User:Paul August/Subpage 25

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This list of major themes of mathematics, lists some major themes in modern mathematics and a gives a few important links to articles by theme. An alphabetical and subclassified list of mathematics articles is available. This list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists.

Contents

[edit] Quantity

Image:Image:1,2, ... .png
Natural numbers
Quantity starts with counting and measurement.
1, 2, \ldots \, \ldots, -1, 0, 1, \ldots \, \frac{1}{2}, \frac{2}{3}, 0.125,\ldots \, \pi, e, \sqrt{2},\ldots \, i, 3i+2, e^{i\pi/3},\ldots \,
Natural numbers Integers Rational numbers Real numbers Complex numbers


[edit] Structure

Pinning down ideas of size, symmetry, and mathematical structure.
36 \div 9 = 4 Image:Rubik float.png
Arithmetic Number theory Abstract algebra Group theory Order theory
Monoids • Rings • Fields • Linear algebra • Algebraic geometry • Universal algebra

[edit] Space

A more visual approach to mathematics.
Geometry Trigonometry Differential geometry Topology Fractal geometry
Algebraic geometry • Coordinate systems • Differential topology • Algebraic topology • Linear algebra • Combinatorial geometry • Manifolds

[edit] Change

Ways to express and handle change in mathematical functions, and changes between numbers.
Image:Vector field.svgg \frac{d^2}{dx^2} y = \frac{d}{dx} y + c
Calculus Vector calculus Differential equations Dynamical systems Chaos theory
Analysis • Real analysis • Complex analysis • Functional analysis • Special functions • Measure theory • Fourier analysis • Calculus of variations

[edit] Foundations and methods

Approaches to understanding the nature of mathematics.
P \Rightarrow Q \,
Mathematical logic Set theory Category theory
Foundations of mathematics • Philosophy of mathematics • Intuitionism • Constructivism • Proof theory • Model theory • Reverse mathematics

[edit] Discrete mathematics

Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values.
\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix}
Combinatorics Theory of computation Cryptography Graph theory
Computability theory • Computational complexity theory • Information theory

[edit] Applied mathematics

Applied mathematics aims to develop new mathematics to help solve real-world problems.
Mathematical physics • Analytical mechanics • Mathematical fluid dynamics • Numerical analysis • Optimization • Probability • Statistics • Mathematical economics • Financial mathematics • Game theory • Mathematical biology • Cryptography • Operations research

[edit] Important theorems

See also list of theorems.

These theorems have interested mathematicians and non-mathematicians alike.
Pythagorean theorem • Fermat's last theorem • Gödel's incompleteness theorems • Fundamental theorem of arithmetic • Fundamental theorem of algebra • Fundamental theorem of calculus • Cantor's diagonal argument • Four color theorem • Zorn's lemma • Euler's identity • Gauss–Bonnet theorem • Quadratic reciprocity • Riemann–Roch theorem.

[edit] Important conjectures

See also list of conjectures

These are some of the major unsolved problems in mathematics.
Goldbach's conjecture • Twin Prime Conjecture • Riemann hypothesis • Poincaré conjecture • Collatz conjecture • P=NP? • open Hilbert problems.

[edit] History and the world of mathematicians

See also list of mathematics history topics

History of mathematics • Timeline of mathematics • Mathematicians • Fields medal • Abel Prize • Millennium Prize Problems (Clay Math Prize) • International Mathematical Union • Mathematics competitions • Lateral thinking • Mathematics education • Mathematical abilities and gender issues