Foundations:

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

Pure mathematics:

• Number theory: properties of integers, especially primes and prime factorization.
• Algebra: numerical quantities and attempts to solve equations.
• Abstract algebra;
• Geometry: figures, objects, and their relationships to each other.
• Differential geometry: distance and curvature on surfaces and manifolds
• Topology: invariant property of objects under continuous transformations.
• Analysis: real- and complex-valued continuous functions.
• Dynamical systems: description of how a complex system changes over time.
• Discrete math: objects that can assume only distinct, separated values.
• Graph;
• Combinatorics;

Applied mathematics:

• Probability and statistics;
• Computation: tasks that are theoretically possible with computing machines; the relative difficulty and complexity of these tasks.
• Optimization and Game theory;

## Miscellaneous

Table: Development of Common Mathematical Structures

Structure Origin/motivation New Operations Feature Abstraction
$\mathbb{N}$ Counting $+, *$
$\mathbb{Z}$ Closed inversion of $+$ $-$
$\mathbb{Q}$ Closed inversion of $*$ $/$ Polynomial (rational function)
$\mathbb{R}$ Closure/completeness ^ Analysis metric, topology
$\mathbb{C}$ root of negative numbers Trigonometrics, Fourier analysis
$\mathbb{F}^n$ Product space $(\cdot,\cdot)$ inner product, norm
$\mathbb{F}^{\infty}$ Discrete process
$L_p$ Approximation of functions

MathWorld Classroom