Mathematics is concerned with numbers, data, quantity, structure, space, models and change.
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 PhD dissertation.
Solving problems with applications in real life
Mathematics that studies entirely abstract concepts.
Professional mathematicians also engage in the teaching of mathematics.
Many careers in mathematics outside of universities involve consulting.