- 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.
- Number theory: properties of integers, especially primes and prime factorization.
- Algebra: numerical quantities and attempts to solve equations.
- 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.
- Probability and statistics;
- Computation: tasks that are theoretically possible with computing machines; the relative difficulty and complexity of these tasks.
- Optimization and Game theory;
Table: Development of Common Mathematical Structures
||Closed inversion of $+$
||Closed inversion of $*$
||Polynomial (rational function)
||root of negative numbers
||Trigonometrics, Fourier analysis
||inner product, norm
||Approximation of functions