Mathematics

Foundations

Mathematical logic
Set theory: formal theories of sets, in varying consistency strengths.
Category theory: algebraic properties of collections of transformations between mathematical objects of the same type.

Number theory: properties of integers, especially primes and prime factorization.
Algebra: numerical quantities, equations, and structures.
Geometry: shapes of and spatial relationships (e.g. distance) between objects.
- Axiomatic geometry: Euclidean geometry, non-Euclidean geometries (e.g. elliptic, hyperbolic).
- Differential Geometry: distance and curvature on manifolds.
Topology: invariant property of objects under continuous transformations.
Analysis: real- and complex-valued continuous functions; operators.
Dynamical system: description of how a complex system changes over time.
Discrete math: objects that can assume only distinct, separated values.
- Graph;
- Combinatorics;

Probability and Statistics;
Optimization and Game theory;
Computation theory: tasks that are theoretically possible with computing machines, and their relative difficulty and complexity.
Numerical methods;

Table: Development of Common Mathematical Structures

Set	Origin/Motivation	New Operator	Feature; Abstraction
$\mathbb{N}$	counting	$+$, $*$
$\mathbb{Z}$	closed inversion of $+$	$-$
$\mathbb{Q}$	closed inversion of $*$	$/$	rational function
$\mathbb{R}$	complete metric	^	analysis; metric, topology
$\mathbb{C}$	root of negative numbers		harmonic analysis
$\mathbb{F}^n$	product space	$\cdot$	inner product, norm
$\mathbb{F}^\infty$	discrete process/function
$L^p$	approximation of functions

Resources: