MA, MS, or PhD?
The Master of Arts (MA) and Master of Science (MS) in mathematics degrees take about two years to complete and prepare you for careers in teaching (at secondary schools, community colleges, and some four-year colleges) as well as in industry. Both degrees require a Master's thesis. The MA degree requires more courses outside mathematics and has a teaching option intended for secondary school teachers. The MS degree requires a greater number of more advanced mathematics courses (technically those which are not dual numbered). The MA and especially the MS degree often serve as a stepping stone towards the PhD.
The Doctor of Philosophy (PhD) degree usually requires about five years to complete. It is the degree for you if you plan to pursue a career in academic research, university teaching, or high-level research in industry or a government laboratory. The most important requirement is a dissertation which represents a substantial new contribution, of publishable quality, to the mathematical sciences. In fact, the point of the PhD degree is to prepare you to make and publish exactly this sort of contribution.
Mathematics or Applied Mathematics?
There are two academic units offering graduate degrees in the mathematical sciences at the University of Arizona: the Graduate Program in Mathematics in the Mathematics Department, and the Graduate Interdisciplinary Graduate Program in Applied Mathematics. These are separate units with distinct degrees and different courses of study. The following guidelines should help you decide which of these two programs is most suitable for you.
The Graduate Program in Mathematics offers core courses in algebra, analysis, and topology-geometry. In addition to studying a broad range of topics in mathematics, students in this program tend to pursue research topics in more theoretical aspects of mathematics, and nearly always write their dissertation under the supervision of a professor in the Mathematics department.
The Graduate Interdisciplinary Program in Applied Mathematics offers core courses in numerical analysis, applied analysis, and mathematical methods. Students in this program study the applications of mathematics, with a strong emphasis on interdisciplinary research at the interface of mathematics and the physical, life and engineering sciences. It is common for students in this program to have advisors from other departments.
Although the two graduate programs are separate, a special feature of the University of Arizona is that they work closely together. Students from both graduate programs share offices, help teach the same courses, and often take advanced level mathematics courses together. Furthermore there are no hard divisions between the types of research projects that they can pursue: sometimes students in the Mathematics program will work on applications oriented problems, and students in the Applied program will work on more theoretical problems. Although the choice of program is usually clear, we would be happy to advise as to which one best matches your interests.
See http://appliedmath.arizona.edu for more on the Graduate Interdisciplinary Program in Applied Mathematics.