Some scientists who research other fields are also considered mathematicians if their research provides insights into mathematics—one notable example is Isaac Newton. Conversely, some mathematicians may provide insights into other fields of research—these people are known as applied mathematicians. Education
Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation. There are notable cases where mathematicians have failed to reflect their ability in their university education, but have nevertheless become remarkable mathematicians. Fermat, for example, is known for having been "Prince of Amateurs", because of his extraordinary achievements with little formal mathematics training. 
Mathematicians do research in fields such as logic, set theory, category theory, abstract algebra, number theory, analysis, geometry, topology, dynamical systems, combinatorics, game theory, information theory, numerical analysis, optimization, computation, probability and statistics. These fields comprise both pure mathematics and applied mathematics, as well as establish links between the two. Some fields, such as the theory of dynamical systems, or game theory, are classified as applied mathematics due to the relationships they possess with physics, economics and the other sciences. Whether probability theory and statistics are of theoretical nature, applied nature, or both, is quite controversial among mathematicians. Other branches of mathematics, however, such as logic, number theory, category theory or set theory are accepted to be a part of pure mathematics, although they do indeed find applications in other sciences (predominantly computer science and physics). Likewise, analysis, geometry and topology, although considered pure mathematics, do find applications in theoretical physics - string theory, for instance.
Although it is true that mathematics finds diverse applications in many areas of research, a mathematician does not determine the value of an idea by the diversity of its applications. Mathematics is interesting in its own right, and a majority of mathematicians investigate the diversity of structures studied in mathematics itself. Furthermore, a mathematician is not someone who merely manipulates formulas, numbers or equations - the diversity of mathematics allows for research concerning how concepts in one area of mathematics can be used in other areas too. For instance, if one graphs a set of solutions of an equation in some higher dimensional space, he may ask about the geometric properties of the graph. Thus one can understand equations by a pure understanding of abstract topology or geometry - this idea is of importance in algebraic geometry. Similarly, a mathematician does not restrict his study of numbers to the integers; rather he considers more abstract structures such as rings, and in particular number rings in the context of algebraic number theory. This exemplifies the abstract nature of mathematics and how it is not restricted to questions one may ask in...