PhD in Mathematics

What Do You Learn in a PhD in Mathematics?

A Doctor of Philosophy in Mathematics, or PhD in Mathematics, is often a final educational path for those who wish to study mathematics more comprehensively. In many cases, students completing this educational path work towards both pure and applied mathematics opportunities.

A large component of the work to obtain a PhD in mathematics is often research-based. This may include research in mathematical biology, combinatorics, matrix, applied dynamical systems, geometry, optics, numerical analysis, and partial differential equations, among various other topics. Students may take a wide range of courses specific to the area they wish to work in, which could extend beyond teaching to include business, science, and other areas.

PhD in Mathematics programs

Mathematics Courses May Include

It is somewhat typical for students to have courses at this level that include real analysis, linear algebra and matrix analysis, and complex analysis. It may also include mathematical neuroscience, functional analysis, operator theory, and various other areas as electives where students may be able to get the level and type of education they desire.

Some of the courses taken may relate to the field a person wishes to work in after completing a PhD, such as working as a professor, software development professional, engineer, or other fields.

Here are some example courses that a student may take in this field as they prepare for their doctorate of mathematics.

1st course Advanced Applied Mathematics

Advanced Applied Mathematics

This class typically involves the use of concepts and strategies in mathematical modeling. These concepts are often developed through the use of investigation and case studies, among other methods. Topics covered may include scaling, consistency of models, and singular effects.

2nd course Linear Algebra

Linear Algebra

This course typically involves the use of concepts and strategies focused on linear systems. Linear spaces, matrices, and diagonalization may also be topics that are covered in this course. A large emphasis is often placed on mathematical rigor.

3rd Course Analysis


This course typically is an extension of calculus, both fundamental and advanced. Topics covered may include the real number system, sequences, and series. Point set topology and uniform convergence may also be covered.

4th Course Methods of Applied Mathematics

Methods of Applied Mathematics

This class typically covers applied mathematics and how it is applied to science and engineering. Differential equations and Fourier and Laplace transform may also be topics. Eigenfunction expansions may also be covered.

5th Course Numerical Methods

Numerical Methods

This course typically requires proficiency in a computer language. This course may focus on accuracy and rigor in regards to numerical solution data. Machine arithmetic, linear system, and nonlinear system may also be topics in this course..

6th course

Complex Variables

This course typically deals in theory. Topics covered may include applications of analytic functions, elementary complex functions, and conformal mapping. Other areas that may be introduced are maximum modulus principle, residue theorem, and Laurent series.

5 Most frequently asked questions (FAQs) about a PhD in Mathematics

Making the decision to get a PhD in math is a personal one. It is important to consider factors such as long-term career goals, costs, and time commitments. Those who complete this degree may find it valuable as it provides access to the skills needed to work as a professor or in some other positions.

Completing a doctoral degree in mathematics may help prepare a student to work in a range of jobs. This may include working in accountancy and finance careers, banking careers, statistician positions, teaching, and engineering positions.

Completing a PhD in math is a necessary move for those who are looking for a career in academia, in many cases. It may also be beneficial to those who wish to research in areas such as engineering, medicine, or technology. Some may also use it to help study and solve complex problems in real-world settings.

Though programs differ, many people spend 3 to 5 years working to earn a PhD in math. This may be shorter for those who complete some of their courses online or in an accelerated program.

Topics range widely but they may include analysis, differential geometry, discrete math, theoretical computer science, probability, statistics, and numerous other areas.

Top 25 Schools Graduating Students with a PhD in Mathematics

Below are the top 24 schools in the U.S. for a PhD in mathematics based on the number of students graduating from the program in 2020, according to NCES data.

College / UniversityGraduatesAcceptance Rate
University of Illinois Urbana-Champaign3159%
Massachusetts Institute of Technology267%
University of California-Los Angeles2512%
Purdue University-Main Campus2360%
Texas A & M University-College Station2158%
University of Wisconsin-Madison2154%
University of California-Berkeley2016%
The Pennsylvania State University1976%
University of Michigan-Ann Arbor1923%
University of Utah1962%
University of Illinois Chicago1873%
Ohio State University-Main Campus1754%
University of California-Irvine1727%
University of Chicago176%
The University of Tennessee-Knoxville1679%
The University of Texas at Austin1632%
University of Maryland-College Park1644%
CUNY Graduate School and University Center15N/A
Auburn University1481%
University of Colorado Boulder1478%
University of South Carolina-Columbia1469%
Columbia University in the City of New York135%
Georgia Institute of Technology-Main Campus1321%
University of California-San Diego1331%
University of Notre Dame1316%

Find Funding

For those thinking about completing a PhD in math , it may be necessary to consider paying for it. Finding funding may be possible for many students pursuing a mathematics doctoral degree. The average cost of a master’s degree program in 2019 to 2020 was $19,792, and some PhD programs are even higher in cost according to NCES. For that reason, it is important to consider all funding options and what they could mean for your long-term educational goals. Take a closer look at some of the funding options that may be available.


Paying for college may be challenging for many people. Scholarships may help to reduce the amount of money a student has to pay for their education. These are funds that typically do not require repayment. Getting them may be challenging as they typically require careful consideration. Students must find, apply for, and be awarded scholarships and then make sure they continue to meet any requirements set for them. To get started, learn more about scholarships for PhD students studying mathematics that may be available.

WISE Scholarship

Who Can Apply: Washingtonian women interested in a career in a STEM field are welcome to apply for the $3,500 scholarship. The scholarship award is to be used toward tuition and educational expenses, including but not limited to: textbooks, supplies (paper, pens, notebooks, etc.), required technology, on-campus dining, housing, and transportation costs.

Amount: $3,500

Deadline: January 7

SHPE Graduate Scholarship

Who Can Apply: This award is for graduate students enrolled full time in a degree program in a STEM (science, technology, engineering, mathematics) field. Students must be members of the Society of Hispanic Professional Engineers and have a grade point average of 3.0 or higher. Students of Hispanic descent will be given priority.

Amount: $3,000

Deadline: April 30

Barry M. Goldwater Scholarship

Who Can Apply: This award is for U.S. sophomores or juniors in college who are majoring in mathematics, the natural sciences, or engineering. The applicant must have a 3.37 grade point average or higher to be eligible.

Amount: $15,000

Deadline: January 29

SHPE Dissertation Scholarship

Who Can Apply: This award is for graduate students enrolled full time in a Ph.D. or Ed.D. degree program in a STEM (science, technology, engineering, mathematics) field who have advanced to candidacy. Students must be members of the Society of Hispanic Professional Engineers and have a grade point average of 3.5 or higher. Students of Hispanic descent will be given priority.

Amount: $5,000

Deadline: April 30

Federal Loans

Federal student loans are one of the options some students have for paying for their graduate degree. These are loans backed by the U.S. federal government. That means these lenders often have to stick with the federal government’s rules for lending to college students. These loans are a bit different at the graduate level than they are at the undergraduate level. However, they may still be accessible to those working on grade school. Here are a few options to consider.

  • Direct Unsubsidized Loans: These are available to graduate schools and professional students. Unlike direct subsidized loans, which are made available to undergraduate students with demonstrated financial need, direct unsubsidized loans often do not have a need based requirement. Many students may be eligible for them.
  • Direct PLUS Loans: These are often made available to professional or graduate level students. They are usually designed to pay for educational expenses that are typically not paid for through other loans. Eligibility for these loans is not based on financial need. However, some require a credit check to be performed, and borrowers may not qualify without meeting other requirements in some cases.
  • Direct Consolidation Loans: These loans may enable a student to combine all of their undergraduate and graduate level federal debt into one new loan. This consolidation loan is typically provided after a student completes their education.

Private Student Loans

Private student loans are usually another opportunity for some students. Unlike federal loans, these often do not have U.S. government backing. That means that the terms and conditions of them may change from one lender to the next. It is important to know what to expect. Learn as much as you can about the private student loans and the lender’s qualifications. Some key questions to ask include:

  • Amount allowed to borrow
  • Credit score qualifications for the loan
  • Income requirements
  • Whether forbearance and refinancing are options
  • Is loan forgiveness a possible option
  • When repayment starts

Learn as much as you can about the private student loan options before you choose one to help cover the cost of your education.

Is mathematician a good career?

A mathematician is a professional who conducts research on mathematics fundamentals or applicatons of math techniques to management, science, or other fields. They may also solve problems in various fields using mathematical methods.

The day-to-day tasks of a person working in this field include addressing the relationships of quantities, forms, and magnitudes through the use of symobols and numbers. They may also utilize research by writing reports and publishing papers. Some may also present at professional conferences. They may also maintain knowledge in the field. They may also apply math theories and techniques to find solutions to problems related to sciences, business, engineering, and other fields. Many also conduct research to further the understanding of math in areas of probability, logic, business, and engineering.

PhD in Mathematics , Important skills for mathematicians
  • Mathematics – 100 Using mathematics to solve problems.
  • Critical Thinking – 81 Using logic and reasoning to identify the strengths and weaknesses of alternative solutions, conclusions or approaches to problems.
  • Complex Problem Solving – 78 Identifying complex problems and reviewing related information to develop and evaluate options and implement solutions.

Some of their work activities may include preparing analytical reports, analyzing data to identify trends in variables. They may also present research results to others in the field. They work to maintain their knowledge about emerging industry and tech trends. Often, they are tasked with finding solutions to problems in applied fields or in science.

To do the work of a mathematician, it may be necessary to have at least a master’s degree. Some employers want individuals to have a doctoral degree in this area. This often requires at least five years of experiences.

2020 Median Salary for a Mathematician

The median salary for people working as a mathematician in 2020 in each of the states is as follows, based on information provided by the Bureau of Labor Statistics:

New Jersey$117,800MissouriN/A
New York$104,550NebraskaN/A
North DakotaN/ANew HampshireN/A
OklahomaN/ANew MexicoN/A
Pennsylvania$99,160North Carolina$124,170
South CarolinaN/AOhioN/A
UtahN/ARhode IslandN/A
Virginia$126,770South DakotaN/A
West VirginiaN/AWyomingN/A

Is statistician a good career?

A person working as a statistician works to develop and apply math or statistical theory and methods in a variety of ways. This may include collecting, organizing, and interpreting data so they can summarize and use that data for various goals or needs. They may have a specific focus in an area such as biostatistics, business statistics, economic statistics, or agricultural statistics. Many work as survey or mathematical statisticians in various fields.

The tasks of a statistician range widely based on where they work. Many times, their work centers around analyzing and interpreting statistical data as a way to identify significant differences in relationships among various sources of information. They may also identify relationships and trends in data as well as other factors and use that to help with completing research. In some situations, statisticians also determine whether various statistical methods are the appropriate choice for the task at hand based on needs and research questions of interest. They often prepare data for processing by organization information and checking for inaccuracies.  Many also present statistical results using graphs, bullets, or charts in a variety of settings such as meetings or conferences.

PhD in Mathematics , important skills for statisticians
  • Mathematics – Using mathematics to solve problems.
  • Critical Thinking – Using logic and reasoning to identify the strengths and weaknesses of alternative solutions, conclusions or approaches to problems.
  • Reading Comprehension – Understanding written sentences and paragraphs in work related documents.

Some of the work activities in this field include analyzing data to identify trends or relationships among various elements and determining the appropriate methods for analyzing data. They may also evaluate technical data to determine the effects on designs or plans. Many also spend their time evaluating data quality and preparing data for further analysis.

To do this work, many employers require individuals to have a master’s degree in education. Some employers may need a student to have a PhD instead. It may be common for organizations to expect individuals to have 5 to 7 years of experienced in the field of interest.

2020 Median Salary for a Statistician

For those working as a statistician in 2020, the following was their median income according to data from the Bureau of Labor Statistics:

New Jersey$117,500Missouri$66,950
New York$98,830Nebraska$70,770
North Dakota$63,090New Hampshire$77,730
Oklahoma$59,340New Mexico$61,690
Pennsylvania$82,670North Carolina$100,480
South Carolina$67,330Ohio$81,280
Utah$70,400Rhode Island$99,710
VirginiaN/ASouth DakotaN/A
West Virginia$73,650WyomingN/A

Sandy Baker


Sandy has extensive experience writing educational articles for topics ranging from online education to college degrees. She’s worked with several Ivy League colleges to create blogs, newsletters, sales material for recruiting as well as “how to manage” college lifestyle pieces. Additionally, she’s written for well-respected study abroad programs helping students to find international opportunities spanning the globe from South America to Africa and Asia.

Sandy’s experience also includes writing about financial aid, FAFSA, scholarship searches, and managing college loans and grants. This includes aiding both students and parents in managing the application and financial aid process from start to finish. Her writing in this area has been featured in The New York Times, Cleveland Magazine, and several blogs.

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