Pursuing a Ph.D. in math can be a challenging path to follow; however, with a lot of hard work and an aptitude for the subject, it can prove rewarding. There are plenty of excellent Ph.D. mathematics programs to attend at universities all over the world. On completion of the math Ph.D., there are plenty of career prospects available in academia, actuary, finance, data science, research, and many other areas.

Entry Requirements for a Ph.D. in Mathematics

As there is such a wide range of universities in the USA with admission standards and requirements that vary considerably between graduate schools, we can only offer a broad overview. A bachelor’s degree in mathematics or a related subject such as economics or natural science is typically required. The mathematics department is often interested in your GRE subject test in mathematics scores and expects them to be in the upper percentile. Applicants should have completed classes in algebra, calculus, real analysis, and linear equations. Other courses of value would be advanced calculus, complex analysis, and topology.

Overall you want to present as strong a case as possible to convince them that you have a chance of success in their graduate program. Strong recommendation letters can be pivotal in admission decisions. The letters should ideally be from mathematics professors who have taught multiple courses to the student.

Here are the typical admission requirements for a Ph.D. program:

  • Bachelor’s degree with a GPA of 3.0
  • Official college transcripts
  • Two to four recommendation letters
  • Curriculum vitae
  • A personal statement that includes why the student wants to pursue a Ph.D. in addition to prior mathematics research and classroom experience and current interests in mathematics.
  • A competitive GMAT or GRE score

Applied vs. Pure Mathematics

One consideration to bear in mind when considering a maths Ph.D. program is which path to take – pure or applied mathematics. Generally, pure math is focused on theoretical use, while applied mathematics has more practical benefits with real-world applications. However, even seasoned mathematicians can’t agree on what is pure math and what is applied math, so please take our definition with a pinch of salt. Some graduate programs offer a course that covers both applied and pure math topics. Which is best? That’s for you to decide. Applied math is often said to lead to more career options, but if you love more abstract and what we’ve heard described as ‘poetic’ math, pure mathematics might better suit you.

Math Ph.D. Course Requirements

Typically a Ph.D. graduate program can be completed between three and six years, but it will vary.

Similar to the application requirements for a math Ph.D. program, the curriculum will vary between institutions. Usually, it will include qualifying exams, coursework, research, and teaching. Under the guidance of an advisor, the student must produce a doctoral dissertation on an original topic. The student must defend their dissertation before a committee of experts.

Typical Course Timeline

While the curriculum may differ between graduate programs, what every full-time Ph.D. student can expect is that they will be totally immersed in the world of mathematics from dawn to dusk. Expect it to be difficult, time-consuming, and energy-draining, but if you love math, which you need to, then you’ll survive and thrive on the challenge.

Coursework and Foundation

First-year students will take between two to five courses a semester. The initial studies will concentrate on the fundamental core materials such as abstract and linear algebra, topology, and real analysis. Time will also be taken doing homework, reading, and attending seminars.

In the second year, students will take more advanced courses online and reading courses. Students should also select an advisor and start preparing to choose a research area for their dissertation. In the second year, a student will probably add teaching classes to their schedule. Students will also be preparing for the qualifying exams by studying the previous exam papers and the exam syllabus topics.

Qualifying Exams

The qualifying exams are based on the core foundation course work, and you must pass them before advancing to candidacy and working with an advisor. Usually, a student is allowed only two chances to pass this exam; if they don’t pass, they will have to leave the course. You can achieve one of three grades: fail, pass, and high pass. For example, to qualify at Stanford, you are required to take and pass two written exams, one in real analysis and the other in algebra. The exams are in two parts, and students are given three hours to complete each part.

Admission to Candidacy Examination

The candidacy examination consists of a written and oral examination; its purpose is to determine if the student is ready to start work on their proposed thesis. An examining committee will commonly consist of the student’s advisor, two other members of the faculties mathematics department, and often chaired by someone outside the faculty.

The written part of the examination involves preparing a report usually of at least 15 pages on a dissertation proposal on the research area they are planning on undertaking. The report will be distributed to the examiners in advance of the oral examination.

For the oral candidacy exam, the student will present a lecture of around twenty minutes on their report’s subject. This is followed by a discussion in which the examiners will field questions to the student. The questions will be about the student’s report and then cover broader mathematical subjects.  The candidacy exam will allow the committee members to ascertain if the student has the aptitude to continue with the Ph.D. and that they are prepared enough to begin work on their dissertation. After completing the admission to candidacy examination commonly, there will be four outcomes; a pass, a pass with conditions, failure but an option to retake the exam and dismissal from the program.

Thesis

At this stage in the course, you will work closely with an adviser’s guidance to prepare your thesis; we will further discuss choosing an advisor shortly. The dissertation is required to be both an original and significant piece of publishable research. You or your advisor can propose the thesis topic; however, principally, the research must be your own idea. The advisor will provide you with feedback and suggestions. You can expect to spend two to four years on your thesis and average about twenty hours a week on research.

Dissertation Exam

The final hurdle is to present your thesis committee. This is typically the same set up as the candidacy exam, with your advisor chairing the committee. You will be asked to explain your thesis and defend its findings. The committee members will receive copies of the written dissertation around two weeks in advance of the exam. The dissertation’s lecture and oral defense are usually twenty to forty minutes, and if you have successfully defended your research, you will graduate with a Ph.D.

Other Considerations

Teaching – You are also likely to have teaching commitments that can take up at least two or three hours of your week, and you’ll need to add preparation time on top of that.

Foreign Language Exam –  Depending on the university, you might need to pass a foreign language exam as part of the graduate program, commonly in French, German or Russian. Often it just involves translating a couple of pages of mathematical literature from the original language into English. The use of a dictionary is usually allowed but not a computer (goodbye, Google Translate!), and it can be taken multiple times until passed. Many universities have phased this requirement out.

Computer Programming Exam – Again this isn’t a requirement for all graduate programs.  For instance, students may need to show that they have the required competence by completing a designated computer programming project. As computer programming often goes hand in hand with mathematics in both industry, research, and teaching, it will look good on your resume at the very least.

Communication Skills Requirement – This is another requirement that some universities require, with tasks to complete in both written and verbal communication skills.

Choosing your Advisor

Picking the right advisor should be your primary focus after passing your qualifying exams. The advisor should be a specialist in the area your research will be focused on. You should research potential advisors, including talking to current and former students. Finding the right advisor isn’t an exact science, but you need to be happy with your decision as you will be working closely with them for many years.

Skills Gained from a Ph.D. in Mathematics

Completing a math Ph.D. program in addition to the obvious math specific expertise obtained has many transferable skills that will be advantageous in many different industries.

To begin with, you will have demonstrated the strength of character to undertake and complete years of intensive and continuous learning to reach your goal. You will have skills in logic, deduction, reasoning, and problem-solving for real-world applications. You will have honed your verbal and written skills and had experience working as part of a team. Don’t underestimate the importance of all these additional skill sets as they could help you secure your dream job.

Ph.D. Mathematics Career Options

A significant advantage of taking a Ph.D. degree in mathematics is the wide choice of career opportunities it opens up. With so many disciplines based on core mathematical principles, science, and finance, to name but two, two fellow students might engage in totally different professional careers after completing a degree. One, for instance, might explore the world of banking while the other might continue with academia. Ph.D. students will gain plenty of theoretical knowledge regarding mathematics, but they also will be schooled in how it can be used in the real world. Ph.D. maths graduates with their talent for problem-solving will be much sought after in many industries.

While the list below is in no way exhaustive, these are a few options open to Ph.D. graduates in mathematics.

Academia

Many mathematics graduates pursue academia as a career choice, with some of history’s most famous mathematicians to follow in the footsteps of can be a rewarding direction to take. There is a wide range of jobs available for students, from teaching to research or more administrative roles such as postsecondary education administrators. If you are a math Ph.D. who lives for mathematics and wants to be around like-minded people, then academia is a good choice.

Actuaries

The role of an actuary can be financially rewarding, with a median salary of $108,350 and an estimated 20% job growth between 2018 -2028, according to the  U.S. Bureau of Labor Statistics.

An Actuary evaluates financial risks using a combination of statistics, economic knowledge, maths, and probability. A strong background in mathematics is a requirement for the job; however, you also need to be certified as an actuary, so even more study is required.

Fortunately, you can still work as an actuary while you prepare for the exam. Good communication skills are required as you will be explaining complicated calculations and analysis to laypeople. An actuary role could mean working in an office or as a consultant where you might need to travel to visit clients.

Data Scientists

An entry-level salary for a data scientist is around $70,000, but once you have some experience behind you, the average is about $120,000 a year.

The role involves exploring data using specialized tools and gaining actionable insights from it to, for instance, help a company make important business decisions. Essentially data scientists are modern-day alchemists turning raw data into gold.

Mathematical skills are required for this role, not just analytical but specialist areas such as linear algebra and multivariable calculus. However, as well as math, soft skills are needed so that the data can be explained to non-technical people. Math graduates who have a strong computer science background and programming languages will especially be suited for this role.

Mathematicians

With an average salary of $82,739, according to salary.com, it’s a logical career path for someone who has completed a Ph.D. program in math is to become a mathematician. Mathematicians will use their math skills to solve real-world problems in fields such as business,  health, economics, engineering, and science. As a mathematician, you will put your math skills to fair use and use the latest computer technologies; math graduates with computer programming skills or computer science knowledge will be the most in-demand.

Statisticians

The median annual salary for a statistician is $91,160. Statisticians use their mathematical skills to analyze and interpret statistical and quantitative data. Statisticians are required in a wide range of industries, including finance, healthcare, and sports, and not forgetting the importance of their role in government.

Besides strong math qualifications, especially analytics, a statistician requires  IT skills as they will be required to use specialized computer software for data analysis. Communication skills are likely to be necessary, written, and oral as statisticians will often be asked to complete reports on their analysis and attend meetings to present their findings.

As previously mentioned, a Ph.D. in mathematics can open many doors; other popular jobs include cryptographer, quantitative analyst, accountant, operations research analyst, computer programmer, to name a few.

Ph.D. in Mathematics Online

While there are plenty of graduate programs available to attend either at an accredited college or university physically, it is not possible to take a Math Ph.D. online. The likely explanation for this is twofold, typically graduates will be involved in teaching, working as research assistants, and other reasons to be on campus. The second reason is that there isn’t a strong enough market for online PH.D.s.

However, you can get a master’s degree in mathematics online, preparing you for a doctorate in mathematics.

If you are in a situation where you must take your Ph.D. online, you can view all the options open to you; we can help you find a suitable Ph.D. here.

Final Thoughts

Completing a Mathematics Ph.D. is by no stretch of the imagination a walk in the park. Still, if you have a passion and aptitude for maths and the tenacity to stay the course, it can be both gratifying and remunerative. We have given a broad overview, as different faculties have varied entry and course requirements.

Career opportunities are boundless, there are not only opportunities in academia, but in almost any industry you can think of. With technological and scientific innovation growing exponentially by the time you have completed your doctorate, there will likely be even more new and exciting job opportunities to explore.