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.
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