Algebraic Areas of Mathematics

The algebraic areas of mathematics developed from abstracting key observations about our counting, arithmetic, algebraic manipulations, and symmetry. Typically these fields define their objects of study by just a few axioms, then consider examples, structure, and application of these objects. We have included here the combinatorial topics and number theory, each of which is arguably a distinctive area of mathematics; the MathMap suggests that these parts of mathematics (in shades of red) share definite affinities.