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This is a rough classification based on what professional mathematicians actually research and teach at the university level, along with some major elementary topics. Topics covered in primary and secondary school mathematics are listed at Math Wiki:Topics by level.

Foundations of mathematics |
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Logic and * mathematical logic |

Set theory |

Category theory |

* Philosophy of mathematics |

Arithmetic |

Number systems |

Number theory |

Algebra |

Elementary algebra |

Abstract algebra |

* Group theory |

Linear algebra |

* Matrix theory |

* Universal algebra |

Clifford algebra |

Analysis |

Ordinary differential equations |

Partial differential equations |

* Integral equations |

Real analysis |

Complex analysis |

* Functional analysis |

* Variational methods |

* Nonstandard analysis |

* Measure theory |

* Vector and tensor analysis |

Graph theory |

Geometry |

Euclidean geometry |

Analytic geometry |

Non-Euclidean geometry |

* Modern geometry |

* Algebraic geometry |

Differential geometry |

Topology |

Applied mathematics |

Discrete mathematics |

Combinatorics |

Probability |

Statistics |

* Actuarial mathematics |

* Decision theory |

* Game theory |

* Mathematical modeling |

Optimization or * Mathematical programming |

Numerical analysis or * Numerical methods |

* Scientific computing |

* Theory of computation |

Related disciplines |

Recreational mathematics (including games and puzzles) |

History of mathematics |

* Mathematics education |

* Information theory |

* Systems theory |

* Operations research |

Mathematical physics |