The University of Arizona

Trajectories through the UA Math Graduate Program

This is a description of the typical trajectory through the program, focusing mostly on PhD students. Of course, every student is different and your program will be uniquely tailored to your interests and career goals.

The Integration Workshop

Your program begins with the “integration workshop,” an intensive five-day program in August, before the semester starts. The goal is to integrate new students into the program by looking at undergraduate mathematics from a more graduate perspective and by introducing them to a few faculty members and senior graduate students. The program is intense, and focuses almost exclusively on mathematics before other demands (like teaching) set it. The workshop also serves to identify strengths and weaknesses in students' preparation, so that they may enroll in appropriate courses for the fall. You can read more about integration workshops.

The first year

Core courses and quals

Advanced work in the program assumes that all students have a solid command of a rigorous foundational core of mathematics, including algebra, analysis, and geometry-topology. The first year program serves to provide this foundation, and culminates in the qualifying exams, which are taken in August following the first year. You can read more about the qualifying exams including copies of past exams.

Research Tutorial Groups (RTGs)

In some PhD programs, student do not encounter research until the third or fourth year. At Arizona, you will participate in a “Research Tutorial Group” (RTG) which provides an early research experience. The first part of the RTG takes place in the spring of the first year, and consists of three or four short lecture series by faculty members on research problems in their areas. In the fall of the second year, you choose a faculty member and work with them in a small group on the research problem. You present your results at a RTG mini-conference at the end of the fall semester. You can read more about recent RTGs.

First teaching experience

At Arizona, teaching assistants are not so much assistants as full-fledged teachers. You have full responsibility for your class, which includes lecturing, writing exams and setting homework assignments, and determining grades. This is a significant responsibility and the department provides many resources to help you meet it. You will be enrolled in a one-unit “Professional Development Workshop in Teaching Mathematics” and you will be assigned a faculty supervisor to guide you through the first teaching experience. Students in the early stages of the program are usually assigned lower division courses such as trigonometry or algebra.

MS and MA theses

If you are pursuing a Master's degree, you will continue with course work in the second year and write a Master's thesis under the supervision of a faculty member. Although a significant piece of work, a Master's thesis is less elaborate than a PhD dissertation. It typically consists of the working out of a particular case of a theory or an interesting example or calculation, rather than the elaboration of a new theory or resolution of an outstanding problem. Master's dissertations are sometimes published in the mathematics literature. You can examine the list of recent MA and MS degrees.

The middle years

Specialized course work and orals

If you are pursuing PhD, after passing the qualifying exams you will take more advanced course work and start participating in one of the research seminars. Your goal at this point is to identify an area of mathematics and a faculty member with whom you will write the thesis. Once you and a potential advisor have found one another, you will prepare and pass an oral and written comprehensive exam. This includes delivering a lecture, typically on a recent paper in the area in which you want to specialize, and answering questions to assess your knowledge of the field.

More advanced teaching

As you progress through the TA ranks, from Grad Assistant I to Grad Associate II, you will be given higher level courses to teach (possibly including calculus, linear algebra, or differential equations) or courses involving more specialized skills or knowledge (like the technology-intensive calculus sequence for business majors, or statistics). You may also be a “super-TA,” assisting a faculty member teaching an advanced undergraduate or even graduate course, for example by running review sessions or assisting with programming projects. You may also become involved in research in mathematics education or outreach to local schools.

An internship

The middle years are also an ideal time to consider an internship, which might be with a company, a government research lab like Los Alamos, or Sandia, or at a government agency like the EPA. This is an ideal way for you to see how mathematics gets applied in the real world, what the environment is like for industrial mathematicians, and not least, make a healthy salary. Arizona has had a close relationship with Los Alamos National Lab for decades, and has a well-developed internship program for mathematics students at both the undergraduate and graduate level. You can read more about internship opportunities.

The PhD dissertation

This is the key component of the PhD degree. It is the culmination of your graduate career and your entry into the world of research in mathematics. You will spend two to three years researching a specialized topic by proving theorems, constructing models, or analyzing data, and then writing up the results. Your thesis will represent a substantial new contribution to the mathematical sciences and will be publishable in a reputable journal. To see some recent PhD dissertations, see the list of recent PhD graduates.

Careers for PhDs

One of the main purposes of graduate school is to prepare yourself for a career. With the training and experiences you receive at Arizona, you will be ready to embark on a career in a variety of mathematically oriented professions such as academic research, teaching at a univesity, four-year college, or community college, or applying mathematics at a company, government lab, or agency. The list of recent PhD graduates includes the initial placement for most of our graduates.