1. Algebra
"In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread in almost all of mathematics." (Wikipedia: Algebra, 7.5.21 UTC 16:30)
2. Arithmetic
"Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on the them - addition, subtraction, multiplication and division, exponentiation and extraction of roots." (Wikipedia: Arithmetic, 8.6.21 UTC 00:20)
3. Calculus
"Calculus... is the mathematical study of continuous change..." (Wikipedia: Calculus, 8.5.21 UTC 10:26)
4. Category theory
"Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms)." (Wikipedia: Category theory, 8.16.21 UTC 08:56)
5. Chaos theory
"Chaos theory is a branch of mathematics focusing on the study of chaos - states of dynamical systems whose apparently-random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions. (Wikipedia: Chaos theory, 8.15.21 UTC 05:49)
6. Combinatorics
"Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures." (Wikipedia: Combinatorics, 7.25.21 UTC 20:08)
7. Game theory
"Game theory is the study of mathematical models of strategic interaction among rational decision-makers." (Wikipedia: Game Theory, 8.10.21 UTC 19:27)
8. Geometry
"[Geometry] is concerned with properties of space that are related with distance, shape, size and relative position of figures." (Wikipedia: Geometry, 7.26.21 UTC 07:51)
9. Graph theory
"Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines)." (Wikipedia: Graph theory, 8.5.21 UTC 01:45)
10. Group theory
"Group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be see as groups endowed with additional operations and axioms." (Wikipedia: Group theory, 5.621 UTC 04:49)
11. Mathematical analysis
"Mathematical analysis is a the branch of mathematics dealing with limits and related theories such as differentiation, integration, measure, infinite series and analytic functions." (Wikipedia: Mathematical analysis, 7.24.21 UTC 07:10)
12. Mathematical logic
"Mathematical logic... is a subfield of mathematics exploring the applications of formal logic to mathematics... The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems." (Wikipedia: Mathematical logic, 8.12.21 UTC 14:04)
13. Number theory
"Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions." (Wikipedia: Number theory, 7.3.21 UTC 19:15)
14. Order theory
"Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as 'this is less than that' or 'this precedes that'." (Wikipedia: Order theory, 4.22.21 UTC 05:51)
15. Probability
"Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true." (Wikipedia: Probability, 8.16.21 UTC 12:52)
16. Representation theory
"Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures." (Wikipedia: Representation theory, 8.15.21 UTC 6:55)
17. Set theory
"Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects." (Wikipedia: Set theory, 8.11.21 UTC 19:54)
18. Topology
"Topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself." (Wikipedia: Topology, 8.5.21 UTC 23:10)
19. Trigonometry
"Trigonometry is a branch of mathematics that studies relationships between side lengths and angels of triangles." (Wikipedia: Trigonometry, 8.9.21 UTC 2:39)
