This is a description of some possible trajectories through the program, focussing mostly on PhD students. Of course every student is different and your program will be unique.
[Coming soon: detailed trajectories, including specific courses, in the trajectories part of the future mentoring section.]
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.
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.
[Coming soon: See the mentoring section for a discussion of the content of the qualifying exams and hints for success.]
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.
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.
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 matematical literature. You can examine the list of recent MA and MS degrees.
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.
[Coming soon: For more details about the oral exam and hints for success, see the mentoring secion.]
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.
VIGRE (which stands for Vertical Integration in Graduate Research and Education) is a prestigious program of the National Science Foundation to support graduate research and education. Among other things, it provides fellowship support for graduate students. You apply for this support to achieve specific goals, such as preparing for the qualifying or comprehensive exams or working with a particular faculty member on an independent study or research project. Your proposal may also include “vertical integration activities” which are meant to foster interaction between faculty, post-docs, graduate students, undergraduates, and even high school students. For example, you might be involved in a research project for undergraduates (an undergraduate version of the RTG), or you might go with faculty to a local high school AP calculus class to present a mathematical enrichment activity. You may be funded by VIGRE fellowships for up to 33 months. See the VIGRE pages for more details on funding and vertical integration activities.
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.
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 dissertation titles, see the list of recent PhD graduates.
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.
[Coming soon: See the alumni page for some examples of careers pursued by Arizona Mathematics Graduates.]