A graduate program in mathematics explores key concepts and principles in advanced mathematics. Mathematics is an increasingly multidisciplinary field, with applications for business, government, engineering, computer science, and more. As a result, Ph.D. programs in mathematics are often interdisciplinary, drawing from principles and perspectives in mathematics, as well as philosophy, psychology, anthropology, sociology, and education. Students will have the opportunity to explore a variety of areas of study while still developing expertise in mathematics specialties and principles.

Students can expect to spend up to five years earning a doctorate in mathematics beyond the bachelor’s degree, although students often enter doctorate programs in mathematics with a master’s degree in mathematics or a related field already under their belt. They will have the opportunity to conduct research in a specialized area, such as nonlinear analysis, applied analysis, or numerical analysis and scientific computing. Graduates often go on to careers in academic research and teaching. They may also work for the government or private industry, or into related fields such as statistics or applied mathematics.

#### About Online PhD Programs in Mathematics

Doctoral programs in mathematics generally have two goals. The first is to produce scholars who can contribute to the mathematics community and communicate their knowledge to students and their peers. The second is to prepare graduates for careers in government, education, or the private sector. Students obtain a fundamental understanding of basic fields in mathematics, as well develop research and problem-solving skills for application in areas such as mathematics teaching, learning, technology, and policy. Graduate programs in mathematics may feature study in education, philosophy, psychology, anthropology, and sociology. For an idea of what types of courses you can expect to encounter during a doctorate in mathematics, here are some of the graduate-level math classes offered by the University of Houston’s Department of Mathematics:

Year One:

**Graph Theory with Applications**. This course examines the mathematical and scientific applications of graph theory. Students in this class study the essential concepts of graph theory, such as colorings, matchings, and domination, as well as its application to neurology, sociology, and computer science.**Abstract Algebra**. Topics of study in this course include groups, rings, and fields, as well as algebra of polynomials, Euclidean rings, and principal ideal domains.**Differential Equations**. This course covers linear and nonlinear systems of ordinary differential equations; existence, uniqueness, and stability of solutions; initial value problems; higher dimensional systems; and Laplace transforms.

Year Two:

**Cardinal and Ordinal Numbers**. Topics covered include ordinal and cardinal number theory; transfinite induction, equivalence of the axiom of choice, well-ordering principle, Zorn’s lemma, and Hausdorff maximality principle; and uses of generalized continuum hypothesis.**Modern Algebra**. This course covers topics from the theory of groups, rings, fields, and modules with special emphasis on universal constructions.**Graph Theory and Combinatorics**. Basic graph-theoretical and combinatorial methods and algorithms with their applications are taught in this course.

A Ph.D. in mathematics is a research degree, so a large part of earning a doctorate in mathematics is conducting original research. Graduate students in a mathematics Ph.D. program are expected to complete original research in the form of a dissertation. Research is conducted under the supervision of an instructor and presented before a dissertation committee prior to completion of the program.

#### Value and Criticisms of a PhD in Mathematics

For students looking to pursue a career in mathematics, conducting original research or teaching at the university level, the Ph.D. is the way to go. As the U.S. Bureau of Labor Statistics (BLS) notes, a doctorate is typically required for a job as a professor of mathematics in a college or university. In addition to fulfilling any professional requirements, graduate study in mathematics offers the opportunity to network with people who can help you reach your career goals. A doctorate also presents the opportunity to advance the field by conducting original research, which can lead to being published.

There are more practical things to consider that make earning a Ph.D. in mathematics difficult. For example, it is incredibly competitive. Some programs may only accept a handful of students. You may have to apply several times before being admitted. There’s also the matter of finances. Ph.D. programs in mathematics can take upwards of five years to complete. That’s five years without a salary, unless you’re teaching or on a fellowship, which are also competitive. Financial assistance may not be enough to cover your needs, especially if you are supporting a family while you earn your degree. Lastly, while opportunities are continually increasing in distance learning, there are limited opportunities to complete fully online PhD programs in mathematics, so you may need to relocate to pursue a doctorate program.

#### Application & Admission Requirements

Applicants to doctorate programs in mathematics must have a bachelor’s or master’s degree in mathematics, although those with graduate degrees in the physical and engineering sciences may be eligible. They may need to take any necessary undergraduate courses before entering the doctoral program. In addition to the standard application and statement of purpose, applicants may need to submit three letters of recommendation and scores from both the general GRE and the Mathematics Subject GRE exams. GRE scores may need to meet a minimum requirement, depending on the competitiveness of the program. International applicants may need to show evidence of competent English-speaking skills.

#### Career Options & Job Market

According to New York University mathematics professor Robert V. Kohn, most mathematics Ph.D.s become college or university professors. This could mean working for a college, university, professional school, or junior college. According to the BLS, post secondary teachers earned a median annual wage of $62,050. Employment of post secondary teachers is expected to grow 17% from 2010 to 2020, and employment expected to grow fastest in for-profit institutions, according to the BLS.

Graduates of mathematics doctoral programs may also pursue careers in finance, software, and, of course, mathematics, developing new mathematical principles and solving problems such fields as business, government, engineering, and the sciences. According to the BLS, mathematicians earned a median annual wage of $99,380 in May 2010. The profession is expected to grow 16% from 2010 to 2020 due to advancements in data collection and processing that will spur demand for mathematicians to analyze data, the BLS reports.

#### Where to Find Information

- American Mathematical Society: This association serves the international mathematics community through meetings, advocacy, and publications, including data on the profession.
- Mathematical Association of America: This 20,000-member national organization was founded in 1894 and provides information on careers in mathematics.
- Society for Industrial and Applied Mathematics: This organization promotes interactions between mathematics and other scientific and technological communities through membership activities, publications, and conferences. It also provides information on careers in mathematics.
*U.S. News & World Report*: The popular college guidebook ranks the top graduate mathematics programs.

#### How to Get Funding

At this stage in your academic career, you know a thing or two about the financial aid process. But it doesn’t hurt to know all of your options, especially since at the Ph.D. level.. For instance, in addition to state-subsidized and federal loans, schools often offer teaching and research fellowships in which you can receive a stipend for your work, and scholarships. If you are applying for federal loans keep in mind that you’ll have to be attending an accredited school to qualify for aid.

**Federal Direct Stafford Loan**. Students can be awarded up to $20,500 per academic year at a fixed interest rate of 6.80%.**Federal Perkins Loan**. Full-time students are awarded an average of $3,500 per academic year at a fixed interest rate of 5.0%.**Campus-Based Aid**. Opportunities through your university can include scholarships, teaching fellowships, and research grants. Check with your school’s financial aid office and mathematics department what they are.

#### Essential Advice

A Ph.D. is a big commitment, one that you will devote several years of your life pursuing. It’s important to consider the long-term effects the degree will have on your career. It’s also important to consider how it will affect your life in the meantime, such as how you’ll support yourself while you earn your degree and make adjustments in your life for studying, attending class, and working on your dissertation.

- Once you’re ready make sure to give yourself enough time for the application process. Obtaining letters of recommendation, taking relevant entrance exams, and drafting statements of purpose for each school you’re applying to can take several months. You’ll want to stay ahead of unforgiving deadlines in case a scheduled test falls through or a recommendation comes in later than expected.
- Financial deadlines are as important as application deadlines. Find out when the application deadlines for any scholarship and fellowship opportunities you’re eligible for are, and be sure to file your FAFSA by the deadline. Your doctorate could earn you more than it cost to get degree itself, but in the meantime you’ll still want to get as much financial aid help as you can get.
- Your sights might be set on graduate school right now, but it’s important to remember the value of work experience. By taking time off from academia after earning your bachelor’s or master’s degree, you can make valuable professional contacts, save up for your degree, and see firsthand what the demand is like in your field and tailor your Ph.D. towards that. You can also knock any prerequisite courses you need to attend the program out of the way.