The following is a list of topics & books that are on calculus, algebra and category theory, ordered in a way to be thorough as possible; intended to be used as preliminary study for any branch of study which has algebra/calculus as prerequisites. This list was first intended as a thorough preliminary study for machine learning, but admittedly, the list still lacks a lot of supplementary topics, most notably:
- Statistics in general
- Measure theory in general
- The topology of R^n
- Details of commutative algebra
Furthermore, this list is, by no means, complete. These are just listed off of a few undergrad/graduate level mathematics books intended to help me relearn what I have either forgotten or haven't properly learned in the first place. I DO NOT condone anyone use this list as a guide for preliminary topics for any topic requiring calculus/linear algebra. The authors of these books most definitely know what they're doing & what they're writing about, but I don't.
- Kunze, Linear Algebra
- Greub, Multilinear Algebra
- Awodey, Category Theory
- Munkres, Analysis on Manifolds
- Dummit, Abstract Algebra
- Atiyah, Introduction to Commutative Algebra
- Kaplansky, Commutative Rings
[2] 1.3 Definition of a category
[4] 1.1 Basic Axioms and examples
[4] 1.3 Symmetric groups
[4] 1.5 Quaternion group
[2] 1.4 Examples of categories
[4] 7.1 Basic definitions and examples
[4] 10.1 Basic definitions and examples
[2] 1.5 Isomorphisms
[4] 3.1 Definitions and examples
[4] 3.2 More on cosets and Lagrange's theorem
[4] 3.3 Isomorphism theorems
[4] 7.3 Ring homomorphisms and Quotient rings
[4] 7.4 Properties of ideals
[4] 7.5 Rings of fractions
[4] 10.2 Quotient modules and module homomorphisms
[4] 4.1 Group actions and permutation representations
[4] 4.2 Groups acting on themselves by left multiplication
(Cayley's Theorem)
[4] 4.3* Groups acting on themselves by conjugation
(the Class Equation)
[4] 4.4* Automorphisms
[4] 4.5* Sylow theorems
[5] 1.7 Extension and contraction
[4] 8.1 Euclidean domains
[4] 8.2 PIDs
[4] 8.3 UFDs
[4] 9.1 Definitions and basic properties
[4] 9.2 Polynomial rings over fields I
[4] 9.4 Irreducability criteria
[4] 9.5 Polynomial rings over fields II
[5] 4 Primary decompositions
[2] 1.6 Constructions on categories
[4] A.2 Zorn's Lemma
[0] A.6 The Axiom of Choice
[2] 1.7 Free categories
[4] 10.3 Generation of modules direct sums and free modules
[2] 3.1 Duality principle
[2] 3.2 Coproducts
[2] 1.8 Large, small, locally small categories
[2] 2.1 Epis and monos
[2] 2.2 Initial and terminal objects
[2] 2.3 Generalized elements
[2] 2.4 Products
[2] 2.6 Categories with products
[2] 2.7 Hom-sets
[2] 4.1 Groups in a category
[2] 4.2 Category of groups
[2] 4.3 Groups as categories
[1] 1.1 Multilinear mappings
[0] 5.1 Determinant Functions
[0] 5.2 Permutations and the Uniqueness of Determinants
[0] 5.3 Additional Properties of Determinants
[4] 11.4 Determinants
[1] 1.2 Tensor product
[1] 1.3 Subspaces and factor spaces
[4] 10.4 Tensor products of modules
[1] 2.1 Tensor product of algebras
[4] 11.5 Tensor algebras, symmetric and exterior algebras
[1] 1.5 Linear mappings
[1] 1.6 Tensor product of several vector spaces
[1] 1.7 Dual spaces
[1] 1.8 Finite dimensional vector spaces
[1] 5.1 Skew symmetric mappings
[1] 5.2 Exterior algebra
[1] 5.3 Homomorphisms, derivations and antiderivations
[3] 6.4 Tangent vectors and differential forms
[3] 6.5 Differential operator
[3] 6.6 Action of a differentiable map
[1] 2.2 Tensor product of G-graded vector spaces
[1] 2.3 Tensor product of differential spaces
[1] 2.4 Tensor product of differential algebras
[2] 9.1 Preliminary definition
[2] 9.2 Hom-set definition
[2] 9.3 Examples of adjoints
[2] 9.8 Adjoint functor theorem
[0] 6.1 Introduction to Canonical Forms
[0] 6.2 Characteristic Values
[0] 6.3 Annihilating Polynomials
[0] 6.4 Invariant Subspaces
[0] 6.5 Simultaneous Triangulation/Diagonalization
[0] 8.1 Inner Products
[0] 8.2 Inner Product Spaces
[0] 8.3 Linear Functionals and Adjoints
[0] 8.4 Unitary Operators
[0] 8.5 Normal Operators
[0] 9.1 Introduction to Operators
[0] 9.2 Forms on IPSes
[0] 9.3 Positive Forms
[0] 9.4 More on Forms
[0] 9.5 Spectral Theory
[0] 7.1 Cyclic Subspaces and Annihilators
[4] 12.1 Basic theory
[0] 7.2 Cyclic Decompositions and the Rational Form
[4] 12.2 Rational canonical form
[1] 1.4 Direct decompositions
[4] 12.3 Jordan canonical form
[0] 7.3 Jordan Form
[3] 3.1 Integral over a rectangle
[3] 3.2 Existence of the integral
[3] 3.3 Evaluation of the Integral
[3] 3.4 Integral over a bounded set
[3] 3.5 Rectifiable sets
[3] 3.6 Improper integrals
[3] 4.1 Partitions of unity
[3] 4.2 Change of variables theorem
[3] 4.3 Diffeomorphisms in R^n
[3] 5.1 Volume of a parallelopiped
[3] 5.2 Volume of a parametrized-manifold
[3] 5.3 Manifolds in R^n
[3] 5.4 Boundary of a manifold
[3] 5.5 Integrating a scalar function over a manifold
[3] 7.1 Integrating forms over parametrized-manifolds
[3] 7.2 Orientable manifolds
[3] 7.3 Integrating forms over oriented manifolds
[3] 7.5 Generalized Stokes' Theorem
[4] 14.1 Basic definitions
[4] 14.2 Fundamental theorem of Galois theory
[4] 14.3 Finite fields
[4] 14.4 Composite extensions and simple extensions
[5] 1.6 Operations on ideals
[6] 1.2 Integral elements
[6] 1.3 G-ideals, Hilbert rings & the Nullstellensatz
[2] 5.1 Subobjects
[2] 5.2 Pullbacks
[2] 5.3 Properties of pullbacks
[2] 5.4 Limits
[2] 5.6 Colimits
I recognize that the progression for learning these topics is not linear, but this is the best I could come up with to have a sense of order without resorting to more visual representations (like graphs). I plan on graphing a more suited progression system for these topics, but that is easier said than done -- I need to finish the chapters at least once in order to have a sense of how to order the concepts.