Product, direct sum, and tensor product structures of algebras
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 a Group
Commutator, normal, composition, and derived series, solvability
Sylow Theorems
p-subgroups of finite groups and Sylow’s three theorems
Ring Theory
Invertible Elements and Zero Divisors
Units, regular elements, and when regular elements become units in finite commutative rings
Integral Domains
Definitions of Euclidean domains, PIDs, and UFDs, and their inclusion relationships
Chinese Remainder Theorem
Chinese remainder theorem for comaximal ideals
Polynomial Rings
Factorization in polynomial rings over commutative rings and Gauss’s lemma
Division Rings
Division rings, quaternions, Wedderburn’s little theorem, and Schur’s lemma
Central Idempotents and Ring Decomposition
Central idempotents, ring product decomposition, and the Chinese remainder theorem
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
Bases
Definition of free modules, bases, and the universal property
Dual Spaces
Hom of modules, dual modules, and bidual maps
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 Product of Matrices
The Kronecker product as matrix realization of tensor products
Tensor Algebra
Tensor algebra, symmetric algebra, exterior algebra
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
Symmetric Tensors
Symmetric group actions, symmetric tensors, and symmetric powers
Derivations
Differential modules
Differential Modules
Differential modules with derivations on graded algebras
Field Theory
Fields
Definition of a field, prime field, and characteristic
Algebraic Extensions
The definition and degree of algebraic extensions of fields
Algebraic Closures
The existence of algebraically closed fields and algebraic closures
Purely Inseparable Extensions
Definition and role of p-radical extensions 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
Definition of Galois extensions satisfying normality and separability
Properties of Galois Groups
Structure of infinite Galois groups with the Krull topology
The Fundamental Theorem of Galois Theory
The Galois correspondence between subgroups and intermediate fields
Homological Algebra
Diagram chasing
Five lemma, snake lemma
Homology
Basic definitions
Long Exact Sequence
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
Commutative Algebra
Basic Notions
Basic conventions and definitions of rings and algebras in commutative algebra
Localization
Localization of rings and modules, and local ring construction
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 of Ideals
Prime avoidance, associated primes, and their properties
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
Lying over and going up theorems for prime ideals in integral extensions
Nullstellensatz
Proofs of Jacobson rings and Hilbert’s Nullstellensatz
Blowup Algebras
Rees algebra and associated graded ring 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
Fractional Ideals
Fractional ideals, invertible modules, and the Picard group
Regular Local Rings
Characterization of regular systems of parameters and regular local rings
Divisors
Cartier divisors and class groups in Dedekind domains
Noether Normalization
Noether normalization theorem and applications for finitely generated algebras
Differentials
The algebraic definition of the Kähler differential module and its universal property
Representation Theory
Representation Theory of Finite Groups
Representation of finite groups and irreducible decomposition
Characters
Definition of character functions and orthogonality relations
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 Convergence of Filters
Characterizing compactness via filter convergence
Compactness and Paracompactness
Tychonoff theorem, paracompactness, and partitions of unity
Proper Maps
The relationship between proper maps as universally closed maps and compactness
Dimension
Definitions of covering dimension and Krull dimension for algebraic geometry
Connected Spaces
Connected spaces, path-connectedness, and connected components
Algebraic Topology
Topological Manifolds
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 spaces, covering spaces, and the Seifert-van Kampen theorem
Computing Homology
Practical homology computations via relative homology and Mayer-Vietoris
Cohomology
Definition of cohomology and the universal coefficient theorem
Acyclic Models Theorem
The acyclic models theorem on categories with models and its applications
Cup Product
The exterior product in cohomology, cup product definition, and ring structure
Poincaré Duality
Duality between homology and cohomology via orientation sheaves and fundamental classes
Stiefel-Whitney Characteristic Classes
Vector bundles, Stiefel-Whitney classes, and the infinite Grassmannian
Characteristic Classes of Vector Bundles
Euler, Chern, and Pontryagin characteristic classes
Classifying Spaces
Classification of principal G-bundles and construction of the classifying space BG
Manifolds
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 smooth 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
Definition of vector bundles and tangent, cotangent bundles
Vector Fields
Vector fields
Differential Forms
Differential form
Lie Derivative
Lie derivative and Lie bracket
Distribution
The definition of a distribution and the Frobenius theorem
Differential Ideal
Differential ideals and Frobenius’s theorem
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
Dynkin diagrams, ADE classification, and flag varieties
Bruhat Decomposition
Cell decomposition of homogeneous spaces, parabolic subgroups, and Schubert varieties on Grassmannians
Richardson Varieties and Peterson Varieties
Richardson varieties as transversal intersections of opposite Schubert varieties, and Peterson varieties from regular nilpotents: definitions, dimensions, in...
Riemannian Geometry
Riemannian Metric
The Riemannian metric as a positive-definite symmetric 2-tensor on the tangent bundle
Connection
Differentiation on a vector bundle
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
Equivalent definitions of dimension for algebraic varieties
Tangent Spaces and Smoothness
Tangent spaces and smoothness of algebraic varieties
Grassmann Varieties
Grassmannians as parameter spaces of linear subspaces
Algebraic Groups
Algebraic group action
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 Line 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
Scheme Theory
From Varieties to Schemes
From varieties to schemes
The Spectrum
Prime spectrum and Zariski topology of a commutative ring
Affine Scheme
The affine scheme defined by the structure sheaf on a ring’s spectrum
Schemes
Definition of a scheme as a locally affine locally ringed space
Projective Schemes
The Proj construction from graded rings and projective spaces
The Topology of Schemes
Generic points, Zariski topology, and irreducible components
Algebraic Structure of Schemes
Definitions and properties of reduced and integral schemes
Morphisms of Schemes
Four perspectives on scheme morphisms as locally ringed space morphisms
Properties of Scheme Morphisms
Basic properties of scheme morphisms: affine, finite, finite type, and rational maps
Closed Subschemes
Closed subschemes and vanishing schemes defined by an ideal sheaf
Closed Subschemes of Projective Space
Correspondence between closed subschemes of projective space and homogeneous ideals
Fiber Products
Definition and existence of fiber products in the category of S-schemes
Dimension
Dimension of schemes and Krull dimension of local rings
Flat Morphisms
Definition, geometric meaning, criteria, and examples of flat morphisms
Valuation Rings
Valuative criteria for separated and proper morphisms
Quasi-coherent Sheaves
Sheaf of O_X-modules, equivalence on affine schemes, and quasi-coherence
Divisors and Linear Systems
Cartier and Weil divisors, the sheaf O_X(D), linear systems, and ampleness
Sheaf Cohomology of Schemes
Cohomology of quasi-coherent sheaves, Serre vanishing, the cohomological criterion for ampleness, and Hilbert polynomials
Kähler Differentials and Cotangent Sheaves
Kähler differentials, cotangent sheaf, tangent sheaf, Euler sequence, and canonical sheaf
Complete Intersections
Codimension of local complete intersections, Koszul resolutions, and Hilbert polynomials
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
Symplectic Geometry
Classical Mechanics
Classical mechanics and phase space
Symplectic Vector Spaces
Definition of symplectic forms
Symplectic Manifolds
Definitions and properties of symplectic manifolds
Mirror Symmetry
Mirror Symmetry: An Overview
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
Givental J-function and Mirror Theorem
Fundamental solution of quantum differential equation and I=J theorem