Category theory is the study of abstracted collections of mathematical objects, such as the category of sets or the category of vector spaces, together with abstracted operations sending one object to another, such as the collection of functions from one set to another or linear transformations from one vector space to another.
classification in mathematics
Classification in mathematics is the goal in a branch of mathematics of providing an exhaustive list of some type of mathematical object with no repetitions. For example, the classification of 3-manifolds is one of the outstanding problems in topology. With the advent of computers, one weak but precise way to state a classification problem is to ask whether there is an algorithm to determine whether two given objects are equivalent.
Mathematical models may be physical objects used to represent mathematical abstractions, or, more frequently, they are mathematical constructions (formulae, functions, graphs, etc.) used to express physical phenomena. Such models occur throughout applied mathematics and physics, their greatest value being heuristic; i.e., the model may suggest the existence of unsuspected properties in the phenomenon.
Potential theory is the study of harmonic functions. Potential theory is so named because 19th century physicists believed that the fundamental forces of nature derived from potentials which satisfied Laplace's equation. Hence, potential theory was the study of functions which could serve as potentials. Nowadays, we know that nature is more complicated – the equations that describe forces are systems of non-linear partial differential equations such as the Einstein equations and the Yang-Mills equations and that the Laplace equation is only valid as a limiting case. Nevertheless, the term "potential theory" has remained as a convenient term for describing the study of functions which satisfy the Laplace equation.
Pure mathematics is mathematics for the sake of its internal beauty or logical strength. The other major division of mathematics is applied mathematics.