phd in maths scope

Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
• Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
• Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
• Complete a seminar, giving a talk of at least one-hour duration
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
• Meet the University residence requirement of two years or four semesters

Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members.  All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

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The department offers the doctor of philosophy degree with a major in mathematics and a master of arts degree in mathematics.

The Ph.D. degree requires proficiency in basic and advanced graduate mathematics and the completion of a dissertation containing a significant piece of original research in some area of mathematics. The scope of the research program in mathematics is broad. The Ph.D specialty and dissertation may be in any area of mathematics, including but not limited to algebra, algebraic geometry, applied mathematics, combinatorics, computational mathematics, complex analysis, differential equations, differential geometry, dynamical systems, harmonic analysis, logic, mathematical biology, number theory, probability, and topology. A complete list of faculty and their areas of expertise is available through the department website .

Students in the Ph.D. program also have the option to earn a master of arts degree .

Please consult the table below for key information about this degree program’s admissions requirements. The program may have more detailed admissions requirements, which can be found below the table or on the program’s website.

Graduate admissions is a two-step process between academic programs and the Graduate School. Applicants must meet the minimum requirements of the Graduate School as well as the program(s). Once you have researched the graduate program(s) you are interested in, apply online .

Admission is competitive. Applicants to the Ph.D. program are automatically considered for financial support. For more information about application to the Ph.D. and M.A. programs, see the department's admission website .

Graduate School Resources

Resources to help you afford graduate study might include assistantships, fellowships, traineeships, and financial aid.  Further funding information is available from the Graduate School. Be sure to check with your program for individual policies and restrictions related to funding.

Program Resources

Prospective students should see the program website for funding information .

Minimum Graduate School Requirements

Major requirements.

Review the Graduate School minimum academic progress and degree requirements , in addition to the program requirements listed below.

MODE OF INSTRUCTION

Mode of instruction definitions.

Accelerated: Accelerated programs are offered at a fast pace that condenses the time to completion. Students typically take enough credits aimed at completing the program in a year or two.

Evening/Weekend: ​Courses meet on the UW–Madison campus only in evenings and/or on weekends to accommodate typical business schedules.  Students have the advantages of face-to-face courses with the flexibility to keep work and other life commitments.

Face-to-Face: Courses typically meet during weekdays on the UW-Madison Campus.

Hybrid: These programs combine face-to-face and online learning formats.  Contact the program for more specific information.

Online: These programs are offered 100% online.  Some programs may require an on-campus orientation or residency experience, but the courses will be facilitated in an online format.

CURRICULAR REQUIREMENTS

Required courses.

Take a total of 51 graduate credits, or generally 18 courses . This includes courses in math and in breadth. In the Ph.D. program, math courses numbered above 700 are for graduate credit. Math courses below 700 must be approved by the academic advisor.

There are five general and overlapping areas of specialization 1 within the department:   

• Algebra, Algebraic Geometry, Combinatorics and Number theory
• Analysis, Differential Equations and Probability
• Applied and Computational Mathematics
• Geometry, Topology, and Dynamics

There is also a specialty in Mathematics Education 1 . The course requirement is the same as for the other specialties except that the required 51 credits should include 18 credits in courses that relate to mathematics education, and at least one of the courses must be on research techniques in education. The 18 credits may come (wholly or in part) from courses included in breadth.

These pathways are internal to the program and represent different curricular paths a student can follow to earn this degree. Pathway names do not appear in the Graduate School admissions application, and they will not appear on the transcript.

Graduate School Policies

The  Graduate School’s Academic Policies and Procedures  provide essential information regarding general university policies. Program authority to set degree policies beyond the minimum required by the Graduate School lies with the degree program faculty. Policies set by the academic degree program can be found below.

Major-Specific Policies

Prior coursework, graduate work from other institutions.

With program approval, students in the Ph.D. program are allowed to count no more than 19 credits of graduate coursework from other institutions. Coursework earned ten years or more prior to admission to a doctoral degree is not allowed to satisfy requirements.

UW–Madison Undergraduate

No more than 7 credits from a UW–Madison undergraduate degree are allowed to count toward the degree. Coursework earned ten years or more prior to admission to a doctoral degree is not allowed to satisfy requirements.

UW–Madison University Special

With program approval, students are allowed to count no more than 15 credits of coursework numbered 300 or above taken as a UW–Madison Special student. Coursework earned ten years or more prior to admission to a doctoral degree is not allowed to satisfy requirements.

This program follows the Graduate School's Probation policy.

• Good standing (progressing according to standards; any funding guarantee remains in place).
• Probation (not progressing according to standards but permitted to enroll; loss of funding guarantee; specific plan with dates and deadlines in place in regard to removal of probationary status).
• Unsatisfactory progress (not progressing according to standards; not permitted to enroll, dismissal, leave of absence or change of advisor or program).

ADVISOR / COMMITTEE

Students who are not yet working with a dissertation advisor are required to meet semiannually with their academic advisor. All students must have a dissertation advisor by the end of the sixth semester.

CREDITS PER TERM ALLOWED

15 credits. Minimum of 6/semester, other than dissertators.

Time Limits

Students must complete all program requirements within eight years of beginning the program . Extensions have to be approved by the program.

A candidate for a doctoral degree who fails to take the final oral examination and deposit the dissertation within five years after passing the preliminary examination may be required to take another preliminary examination and to be admitted to candidacy a second time.

Doctoral degree students who have been absent for ten or more consecutive years lose all credits that they have earned before their absence. Individual programs may count the coursework students completed prior to their absence for meeting program requirements; that coursework may not count toward Graduate School credit requirements.

grievances and appeals

These resources may be helpful in addressing your concerns:

• Bias or Hate Reporting  
• Graduate Assistantship Policies and Procedures
• Office of the Provost for Faculty and Staff Affairs
• Dean of Students Office (for all students to seek grievance assistance and support)
• Employee Assistance (for personal counseling and workplace consultation around communication and conflict involving graduate assistants and other employees, post-doctoral students, faculty and staff)
• Employee Disability Resource Office (for qualified employees or applicants with disabilities to have equal employment opportunities)
• Graduate School (for informal advice at any level of review and for official appeals of program/departmental or school/college grievance decisions)
• Office of Compliance (for class harassment and discrimination, including sexual harassment and sexual violence)
• Office of Student Conduct and Community Standards (for conflicts involving students)
• Ombuds Office for Faculty and Staff (for employed graduate students and post-docs, as well as faculty and staff)
• Title IX (for concerns about discrimination)

Students should contact the department chair or program director with questions about grievances. They may also contact the L&S Academic Divisional Associate Deans, the L&S Associate Dean for Teaching and Learning Administration, or the L&S Director of Human Resources.

Take advantage of the Graduate School's  professional development resources to build skills, thrive academically, and launch your career. 

• Learn a substantial body of mathematics in introductory and research level graduate courses in mathematics.
• Complete a dissertation under the guidance of an advisor. The dissertation should make an original and substantive contribution to its subject matter.
• Demonstrate breadth within the learning experiences.
• Present research in seminar talks, conferences or publications.
• Communicate complex ideas in a clear and understandable manner.
• Foster ethical and professional conduct.

Please visit the Math Department website for a complete list of faculty and instructional academic staff .

• Requirements
• Professional Development
• Learning Outcomes

Contact Information

Mathematics College of Letters & Science math.wisc.edu

Kathie Brohaugh, Graduate Coordinator [email protected] 608-263-8884 203 Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706

Autumn Kent, Director of Graduate Studies [email protected]

Tullia Dymarz, Director of Admissions [email protected]

Graduate Program Handbook View Here

Graduate School grad.wisc.edu

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• Doing a PhD in Mathematics
• Doing a PhD

What Does a PhD in Maths Involve?

Maths is a vast subject, both in breadth and in depth. As such, there’s a significant number of different areas you can research as a math student. These areas usually fall into one of three categories: pure mathematics, applied mathematics or statistics. Some examples of topics you can research are:

• Number theory
• Numerical analysis
• String theory
• Random matrix theory
• Graph theory
• Quantum mechanics
• Statistical forecasting
• Matroid theory
• Control theory

Besides this, because maths focuses on addressing interdisciplinary real-world problems, you may work and collaborate with other STEM researchers. For example, your research topic may relate to:

• Biomechanics and transport processes
• Evidence-based medicine
• Fluid dynamics
• Financial mathematics
• Machine learning
• Theoretical and Computational Optimisation

What you do day-to-day will largely depend on your specific research topic. However, you’ll likely:

• Continually read literature – This will be to help develop your knowledge and identify current gaps in the overall body of knowledge surrounding your research topic.
• Undertake research specific to your topic – This can include defining ideas, proving theorems and identifying relationships between models.
• Collect and analyse data – This could comprise developing computational models, running simulations and interpreting forecasts etc.
• Liaise with others – This could take many forms. For example, you may work shoulder-to-shoulder with individuals from different disciplines supporting your research, e.g. Computer scientists for machine learning-based projects. Alternatively, you may need frequent input from those who supplied the data for your research, e.g. Financial institutions or biological research colleagues.
• Attend a wide range of lectures, seminars and events.

Browse PhD Opportunities in Mathematics

Application of artificial intelligence to multiphysics problems in materials design, study of the human-vehicle interactions by a high-end dynamic driving simulator, physical layer algorithm design in 6g non-terrestrial communications, machine learning for autonomous robot exploration, detecting subtle but clinically significant cognitive change in an ageing population, how long does it take to get a phd in maths.

The average programme duration for a mathematics PhD in the UK is 3 to 4 years for a full-time studying. Although not all universities offer part-time maths PhD programmes, those that do have a typical programme duration of 5 to 7 years.

Again, although the exact arrangement will depend on the university, most maths doctorates will require you to first register for an MPhil . At the end of your first year, your supervisor will assess your progress to decide whether you should be registered for a PhD.

Additional Learning Modules

Some Mathematics departments will require you to enrol on to taught modules as part of your programme. These are to help improve your knowledge and understanding of broader subjects within your field, for example, Fourier Analysis, Differential Geometry and Riemann Surfaces. Even if taught modules aren’t compulsory in several universities, your supervisor will still encourage you to attend them for your development.

Most UK universities will also have access to specialised mathematical training courses. The most common of these include Pure Mathematics courses hosted by Mathematics Access Grid Conferencing ( MAGIC ) and London Taught Course Centre ( LTCC ) and Statistics courses hosted by Academy for PhD Training in Statistics ( APTS ).

What Are the Typical Entry Requirements for A PhD in Maths?

In the UK, the typical entry requirements for a Maths PhD is an upper second-class (2:1) Master’s degree (or international equivalent) in Mathematics or Statistics [1] .

However, there is some variation on this. From writing, the lowest entry requirement is an upper second-class (2:1) Bachelor’s degree in any math-related subject. The highest entry requirement is a first-class (1st) honours Master’s degree in a Mathematics or Statistics degree only.

It’s worth noting if you’re applying to a position which comes with funding provided directly by the Department, the entry requirements will usually be on the higher side because of their competitiveness.

In terms of English Language requirements, most mathematics departments require at least an overall IELTS (International English Language Testing System) score of 6.5, with no less than 6.0 in each individual subtest.

Tips to Consider when Making Your Application

When applying to any mathematics PhD, you’ll be expected to have a good understanding of both your subject field and the specific research topic you are applying to. To help show this, it’s advisable that you demonstrate recent engagement in your research topic. This could be by describing the significance of a research paper you recently read and outlining which parts interested you the most, and why. Additionally, you can discuss a recent mathematics event you attended and suggest ways in how what you learnt might apply to your research topic.

As with most STEM PhDs, most maths PhD professors prefer you to discuss your application with them directly before putting in a formal application. The benefits of this is two folds. First, you’ll get more information on what their department has to offer. Second, the supervisor can better discover your interest in the project and gauge whether you’d be a suitable candidate. Therefore, we encourage you to contact potential supervisors for positions you’re interested in before making any formal applications.

How Much Does a Maths PhD Typically Cost?

The typical tuition fee for a PhD in Maths in the UK is £4,407 per year for UK/EU students and £20,230 per year for international students. This, alongside the range in tuition fees you can expect, is summarised below:

Note: The above tuition fees are based on 12 UK Universities [1]  for 2020/21 Mathematic PhD positions. The typical fee has been taken as the median value.

In addition to the above, it’s not unheard of for research students to be charged a bench fee. In case you’re unfamiliar with a bench fee, it’s an annual fee additional to your tuition, which covers the cost of specialist equipment or resources associated with your research. This can include the upkeep of supercomputers you may use, training in specialist analysis software, or travelling to conferences. The exact fee will depend on your specific research topic; however, it should be minimal for most mathematic projects.

What Specific Funding Opportunities Are There for A PhD in Mathematics?

Alongside the usual funding opportunities available to all PhD Research students such as doctoral loans, departmental scholarships, there are a few other sources of funding available to math PhD students. Examples of these include:

You can find more information on these funding sources here: DiscoverPhDs funding guide .

What Specific Skills Do You Gain from Doing a PhD in Mathematics?

A doctorate in Mathematics not only demonstrates your commitment to continuous learning, but it also provides you with highly marketable skills. Besides subject-specific skills, you’ll also gain many transferable skills which will prove useful in almost all industries. A sample of these skills is listed below.

• Logical ability to consider and analyse complex issues,
• Commitment and persistence towards reaching research goals,
• Outstanding verbal and written skills,
• Strong attention to detail,
• The ability to liaise with others from unique disciple backgrounds and work as part of a team
• Holistic deduction and reasoning skills,
• Forming and explaining mathematical and logical solutions to a wide range of real-world problems,
• Exceptional numeracy skills.

What Jobs Can You Get with A Maths PhD?

One of the greatest benefits maths PostDocs will have is the ability to pursue a wide range of career paths. This is because all sciences are built on core principles which, to varying extents, are supported by the core principles of mathematics. As a result, it’s not uncommon to ask students what path they intend to follow after completing their degree and receive entirely different answers. Although not extensive by any means, the most common career paths Math PostDocs take are listed below:

• Academia – Many individuals teach undergraduate students at the university they studied at or ones they gained ties to during their research. This path is usually the preferred among students who want to continue focusing on mathematical theories and concepts as part of their career.
• Postdoctoral Researcher – Others continue researching with their University or with an independent organisation. This can be a popular path because of the opportunities it provides in collaborative working, supervising others, undertaking research and attending conferences etc.
• Finance – Because of their deepened analytical skills, it’s no surprise that many PostDocs choose a career in finance. This involves working for some of the most significant players in the financial district in prime locations including London, Frankfurt and Hong Kong. Specific job titles can include Actuarial, Investment Analyst or Risk Modeller.
• Computer Programming – Some students whose research involves computational mathematics launch their career as a computer programmer. Due to their background, they’ll typically work on specialised projects which require high levels of understanding on the problem at hand. For example, they may work with physicists and biomedical engineers to develop a software package that supports their more complex research.
• Data Analyst – Those who enjoy number crunching and developing complex models often go into data analytics. This can involve various niches such as forecasting or optimisation, across various fields such as marketing and weather.

What Are Some of The Typical Employers Who Hire Maths PostDocs?

As mentioned above, there’s a high demand for skilled mathematicians and statisticians across a broad range of sectors. Some typical employers are:

• Education – All UK and international universities
• Governments – STFC and Department for Transport
• Healthcare & Pharmaceuticals – NHS, GSK, Pfizer
• Finance & Banking – e.g. Barclays Capital, PwC and J. P. Morgan
• Computing – IBM, Microsoft and Facebook
• Engineering – Boeing, Shell and Dyson

The above is only a small selection of employers. In reality, mathematic PostDocs can work in almost any industry, assuming the role is numerical-based or data-driven.

How Much Can You Earn with A PhD in Maths?

As a mathematics PhD PostDoc, your earning potential will mostly depend on your chosen career path. Due to the wide range of options, it’s impossible to provide an arbitrary value for the typical salary you can expect.

However, if you pursue one of the below paths or enter their respective industry, you can roughly expect to earn [3] :

Academic Lecturer

• Approximately £30,000 – £35,000 starting salary
• Approximately £40,000 with a few years experience
• Approximately £45,000 – £55,000 with 10 years experience
• Approximately £60,000 and over with significant experience and a leadership role. Certain academic positions can earn over £80,000 depending on the management duties.

Actuary or Finance

• Approximately £35,000 starting salary
• Approximately £45,000 – £55,000 with a few years experience
• Approximately £70,000 and over with 10 years experience
• Approximately £180,000 and above with significant experience and a leadership role.

Aerospace or Mechanical Engineering

• Approximately £28,000 starting salary
• Approximately £35,000 – £40,000 with a few years experience
• Approximately £60,000 and over with 10 years experience

Data Analyst

• Approximately £45,000 – £50,000 with a few years experience
• Approximately £90,000 and above with significant experience and a leadership role.

Again, we stress that the above are indicative values only. Actual salaries will depend on the specific organisation and position and responsibilities of the individual.

Facts and Statistics About Maths PhD Holders

The below chart provides useful insight into the destination of Math PostDocs after completing their PhD. The most popular career paths from other of highest to lowest is education, information and communication, finance and scientific research, manufacturing and government.

Note: The above chart is based on ‘UK Higher Education Leavers’ data [2] between 2012/13 and 2016/17 and contains a data size of 200 PostDocs. The data was obtained from the Higher Education Statistics Agency ( HESA ).

Which Noteworthy People Hold a PhD in Maths?

Alan turing.

Alan Turing was a British Mathematician, WW2 code-breaker and arguably the father of computer science. Alongside his lengthy list of achievements, Turning achieved a PhD in Mathematics at Princeton University, New Jersey. His thesis titled ‘Systems of Logic Based on Ordinals’ focused on the concepts of ordinal logic and relative computing; you can read it online here . To this day, Turning pioneering works continues to play a fundamental role in shaping the development of artificial intelligence (AI).

Ruth Lawrence

Ruth Lawrence is a famous British–Israeli Mathematician well known within the academic community. Lawrence earned her PhD in Mathematics from Oxford University at the young age of 17! Her work focused on algebraic topology and knot theory; you can read her interesting collection of research papers here . Among her many contributions to Maths, her most notable include the representation of the braid groups, more formally known as Lawrence–Krammer representations.

Emmy Noether

Emmy Noether was a German mathematician who received her PhD from the University of Erlangen, Germany. Her research has significantly contributed to both abstract algebra and theoretical physics. Additionally, she proved a groundbreaking theorem important to Albert Einstein’s general theory of relativity. In doing so, her theorem, Noether’s theorem , is regarded as one of the most influential developments in physics.

Other Useful Resources

Institute of Mathematics and its Applications (IMA) – IMA is the UK’s professional body for mathematicians. It contains a wide range of useful information, from the benefits of further education in Maths to details on grants and upcoming events.

Maths Careers – Math Careers is a site associated with IMA that provides a wide range of advice to mathematicians of all ages. It has a section dedicated to undergraduates and graduates and contains a handful of information about progressing into research.

Resources for Graduate Students – Produced by Dr Mak Tomford, this webpage contains an extensive collection of detailed advice for Mathematic PhD students. Although the site uses US terminology in places, don’t let that put you off as this resource will prove incredibly helpful in both applying to and undertaking your PhD.

Student Interviews – Still wondering whether a PhD is for you? If so, our collection of PhD interviews would be a great place to get an insider perspective. We’ve interviewed a wide range of PhD students across the UK to find out what doing a PhD is like, how it’s helped them and what advice they have for other prospective students who may be thinking of applying to one. You can read our insightful collection of interviews here .

[1] Universities used to determine the typical (median) and range of entry requirements and tuition fees for 2020/21 Mathematics PhD positions.

• http://www.lse.ac.uk/study-at-lse/Graduate/Degree-programmes-2020/MPhilPhD-Mathematics
• https://www.ox.ac.uk/admissions/graduate/courses/dphil-mathematics?wssl=1
• https://www.graduate.study.cam.ac.uk/courses/directory/mapmpdpms
• https://www.ucl.ac.uk/prospective-students/graduate/research-degrees/mathematics-mphil-phd
• http://www.bristol.ac.uk/study/postgraduate/2020/sci/phd-mathematics/
• https://www.surrey.ac.uk/postgraduate/mathematics-phd
• https://www.maths.ed.ac.uk/school-of-mathematics/studying-here/pgr/phd-application
• https://www.lancaster.ac.uk/study/postgraduate/postgraduate-courses/mathematics-phd/
• https://www.sussex.ac.uk/study/phd/degrees/mathematics-phd
• https://www.manchester.ac.uk/study/postgraduate-research/programmes/list/05325/phd-pure-mathematics/
• https://warwick.ac.uk/study/postgraduate/research/courses-2020/mathematicsphd/
• https://www.exeter.ac.uk/pg-research/degrees/mathematics/

[2] Higher Education Leavers Statistics: UK, 2016/17 – Outcomes by subject studied – https://www.hesa.ac.uk/news/28-06-2018/sfr250-higher-education-leaver-statistics-subjects

[3] Typical salaries have been extracted from a combination of the below resources. It should be noted that although every effort has been made to keep the reported salaries as relevant to Math PostDocs as possible (i.e. filtering for positions which specify a PhD qualification as one of their requirements/preferences), small inaccuracies may exist due to data availability.

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PhD in Mathematics

Here are the requirements for earning the PhD degree in Mathematics offered by the School of Math. For requirements of other PhD programs housed within the School, please see their specific pages at  Doctoral Programs . The requirements for all these programs consist of three components:  coursework ,  examinations , and  dissertation  in accordance to the guidelines described in the  GT Catalogue .

Completion of required coursework, examinations, and dissertation normally takes about five years. During the first one or two years, students concentrate on coursework to acquire the background necessary for the comprehensive examinations. By the end of their third year in the program, all students are expected to have chosen a thesis topic, and begin work on the research and writing of the dissertation.

The program of study must contain at least 30 hours of graduate-level coursework (6000-level or above) in mathematics and an additional 9 hours of coursework towards a minor. The minor requirement consists of graduate or advanced undergraduate coursework taken entirely outside the School of Mathematics, or in an area of mathematics sufficiently far from the students area of specialization.

Prior to admission to candidacy for the doctoral degree, each student must satisfy the School's comprehensive examinations (comps) requirement. The first phase is a written examination which students must complete by the end of their second year in the graduate program. The second phase is an oral examination in the student's proposed area of specialization, which must be completed by the end of the third year.

Research and the writing of the dissertation represent the final phase of the student's doctoral study, and must be completed within seven years of the passing of comps. A final oral examination on the dissertation (theses defense) must be passed prior to the granting of the degree.

The Coursework

The program of study must satisfy the following  hours ,  minor , and  breadth  requirements. Students who entered before Fall 2015 should see  the old requirements , though they may opt into the current rules described below, and are advised to do so.

Hours requirements.  The students must complete 39 hours of coursework as follows:

• At least 30 hours must be in mathematics courses at the 6000-level or higher.
• At least 9 hours must form the doctoral minor field of study.
• The overall GPA for these courses must be at least 3.0.
• These courses must be taken for a letter grade and passed with a grade of at least C.

Minor requirement.  The minor field of study should consist primarily of 6000-level (or higher) coursework in a specific area outside the School of Math, or in a mathematical subject sufficiently far from the student’s thesis work. A total of 9 credit hours is required and must be passed with a grade of B or better. These courses should not include MATH 8900, and must be chosen in consultation with the PhD advisor and the Director of Graduate Studies to ensure that they form a cohesive group which best complements the students research and career goals. A student wishing to satisfy the minor requirement by mathematics courses must petition the Graduate Committee for approval.  Courses used to fulfill a Basic Understanding breadth requirement in Analysis or Algebra should not be counted towards the doctoral minor. Upon completing the minor requirement, a student should immediately complete the  Doctoral Minor form .

Breadth requirements.  The students must demonstrate:

• Basic understanding of 2 subjects must be demonstrated through passing the subjects' written comprehensive exams.  At least 1 of these 2 exams must be in Algebra or Analysis.
• Basic understanding of the third subject may be demonstrated either by completing two courses in the subject (with a grade of A or B in each course) or by passing the subject's written comprehensive exam.
• A basic understanding of both subjects in Area I (analysis and algebra) must be demonstrated.
• Earning a grade of A or B in a one-semester graduate course in a subject demonstrates exposure to the subject.
• Passing a subject's written comprehensive exam also demonstrates exposure to that subject.

The subjects.  The specific subjects, and associated courses, which can be used to satisfy the breadth requirements are as follows.

• Area I subjects:​
• Area II subjects:​

Special Topics and Reading Courses.

• Special topics courses may always be used to meet hours requirements.
• Special topics courses may be used to meet breadth requirements, subject to the discretion of the Director of Graduate Studies.
• Reading courses may be used to meet hours requirements but not breadth requirements.

Credit Transfers

Graduate courses completed at other universities may be counted towards breadth and hours requirements (courses designated as undergraduate or Bachelors' level courses are not eligible to transfer for graduate credit).  These courses do not need to be officially transferred to Georgia Tech. At a student’s request, the Director of Graduate Studies will determine which breadth and hours requirements have been satisfied by graduate-level coursework at another institution.  

Courses taken at other institutions may also be counted toward the minor requirement, subject to the approval of the Graduate Director; however, these courses must be officially transferred to Georgia Tech.

There is no limit for the transfer of credits applied toward the breadth requirements; however, a maximum of 12 hours of coursework from other institutions may be used to satisfy hours requirements. Thus at least 27 hours of coursework must be completed at Georgia Tech, including at least 18 hours of 6000-level (or higher) mathematics coursework.

Students wishing to petition for transfer of credit from previous graduate level work should send the transcripts and syllabi of these courses, together with a list of the corresponding courses in the School of Math, to the Director of Advising and Assessment for the graduate program.

Comprehensive Examinations

The comprehensive examination is in two phases. The first phase consists of passing two out of seven written examinations. The second phase is an oral specialty examination in the student's planned area of concentration. Generally, a student is expected to have studied the intended area of research but not necessarily begun dissertation research at the time of the oral examination.

Written examinations.  The written examinations will be administered twice each year, shortly after the beginning of the Fall and Spring semesters. The result of the written examination is either pass or fail. For syllabi and sample exams see the  written exams page .

All students must adhere to the following rules and timetables, which may be extended by the Director of Graduate Studies, but only at the time of matriculation and only when certified in writing. Modifications because of leaves from the program will be decided on a case-by-case basis.

After acceptance into the PhD Program in Mathematics, a student must pass the written examinations no later than their fourth administration since the student's doctoral enrollment. The students can pass each of the two written comprehensive exams in separate semesters, and are allowed multiple attempts.

The Director of Graduate Studies (DGS) will be responsible for advising each new student at matriculation of these rules and procedures and the appropriate timetable for the written portion of the examination. The DGS will also be responsible for maintaining a study guide and list of recommended texts, as well as a file of previous examinations, to be used by students preparing for this written examination.

Oral examination.  A student must pass the oral specialty examination within three years since first enrolling in the PhD program, and after having passed the written portion of the comprehensive exams. The examination will be given by a committee consisting of the student's dissertation advisor or probable advisor, two faculty members chosen by the advisor in consultation with the student, and a fourth member appointed by the School's Graduate Director. The scope of the examination will be determined by the advisor and will be approved by the graduate coordinator. The examining committee shall either (1) pass the student or (2) fail the student. Within the time constraints of which above, the oral specialty examination may be attempted multiple times, though not more than twice in any given semester. For more details and specific rules and policies see the  oral exam page .

Dissertation and Defense

A dissertation and a final oral examination are required. For details see our  Dissertation and Graduation  page, which applies to all PhD programs in the School of Math.

Ph.D. Program Overview

Description.

The graduate program in the field of mathematics at Cornell leads to the Ph.D. degree, which takes most students five to six years of graduate study to complete. One feature that makes the program at Cornell particularly attractive is the broad range of  interests of the faculty . The department has outstanding groups in the areas of algebra, algebraic geometry,  analysis, applied mathematics, combinatorics, dynamical systems, geometry, logic, Lie groups, number theory, probability, and topology. The field also maintains close ties with distinguished graduate programs in the fields of  applied mathematics ,  computer science ,  operations research , and  statistics .

Core Courses

A normal course load for a beginning graduate student is three courses per term. 

There are no qualifying exams, but the program requires that all students pass four courses to be selected from the six core courses. First-year students are allowed to place out of some (possibly, all) of the core courses. In order to place out of a course, students should contact the faculty member who is teaching the course during the current academic year, and that faculty member will make a decision. The minimum passing grade for the core courses is B-; no grade is assigned for placing out of a core course.

At least two core courses should be taken (or placed out) by the end of the first year. At least four core courses should be taken (or placed out) by the end of the second year (cumulative). These time requirements can be waived for students with health problems or other significant non-academic problems. They can be also waived for students who take time-consuming courses in another area (for example, CS) and who have strong support from a faculty; requests from such students should be made before the beginning of the spring semester. 

The core courses  are distributed among three main areas: analysis, algebra and topology/geometry. A student must pass at least one course from each group. All entering graduate students are encouraged to eventually take all six core courses with the option of an S/U grade for two of them. 

The six core courses are:

MATH 6110, Real Analysis

MATH 6120, Complex Analysis

MATH 6310, Algebra 1

MATH 6320, Algebra 2

MATH 6510, Introductory Algebraic Topology

MATH 6520, Differentiable Manifolds.

Students who are not ready to take some of the core courses may take MATH 4130-4140, Introduction to Analysis, and/or MATH 4330-4340, Introduction to Algebra, which are the honors versions of our core undergraduate courses.

"What is...?" Seminar

The "What Is...?" Seminar is a series of talks given by faculty in the graduate field of Mathematics. Speakers are selected by an organizing committee of graduate students. The goal of the seminar is to aid students in finding advisors.

Schedule for the "What Is...?" seminar

Special Committee

The Cornell Graduate School requires that every student selects a special committee (in particular, a thesis adviser, who is the chair or the committee) by the end of the third semester.

The emphasis in the Graduate School at Cornell is on individualized instruction and training for independent investigation. There are very few formal requirements and each student develops a program in conjunction with his or her special committee, which consists of three faculty members, some of which may be chosen from outside the field of mathematics. 

Entering students are not assigned special committees. Such students may contact any of the members on the Advising Committee if they have questions or need advice.

Current Advising Committee

Analysis / Probability / Dynamical Systems / Logic: Lionel Levine Geometry / Topology / Combinatorics: Kathryn Mann Probability / Statistics:  Philippe Sosoe Applied Mathematics Liaison: Richard Rand

Admission to Candidacy

To be admitted formally to candidacy for the Ph.D. degree, the student must pass the oral admission to candidacy examination or A exam. This must be completed before the beginning of the student's fourth year. Upon passing the A exam, the student will be awarded (at his/her request) an M.S. degree without thesis.

The admission to candidacy examination is given to determine if the student is “ready to begin work on a thesis.” The content and methods of examination are agreed on by the student and his/her special committee before the examination. The student must be prepared to answer questions on the proposed area of research, and to pass the exam, he/she must demonstrate expertise beyond just mastery of basic mathematics covered in the core graduate courses. 

To receive an advanced degree a student must fulfill the residence requirements of the Graduate School. One unit of residence is granted for successful completion of one semester of full-time study, as judged by the chair of the special committee. The Ph.D. program requires a minimum of six residence units. This is not a difficult requirement to satisfy since the program generally takes five to six years to complete. A student who has done graduate work at another institution may petition to transfer residence credit but may not receive more than two such credits.

The candidate must write a thesis that represents creative work and contains original results in that area. The research is carried on independently by the candidate under the supervision of the chairperson of the special committee. By the time of the oral admission to candidacy examination, the candidate should have selected as chairperson of the committee the faculty member who will supervise the research. When the thesis is completed, the student presents his/her results at the thesis defense or B Exam. All doctoral students take a Final Examination (the B Exam, which is the oral defense of the dissertation) upon completion of all requirements for the degree, no earlier than one month before completion of the minimum registration requirement.

Masters Degree in the Minor Field

Ph.D. students in the field of mathematics may earn a Special Master's of Science in Computer Science. Interested students must apply to the Graduate School using a form available for this purpose. To be eligible for this degree, the student must have a member representing the minor field on the special committee and pass the A-exam in the major field. The rules and the specific requirements for each master's program are explained on the referenced page.

Cornell will award at most one master's degree to any student. In particular, a student awarded a master's degree in a minor field will not be eligible for a master's degree in the major field.

Graduate Student Funding

Funding commitments made at the time of admission to the Ph.D. program are typically for a period of five years. Support in the sixth year is available by application, as needed. Support in the seventh year is only available by request from an advisor, and dependent on the availability of teaching lines. Following a policy from the Cornell Graduate School, students who require more than seven years to complete their degree shall not be funded as teaching assistants after the 14th semester.

Special Requests

Students who have special requests should first discuss them with their Ph.D. advisor (or with a field member with whom they work, if they don't have an advisor yet). If the advisor (or field faculty) supports the request, then it should be sent to the Director of Graduate Studies.  

PhD Program

During their first year in the program, students typically engage in coursework and seminars which prepare them for the  Qualifying Examinations .  Currently, these two exams test the student’s breadth of knowledge in algebra and real analysis. Starting in Autumn 2023, students will choose 2 out of 4 qualifying exam topics: (i) algebra, (ii) real analysis, (iii) geometry and topology, (iv) applied mathematics.

Current Course Requirements: To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297.

Within the 27 units, students must satisfactorily complete a course sequence. This can be fulfilled in one of the following ways:

Math 215A, B, & C: Algebraic Topology, Differential Topology, and Differential Geometry

• Math 216A, B, & C: Introduction to Algebraic Geometry
• Math 230A, B, & C: Theory of Probability
• 3 quarter course sequence in a single subject approved in advance by the Director of Graduate Studies.

Course Requirements for students starting in Autumn 2023 and later: 

To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297. (The course sequence requirement is discontinued for students starting in Autumn 2023 and later.)

By the end of Spring Quarter of their second year in the program, students must have a dissertation advisor and apply for Candidacy.

During their third year, students will take their Area Examination, which must be completed by the end of Winter Quarter. This exam assesses the student’s breadth of knowledge in their particular area of research. The Area Examination is also used as an opportunity for the student to present their committee with a summary of research conducted to date as well as a detailed plan for the remaining research.

Typically during the latter part of the fourth or early part of the fifth year of study, students are expected to finish their dissertation research. At this time, students defend their dissertation as they sit for their University Oral Examination. Following the dissertation defense, students take a short time to make final revisions to their actual papers and submit the dissertation to their reading committee for final approval.

All students continue through each year of the program serving some form of Assistantship: Course, Teaching or Research, unless they have funding from outside the department.

Our graduate students are very active as both leaders and participants in seminars and colloquia in their chosen areas of interest.

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PhD in Mathematics

The PhD in Mathematics consists of preliminary coursework and study, qualifying exams, a candidacy exam with an adviser, and creative research culminating in a written dissertation and defense. All doctoral students must also do some teaching on the way to the PhD. There are minimal course requirements, and detailed requirements and procedures for the PhD program are outlined in the  PhD Handbook .

Please note that our department alternates recruiting in-coming classes that are focused on either applied or pure mathematics. For the Fall 2024 admissions (matriculation in September 2024), we are focusing on students interested in areas of applied mathematics.

All our professors are active in research, and are devoted to teaching and mentoring of students. Thus, there are many opportunities to be involved in cutting-edge research in pure and applied mathematics. Moreover, the seven other research universities in the Boston area are all within easy reach, providing access to many more classes, seminars and colloquia in diverse areas of mathematical research.

Teaching assistantships are available for incoming PhD students, as well as a limited number of University-wide fellowships. Tufts has on-campus housing for graduate students, but many choose to live off-campus instead.

In addition to the above, PhD students often:

• Mentor undergraduates as teaching assistants and course instructors, and through graduate-student run programs like the Directed Reading Program.
• Meet with advisors and fellow students to share research and collaborate with scholars across disciplines
• Attend professional development workshops and present research at conferences

Welcome to the Math PhD program at Harvard University and the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Learn more about Harvard’s Math community and our statement on diversity and inclusion.

The Harvard Griffin GSAS Office of Equity, Diversity, Inclusion & Belonging offers student affinity groups for graduate students and many other resources.

The Harvard University Office for Gender Equity has dedicated GSAS Title IX resource coordinators who work with and support graduate students.

open. The application deadline is December 15, 2021. -->

The application deadline for fall 2024 admission has passed. Applications for fall 2025 admission will open in September 2024.

For information on admissions and financial support, please visit the Harvard Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Harvard Griffin GSAS is committed to ensuring that our application fee does not create a financial obstacle. Applicants can determine eligibility for a fee waiver by completing a series of questions in the Application Fee section of the application. Once these questions have been answered, the application system will provide an immediate response regarding fee waiver eligibility.

PhD in Mathematics

The PhD in Mathematics provides training in mathematics and its applications to a broad range of disciplines and prepares students for careers in academia or industry. It offers students the opportunity to work with faculty on research over a wide range of theoretical and applied topics.

Degree Requirements

The requirements for obtaining an PhD in Mathematics can be found on the associated page of the BU Bulletin .

• Courses : The courses mentioned on the BU Bulletin page can be chosen from the graduate courses we offer here . Half may be at the MA 500 level or above, but the rest must be at the MA 700 level or above. Students can also request to use courses from other departments to satisfy some of these requirements. Please contact your advisor for more information about which courses can be used in this way. All courses must be passed with a grade of B- or higher.
• Analysis (examples include MA 711, MA 713, and MA 717)
• PDEs and Dynamical Systems (examples include MA 771, MA 775, and MA 776)
• Algebra and Number Theory (examples include MA 741, MA 742, and MA 743)
• Topology (examples include MA 721, MA 722, and MA 727)
• Geometry (examples include MA 725, MA 731, and MA 745)
• Probability and Stochastic Processes (examples include MA 779, MA 780, and MA 783)
• Applied Mathematics (examples include MA 750, MA 751, and MA 770)
• Comprehensive Examination : This exam has both a written and an oral component. The written component consists of an expository paper of typically fifteen to twenty-five pages on which the student works over a period of a few months under the guidance of the advisor. The topic of the expository paper is chosen by the student in consultation with the advisor. On completion of the paper, the student takes an oral exam given by a three-person committee, one of whom is the student’s advisor. The oral exam consists of a presentation by the student on the expository paper followed by questioning by the committee members. A student who does not pass the MA Comprehensive Examination may make a second attempt, but all students are expected to pass the exam no later than the end of the summer following their second year.
• Oral Qualifying Examination: The topics for the PhD oral qualifying exam correspond to the two semester courses taken by the student from one of the 3 subject areas and one semester course each taken by the student from the other two subject areas. In addition, the exam begins with a presentation by the student on some specialized topic relevant to the proposed thesis research. A student who does not pass the qualifying exam may make a second attempt, but all PhD students are expected to pass the exam no later than the end of the summer following their third year.
• Dissertation and Final Oral Examination: This follows the GRS General Requirements for the Doctor of Philosophy Degree .

Admissions information can be found on the BU Arts and Sciences PhD Admissions website .

Financial Aid

Our department funds our PhD students through a combination of University fellowships, teaching fellowships, and faculty research grants. More information will be provided to admitted students.

More Information

Please reach out to us directly at [email protected] if you have further questions.

Doctor of Philosophy (PhD) in Mathematics Education

Graduate Programs

The Ph.D. program emphasizes research and requires a written dissertation for completion. The program is individualized to meet the needs of graduate students. The student must develop, with the guidance from the major professor and committee, a program that is applicable to their background and interest. The average Ph.D. program requires 4-6 years beyond a master’s degree. The program is comprised of coursework in four major areas.

• Mathematics Education
• Mathematics or a related area
• Cognate Area
• Research Core

This residential program has rolling admission Applications must be fully complete and submitted (including all required materials) and all application fees paid prior to the deadline in order for applications to be considered and reviewed. For a list of all required materials for this program application, please see the “Admissions” section below.

• July 1 is the deadline for Fall applications.
• November 15 is the deadline for Spring applications.
• March 15 is the deadline for Summer applications.

This program does not lead to licensure in the state of Indiana or elsewhere. Contact the College of Education Office of Teacher Education and Licensure (OTEL) at [email protected] before continuing with program application if you have questions regarding licensure or contact your state Department of Education about how this program may translate to licensure in your state of residence.

APPLICATION PROCEDURE

Application Instructions for the Mathematics Education PhD program from the Office of Graduate Studies:

In addition to a submitted application (and any applicable application fees paid), all completed materials must be submitted by the application deadline in order for an application to be considered complete and forwarded on to faculty and the Purdue Graduate School for review.

Here are the materials required for this application:

• Transcripts (from all universities attended, including an earned bachelor’s degree from a college or university of recognized standing)
• Minimum undergraduate GPA of 3.0 on a 4.0 scale
• 3 Recommendations
• Academic Statement of Purpose
• Personal History Statement
• Writing Sample
• International Applicants must meet English Proficiency Requirements set by the Purdue Graduate School

We encourage prospective students to submit an application early, even if not all required materials are uploaded. Applications are not forwarded on for faculty review until all required materials are uploaded.

When submitting your application for this program, please select the following options:

• Select a Campus: Purdue West Lafayette (PWL)
• Select your proposed graduate major: Curriculum and Instruction
• Please select an Area of Interest: Mathematics Education
• Please select a Degree Objective: Doctor of Philosophy (PhD)
• Primary Course Delivery: Residential

Program Requirements

I. mathematics education courses (15 – 18 hours).

In mathematics education, students engage in courses that cover topics in the cognitive and cultural theories of learning and teaching mathematics, and the role of curriculum in mathematics education. A three (3) course sequence is required that consists of:

• EDCI 63500 – Goals and Content in Mathematics Education
• EDCI 63600 – The Learning of Mathematics: Insights and Issues
• EDCI 63700 – The Teaching of Mathematics: Insights and Issues

In addition, students are encouraged to take (6 – 9) hours of EDCI 620: Developing as a Mathematics Education Researcher

II. Related Course Work (minimum 6 hours)

All students should have appropriate course work in mathematics, statistics, educational technology, or a related field. Students without a master’s level background in mathematics may be required to take more courses in mathematics. This will be determined by the student’s major professor and advisory committee.

III. Cognate (9 hours)

Students will take three graduate courses in a self-selected cognate area. Cognate area selection should be discussed with the student’s major professor and advisory committee. Possible cognate areas include: mathematics, psychology, philosophy, sociology, technology.

IV. Research Core Courses (15 hours)

All doctoral students in the Department of Curriculum and Instruction must complete five (5) courses from areas in research methodology and analysis before beginning their dissertation:

• EDPS 53300 – Introduction to Research in Education
• EDCI 61500 – Qualitative Research Methods in Education
• MA 51200 – Introductory Statistics
• Advance electives in either quantitative or qualitative methods
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• Counseling and Development
• Educational Leadership and Policy Studies
• Educational Psychology and Research Methodology
• Gifted Education
• Special Education

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Department of Mathematics

Requirements for the ph.d. degree.

In order to qualify for the Mathematics Ph.D., all students are required to:

• Complete eight term courses at the graduate level, at least two with Honors grades.
• Pass qualifying examinations on their general mathematical knowledge;
• Submit a dissertation prospectus;
• Participate in the instruction of undergraduates ;
• Be in residence for at least three years;
• Complete a dissertation that clearly advances understanding of the subject it considers.

All students must also complete any other  Graduate School of Arts and Sciences degree requirements  as they appear in the Programs and Policies bulletin.

The normal time for completion of the Ph.D. program is five to six years. Requirement (1) normally includes basic courses in algebra, analysis, and topology.  Students typically complete the eight-course requirement by the end of their third year.  The Honors grades of (1) must be achieved within the first two years.  A sequence of three qualifying examinations (algebra and number theory, real and complex analysis, topology) is offered each term.  All qualifying examinations must be passed by the end of the second year.  There is no limit to the number of times that students can take the exams, and so they are encouraged to take them as soon as possible.

The dissertation prospectus should be submitted during the third year. 

The thesis is expected to be independent work, done under the guidance of an adviser. This adviser should be contacted not long after the student passes the qualifying examinations. A student is admitted to candidacy after completing requirements (1)–(5) and obtaining an adviser.

In addition to all other requirements, students must successfully complete MATH 991a, Ethical Conduct of Research, prior to the end of their first year of study. This requirement must be met prior to registering for a second year of study.

Master’s Degrees :

The M.Phil. and M.S. degrees are conferred only en route to the Ph.D.; there is no separate master’s program in Mathematics.

M.Phil.   Please refer to the Graduate School Degree Requirements

M.S.   A student must complete six term courses with at least one Honors grade, perform adequately on the general qualifying examination, and be in residence at least one year.

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Ph.D. Program in Mathematics

Degree requirements.

A candidate for the Ph.D. degree in mathematics must fulfill a number of different departmental requirements .

NYU Shanghai Ph.D. Track

The Ph.D. program also offers students the opportunity to pursue their study and research with Mathematics faculty based at NYU Shanghai. With this opportunity, students generally complete their coursework in New York City before moving full-time to Shanghai for their dissertation research. For more information, please visit the  NYU Shanghai Ph.D. page .

Sample course schedules (Years 1 and 2) for students with a primary interest in:

Applied Math (Math Biology, Scientific Computing, Physical Applied Math, etc.)

Additional information for students interested in studying applied math is available here .

Probability

PDE/Analysis

The Written Comprehensive Examination

The examination tests the basic knowledge required for any serious mathematical study. It consists of the three following sections: Advanced Calculus, Complex Variables, and Linear Algebra. The examination is given on three consecutive days, twice a year, in early September and early January. Each section is allotted three hours and is written at the level of a good undergraduate course. Samples of previous examinations are available in the departmental office. Cooperative preparation is encouraged, as it is for all examinations. In the fall term, the Department offers a workshop, taught by an advanced Teaching Assistant, to help students prepare for the written examinations.

Entering students with a solid preparation are encouraged to consider taking the examination in their first year of full-time study. All students must take the examinations in order to be allowed to register for coursework beyond 36 points of credit; it is recommended that students attempt to take the examinations well before this deadline. Graduate Assistants are required to take the examinations during their first year of study.

For further details, consult the page on the written comprehensive exams .

The Oral Preliminary Examination

This examination is usually (but not invariably) taken after two years of full-time study. The purpose of the examination is to determine if the candidate has acquired sufficient mathematical knowledge and maturity to commence a dissertation. The phrase "mathematical knowledge" is intended to convey rather broad acquaintance with the basic facts of mathematical life, with emphasis on a good understanding of the simplest interesting examples. In particular, highly technical or abstract material is inappropriate, as is the rote reproduction of information. What the examiners look for is something a little different and less easy to quantify. It is conveyed in part by the word "maturity." This means some idea of how mathematics hangs together; the ability to think a little on one's feet; some appreciation of what is natural and important, and what is artificial. The point is that the ability to do successful research depends on more than formal learning, and it is part of the examiners' task to assess these less tangible aspects of the candidate's preparation.

The orals are comprised of a general section and a special section, each lasting one hour, and are conducted by two different panels of three faculty members. The examination takes place three times a year: fall, mid-winter and late spring. Cooperative preparation of often helpful and is encouraged. The general section consists of five topics, one of which may be chosen freely. The other four topics are determined by field of interest, but often turn out to be standard: complex variables, real variables, ordinary differential equations, and partial differential equations. Here, the level of knowledge that is expected is equivalent to that of a one or two term course of the kind Courant normally presents. A brochure containing the most common questions on the general oral examination, edited by Courant students, is available at the Department Office.

The special section is usually devoted to a single topic at a more advanced level and extent of knowledge. The precise content is negotiated with the candidate's faculty advisor. Normally, the chosen topic will have a direct bearing on the candidate's Ph.D. dissertation.

All students must take the oral examinations in order to be allowed to register for coursework beyond 60 points of credit. It is recommended that students attempt the examinations well before this deadline.

The Dissertation Defense

The oral defense is the final examination on the student's dissertation. The defense is conducted by a panel of five faculty members (including the student's advisor) and generally lasts one to two hours. The candidate presents his/her work to a mixed audience, some expert in the student's topic, some not. Often, this presentation is followed by a question-and-answer period and mutual discussion of related material and directions for future work.

Summer Internships and Employment

The Department encourages Ph.D. students at any stage of their studies, including the very early stage, to seek summer employment opportunities at various government and industry facilities. In the past few years, Courant students have taken summer internships at the National Institute of Health, Los Alamos National Laboratory, Woods Hole Oceanographic Institution, Lawrence Livermore National Laboratory and NASA, as well as Wall Street firms. Such opportunities can greatly expand students' understanding of the mathematical sciences, offer them possible areas of interest for thesis research, and enhance their career options. The Director of Graduate Studies and members of the faculty (and in particular the students' academic advisors) can assist students in finding appropriate summer employment.

Mentoring and Grievance Policy

For detailed information, consult the page on the Mentoring and Grievance Policy .

Visiting Doctoral Students

Information about spending a term at the Courant Institute's Department of Mathematics as a visiting doctoral student is available on the Visitor Programs  page.

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Ph.D. in Applied Mathematics

See the catalog copy of the description of the Ph.D. in Applied Mathematics program.

1. Overview

A student in the Ph.D. in Applied Mathematics degree program must maintain satisfactory academic progress towards completion of the degree. Student satisfactory academic progress is primarily assessed by: (a) satisfactory coursework performance, (b) the Qualifying Examination, (c) the Dissertation Topic Approval Defense, and (d) the Dissertation Defense. Courses and the Qualifying Examination are used to ensure that the student has the breadth as well as the depth of knowledge needed for research success. The Dissertation Topic Approval Defense is used to ensure that the scope of dissertation research is important, that the plan is well thought out, and that the student has sufficient skills and thoughtfulness needed for success. The Dissertation Defense is used to assess the outcomes of the dissertation research, and whether or not the plan agreed upon by the Dissertation Committee has been appropriately followed.

The key requirements and milestones for the Ph.D. in Applied Mathematics degree are provided below. Failure to satisfy the requirements can result in suspension or dismissal from the program.

• Minimum Hours
• Interdisciplinary Minor
• Core Courses
• Additional “Core” Courses
• Qualifying Examination
• Dissertation Committee
• Dissertation Topic Approval Defense
• Dissertation Defense

2. Minimum Hours

To earn a Ph.D. in Applied Mathematics degree, a student must complete at least 56 approved post baccalaureate credit hours. This includes 2 hours of Responsible Conduct of Research (GRAD 8302), at least 18 hours of dissertation research and reading (MATH 8994), and the hours for the interdisciplinary minor. Graduation requirements mandate that students must achieve a minimum grade point average of 3.0 to graduate. Receiving more than two grades of C or a single grade of U in any graduate course will result in a suspension from the program.

A limited amount of transfer credit is allowed. In accordance with rules of the UNC Charlotte Graduate School, students are allowed to transfer up to 30 semester hours of graduate credit earned at UNC Charlotte or other recognized graduate programs. Only courses with grades A or B may be accepted for transfer credit. To receive transfer credit, students must file an online request (and submit all necessary documents including copies of transcripts and course syllabi if requesting to transfer non-UNC Charlotte courses).

File an online request to transfer post-Baccalaureate credits at http://gpetition.uncc.edu .

3. Interdisciplinary Minor

The interdisciplinary minor may be satisfied by 9 hours of graduate work outside the mathematics department, by 6 credit hours for a directed project in an area of application (MATH 8691/8692), or by a combination of external coursework and a directed project in an area of application totaling 9 credit hours.

It is expected that interdisciplinary minor courses shall in general be in STEM disciplines, but if there are applications in the student’s dissertation work towards the social sciences, courses in those fields are allowed too. The following is a non-exhaustive list of interdisciplinary minor courses allowed for several fields.

Physics: PHYS 5222, 5232, 5242, 5271, 6101 through 6201, 6203 through 6211, 6221 through 6271. A common example is PHYS 6210, but 5242 and 5271 would also be along the same lines.

Optics: OPTI 8101, 8102, 8104, 8105, 8211 with 8102, 8104, and 8211 being particularly relevant.

Molecular Biophysics: PHYS 6108/OPTI 8000, PHYS 6204, PHYS 6610 ( https://mbp.charlotte.edu/ )

Mechanical Engineering: MEGR 6116, 7113, 7164 for students who have specialized in math of fluids, while 6141, 6125, 7102, 7142, and 7143 for those specializing in continuum mechanics and elasticity.

Computer Science: ITCS 6111, 6114, 6150, 6153, 6155, 6165, 6170, 6171, 6220, 6226 with 6114 commonly taken.

Finance and Economics: Any of FINN or ECON courses listed under the MS Mathematical Finance program. Common examples include FINN 6203, 6210, 6211, and ECON 6206, 6113, 6219.

Mathematics Education: Any graduate level MAED courses such as MAED 6122, 6123, 6124.

4. Core Courses

All students in the Ph.D. in Applied Mathematics degree program must take the following courses, regardless of their intended area of study:

• GRAD 8302 Responsible Conduct of Research (2 hours, usually required to take within the first year in the program)
• MATH 8143 Real Analysis I (3 hours)
• MATH 8144 Real Analysis II (3 hours)
• MATH 8994 Doctoral Research and Reading (at least 18 hours)

Students whose intended area of study is statistics or mathematical finance are also required to take

• MATH 8120 Theory of Probability I (3 hours)

5. Additional “Core” Courses

The following courses, though not explicitly required, are strongly recommended for each area of study.

Statistics: STAT 5123, 5124, 5126, 5127, 6115, 8127, 8133, 8135, 8137, 8139, 8122, 8123, 8027 (at least once)

Computational Math: MATH 5165, 5171, 5172, 5173, 5174, 5176, 8172, 8176

PDE and Mathematical Physics: MATH 5173, 5174, 8172

Probability: MATH 5128, 5129, 8120, 8125

Dynamical Systems: MATH 5173, 5174, 7275, 7276, 7277

Topology: MATH 5181, 8171, 8172 and independent study

Algebra: MATH 5163, 5164, 8163, 8164, and 8065 and/or independent study

Mathematical Finance: MATH 6202, 6203, 6204, 6205, 6206

6. Qualifying Examination

After being admitted to the Ph.D. program, a student is expected to take the qualifying examination within three semesters. This time limit may be extended up to two additional semesters in certain cases, depending on the background of the student and with program approval. The qualifying examination consists of two parts: the first part is a written examination based on Real Analysis I and II (MATH 8143/8144) or Theory of Probability I and Real Analysis I (MATH 8120/8143), the latter intended for a student with intended area of study in statistics or mathematical finance . The second part is a written examination based on two other courses chosen by the student to be specifically related to the student’s intended area of study and approved by the Graduate Coordinator. Typical choices for Part II are STAT 5126/5127, MATH 5173/5174, MATH 5172/5176, MATH 5163/5164, MATH 6205/6206, etc. The student may be allowed to retake a portion of the qualifying examination a second time if the student does not pass that portion on the first attempt within the guidelines of the Graduate School regulations pertaining to the qualifying examination and as overseen by the department Graduate Committee. A student who does not complete the qualifying examination as per the regulations of the Graduate School will be terminated from the Ph.D. program.

Complete and submit the following form after taking the Qualifying Examination. (Qualifying Exam Report Form) -> Graduate School form.

7. Dissertation Committee

After passing the Qualifying Examination, the student must set up a Dissertation Committee of at least four graduate faculty members, which must include at least three graduate faculty members from the Department of Mathematics and Statistics and one member appointed by the Graduate School. The committee is chaired by the student’s dissertation advisor. If the dissertation advisor is a graduate faculty member from an outside department or institution, a graduate faculty member from the Department of Mathematics and Statistics must be a co-chair of the committee. The Dissertation Committee must be approved by the Graduate Coordinator. After identifying and obtaining the signatures of the Dissertation Committee faculty, the Appointment of Doctoral Dissertation Committee Form must be sent to the Graduate School for the appointment of the Graduate Faculty Representative.

The Dissertation Committee should be appointed as soon as it is feasible, usually within a year after passing the Qualifying Examination.

Complete and submit the following form within a year of passing the Qualifying Examination. (Appointment of Doctoral Dissertation Committee Form) -> Graduate School form.

8. Dissertation Topic Approval Defense

Each student must present and orally defend a Ph.D. dissertation proposal after passing the Qualifying Examination and within ten semesters of entering the Program. The Dissertation Topic Approval Defense will be conducted by the student’s Dissertation Committee, and will be open to faculty and students. The dissertation proposal must address a significant, original and substantive piece of research. The proposal must include sufficient preliminary data and a timeline such that the Dissertation Committee can assess its feasibility.

The student should provide copies of the written dissertation proposal to the Dissertation Committee at least two weeks prior to the oral defense. At the discretion of the Dissertation Committee, the defense may include questions that cover the student’s program of study and background knowledge and techniques in the research area. The Dissertation Committee will unanimously grade the Dissertation Topic Approval Defense as pass/fail according to the corresponding rubrics. A student may retake the Dissertation Topic Approval Defense if he/she fails the first time. The second failed attempt will result in the termination of the student’s enrollment in the Ph.D. program. It is expected that the student first take the proposal defense by the ninth semester after enrollment to provide time for a second try should the first one fail. A doctoral student advances to Ph.D. candidacy after the dissertation proposal has been successfully defended. Candidacy must be achieved at least six months before the degree is conferred (so if you plan to graduate in a spring semester with the commencement on May 14, then you would need to successfully defend your dissertation topic by November 13 the prior year.)

The student must follow the following procedure in order to defend the dissertation proposal.

• Communicate with the Dissertation Committee to set up a date/time for the oral defense, and reserve a defense room for at least two hours .
• Send an electronic or written copy of the dissertation proposal to each member of the Dissertation Committee at least two weeks prior to the oral defense.
• Inform the Graduate Coordinator the schedule at least one week prior to the oral defense.

Complete and submit the following form only after successfully passing the Dissertation Topic Approval Defense. (Petition for Topic Approval Form) -> Graduate School Form.

9. Dissertation

Each student must complete and defend a dissertation based on a research program approved by the student’s dissertation advisor which results in a high quality, original and substantial piece of research. The student must orally present and successfully defend the dissertation before the student’s doctoral dissertation committee in a defense that is open to the public. The Dissertation will be unanimously graded as pass/fail based on the corresponding rubrics by the Dissertation Committee and must be approved by the Dean of the Graduate School. Two attempts of the Dissertation Defense are permitted. The second failed attempt will result in the termination of the student’s enrollment in the Ph.D. program.

The student must follow the following procedure in order to defend the dissertation.

• Communicate with the Dissertation Committee to set up a date/time for the public defense, and reserve a defense room for at least two hours with the help of the Graduate Coordinator.
• Send an electronic or written copy of the dissertation to each member of the Dissertation Committee at least three weeks prior to the public defense.
• Send an electronic copy of the dissertation in PDF as well as an abstract in a separate word file to the Graduate Coordinator at least two weeks prior to the public defense. The abstract is limited to 200 words, and does not have to be the same as the abstract included in the dissertation.
• Prepare a presentation that should be at least 45 minutes long.

Complete and submit the following forms after defending your Dissertation. (Dissertation Report for Doctoral Candidates Form) -> Graduate School Form.

Also, submit the Dissertation Title Page with Original Committee Signatures.

In addition, submit ETD Signature Form with original committee and student signatures to the Graduate School within 24 hours after defense.

10. Graduation

Detailed information about graduation including the dissertation manual can be found on the Graduate School’s Graduation website . The Graduation process consists of the following steps and all three are required to complete the graduation process.

• Complete the online graduation application – directions below – deadlines in the academic calendar.
• Submit an approved candidacy application to the Graduate School – directions below – deadlines in the academic calendar.
• Register for the term of graduation – deadlines in the academic calendar.

How to Apply for Graduation: Log into My UNCC. Select Banner Self Service, Student Services, Student Records and Online Graduation Application. Read the directions. Click “Continue” if this is the first time you have applied for graduation or “Create a New Application” if you have applied previously. After you complete all sections, remember to click the “Submit” button.

How to Access the Electronic Candidacy Application: Log into My UNCC. Select Banner Self Service, Student Services, Student Records and Apply for Candidacy for Graduate Students. Carefully read and follow all directions for each section. The total number of credit hours selected must be at least the minimum required for your degree or certificate. After you complete all sections, click the “Print” button. Do not change the page layout or formatting in any way. Take the printed document to your department for the Graduate Coordinator’s approval signature and then submit it to the Graduate School. Electronically submitted forms are not accepted.

Pay attention to the various deadlines in the official UNC Charlotte academic calendar , in particular, the following deadlines if you are planning to graduate.

• Deadline for graduate students to file candidacy form
• Deadline for graduate students to apply for graduation
• Doctoral dissertation pre-defense formatting consultation deadline
• Doctoral dissertation defense deadline
• Doctoral dissertation post-defense formatting consultation deadline
• Last day to submit doctoral dissertations to Graduate School

In addition, complete and submit the following forms to the graduate school.

• Complete Survey of Earned Doctorates at https://sed.norc.org/doctorate . Print the Certificate of Completion and submit to the Graduate School within 24 hours after defense.
• Contact Information Form
• Copyright and Open Access Publishing Payment Form (optional)

Finally, a sample graduation checklist for doctotal graduates can be found here . Make sure

• All courses with In Progress grades have been assigned grades; check with your advisor to ensure this happens. This must be completed 10 days prior to the commencement.
• All courses in the current term have been assigned grades; check with your advisor or instructor to ensure this happens. This must be completed 10 days prior to the commencement.

Mathematics Education PhD

Doctor of philosophy.

The Doctor of Philosophy (Ph.D.) degree emphasizes research competencies. The degree requires a scholarly dissertation of intellectual merit and sound research methodology. Dissertation research may include analytical studies of the process of teaching or experimental studies of the teaching-learning process, including studies of verbal learning and laboratory practice or historical studies.

Admissions Information

Displaying requirements for the Spring 2024, Summer 2024, and Fall 2024 terms.

• Points/Credits: 75
• Entry Terms: Spring/Summer/Fall

Application Deadlines

• Spring: November 15
• Summer/Fall (Priority): December 1
• Summer/Fall (Final): Rolling

* For details about rolling deadlines , visit our admission deadlines page.

Supplemental Application Requirements/Comments

• Online Degree Application , including Statement of Purpose and Resume
• Transcripts and/or Course-by-Course Evaluations for all Undergraduate/Graduate Coursework Completed
• Results from an accepted English Proficiency Exam (if applicable)
• $75 Application Fee
• Two (2) Letters of Recommendation

Requirements from the TC Catalog (AY 2023-2024)

Displaying catalog information for the Fall 2023, Spring 2024 and Summer 2024 terms.

View Full Catalog Listing

Doctoral Degrees

All candidates for the Ed.D., Ed.D.C.T., or Ph.D. degrees are expected to demonstrate both mathematics and mathematics education competencies through a series of certification examinations taken upon the completion of 60 graduate points. Certification examinations test the student’s knowledge of current research and theory in mathematics education and mathematics content. Examinations are offered once in the fall, spring, and summer terms. Courses recommended as preparation for the examinations in mathematics education include MSTM 6037,  MSTM 4019, and other mathematics education courses; Courses recommended as preparation for the examinations in mathematics are 6000- level mathematics content courses.

Students must demonstrate acceptable proficiency in at least three of the following six mathematics content areas: algebra, analysis, discrete mathematics, foundations of mathematics, geometry and topology, and probability and statistics. Students may sit for the examination in mathematics content during the regular certification examination times. Alternatively, they may register for advanced content courses and, with permission of the program, sit for the content area certification examination upon completion of the course. Incoming doctoral candidates should register for MSTM 6037 Professional Seminar in Mathematics during the first year of doctoral studies.

Doctoral students whose dissertations require statistical analysis should include appropriate statistics courses in their programs. These points can be included either in the mathematics/mathematics education requirement or can be taken as research electives.

Doctor of Philosophy in Mathematics Education

The Doctor of Philosophy (Ph.D.) degree emphasizes research competencies. The degree program requires a scholarly dissertation of intellectual merit and sound research methodology. Dissertation research may include analytical studies of the process of teaching or experimental studies of the teaching-learning process, including studies of verbal learning and laboratory practice or historical studies.

Candidates are encouraged to develop an association with a faculty member early in their studies to identify a problem area of mutual interest to plan a course of studies that leads to the competencies needed to complete dissertation research and prepare for a professional role. Further details are available in the brochures on doctoral studies and in the general descriptions of doctoral programs available from the Office of Doctoral Studies (ODS).

A program of study for the Doctor of Philosophy degree must include at least 45 points taken under Teachers College registration. In order to permit the acquisition of broad and basic scholarship, each program of study should include at least 60 points in mathematics, mathematics education, statistics, and computing. At least 35 points should be in advanced courses – including research courses (MSTM 6500 or 6501 and MSTM 7500). (Any Teachers College course at the 6000 level or above, any Columbia University Graduate School of Arts and Sciences course with a “G” prefix, any “W” course numbered above 4000, or any transferred course with a graduate-level prerequisite will be considered an advanced course.) Further, 15 points in the philosophical, psychological, and curricular foundations of education must be included in every Ph.D. degree program. Students whose dissertations require statistical analysis should include appropriate statistics courses in their programs. These points can be included either in the mathematics/mathematics education requirement or can be taken as research electives.

Candidates for the Ph.D. degree are required to demonstrate competency in two languages chosen from among French, German, and Russian. Students who require other languages for the preparation of their dissertation may petition the program to request one substitution. Students in mathematics may not use computer languages or statistics to satisfy the language requirement.

The Ph.D. dissertation is a scholarly study contributing new theoretical knowledge to the field and should be planned early in the program when sufficient advanced courses have been completed to permit the candidate to enroll in relevant research courses. Ph.D. dissertations in mathematics education should be (1) experimental studies in learning, (2) analytical studies in policy theory in mathematics education, or (3) other scholarly investigations of problems and issues of broad significance in the field.

The website of the Program offers a list of Topic study groups which doctoral students are recommended to join.

• View Other Degrees

Program Director : Professor Alexander Karp

Teachers College, Columbia University 323 Thompson

Phone: (212) 678-3381 Fax: (212) 678-8319

Email: tcmath@tc.edu

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Best Doctorates in Mathematics: Top PhD Programs, Career Paths, and Salaries

Given that only a small proportion of the general population holds a PhD in Mathematics, there aren’t many people pursuing doctoral studies in the math field. It requires a high level of intelligence and mathematical ability which a typical person doesn’t possess. It also requires a decent work ethic and the ability to commit five to seven years to research and study.

However, PhD in Mathematics salaries are high. If you’re interested in pursuing a PhD in Mathematics, check out our article to find the best mathematics PhDs and what mathematics jobs you can get. First, let’s find out what a PhD in Mathematics is.

Find your bootcamp match

What is a phd in mathematics.

A PhD in Mathematics is a doctoral degree obtained by a graduate student in mathematics. This kind of mathematics graduate program allows students to develop their research capabilities in mathematics and its potential applications.

PhD in Mathematics can be a Doctor of Philosophy, Doctor of Science, or Doctor of Applied Mathematics degree. Students need to submit their PhD thesis in order to complete the math degree program they are pursuing. Math doctorate programs can take between three and six years to complete, depending on how much time is dedicated to the thesis and its required research.

How to Get Into a Mathematics PhD Program: Admission Requirements

The requirements to get into a mathematics PhD program typically include application fees, school transcripts, a professional resume, three letters of recommendation, and a statement of academic purpose. The admission process for most mathematical PhD programs is similar.

Some schools also require that students have mastered the material roughly equivalent to the undergraduate mathematics major, such as several semesters of calculus and experience with differential equations. Other doctoral math programs require applicants to submit essays or a minimum GPA score as entrance requirements.

PhD in Mathematics Admission Requirements

• Professional CV or resume
• Statement of academic purpose
• Application fee
• At least three recommendation letters
• Undergraduate and graduate school transcripts

Mathematics PhD Acceptance Rates: How Hard Is It to Get Into a PhD Program in Mathematics?

It is extremely hard to get into a PhD program in Mathematics. For example, the Harvard PhD acceptance rate is 4.59 percent and the University of Wisconsin has a zero percent PhD acceptance rate over a five-year median. Doctoral studies in mathematics are rigorous and have a low acceptance rate, making it difficult to get into and obtain a Doctorate in Mathematics.

How to Get Into the Best Universities

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Best PhDs in Mathematics: In Brief

Best universities for mathematics phds: where to get a phd in mathematics.

The best universities for mathematics PhDs are Harvard University, Massachusetts Institute of Technology, Princeton University, Stanford University, University of California, Berkeley, and Yale University.

Other universities also have excellent PhD in Mathematics programs that we haven’t mentioned in our list, like Johns Hopkins University, Colorado State University, University of California, Los Angeles, and University of Chicago.

Columbia University is a private Ivy League research university founded in 1754 as King's College. It is among the top five national universities and is known as the oldest institution of higher education in New York. At Columbia University, students can find doctoral studies in biological sciences, astronomy, chemistry, economics, music, psychology, and religion.

Mathematics PhD

The PhD mathematics program is designed for full-time students interested in research and teaching at the university level. It is an intense course also meant for students that seek careers in quantitative research and development in business or government. It covers subjects like analysis and probability, Riemann surfaces, cumulative algebra, and modern geometry.

Mathematics PhD Overview

• Program Length: 4 years
• Acceptance Rate: 6%
• Tuition and Fees: $25,248/semester
• PhD Funding Opportunities: Department of Defense Funding, NIH Fellowship Parent Funding 

Mathematics PhD Admission Requirements

• A completed online application form at the Online Application System
• $120 application fee
• Statement of purpose
• A video essay (to be completed in the Online Application System)
• A current resume or CV
• Official transcripts from each university you attended
• Two letters of reference

Harvard University was established in 1636 by the Puritan clergyman John Harvard. It is known as the oldest institution of higher learning in the US and as one of the most prestigious universities in the world. It offers many PhD programs in subjects such as anthropology, statistics, virology, physics, neuroscience, mathematics, immunology, and history of science.

This doctoral program is intended for students who aspire to be research mathematicians and have demonstrated a strong interest in this field. The math graduate courses cover topics like commutative algebra, advanced real analysis, differential geometry evolution dynamics, Riemann surfaces, and Hodge theory.

• Program Length: 4-5 years
• Acceptance Rate: 7%
• Tuition and Fees: $50,928/first two years of study; $13,240/second two years of study
• PhD Funding Opportunities: Hertz Foundation, National Defense Science and Engineering Graduate Fellowship Program, National Physical Science Consortium for Minorities and Women, National Science Foundation
• Application
• $105 application fee 
• Transcripts
• Letters of recommendation
• Demonstration of English proficiency (for non-native speakers)

Massachusetts Institute of Technology (MIT) is a land-grant research university in Cambridge. Founded in 1861, MIT is best known for its programs in physical science and engineering. It offers a wide array of doctoral studies in aeronautics and astronautics, physics, political science, economics, chemical engineering, biology, brain, and cognitive sciences.

Doctor of Philosophy or Doctor of Science Mathematics Degree

In addition to choosing between a Doctor of Philosophy and a Doctor of Science program, students can choose to apply to the Pure or Applied Mathematics programs. Geometry, number theory, algebra, logic, statistics, topology, astrophysics, combinatorics, analysis, probability, numerical analysis, and theoretical physics, are covered in the Pure and Applied Mathematics curricula.

Doctor of Philosophy or Doctor of Science Mathematics Degree Overview

• Acceptance Rate: N/A
• Tuition and Fees: $27,755/term
• PhD Funding Opportunities: Postdoctoral and Senior Research Awards, Faculty Early Career Development Program, Mathematical Sciences Postdoctoral Research Fellowships (MSPRF)

Doctor of Philosophy or Doctor of Science Mathematics Degree Admission Requirements

• Fill out the online application
• Transcripts including grades in math/science/engineering courses
• At least three names and email addresses of recommendation letter writers
• Educational and work history
• Statement of objectives
• Outside financial support and potential outside support
• Credit/debit card payment of the $75 application fee
• Self-reported grades
• Arrange for submission of official transcripts and letters of recommendation

Princeton University is a private Ivy League research university that was founded in 1746. It is known as the fourth-oldest institution of higher education in the US. Princeton University has many PhD programs in subjects like philosophy, mechanical and aerospace engineering, electrical and computer engineering, and neuroscience.

Applied and Computational Math PhD

In this program, students gain a thorough understanding of areas of mathematics critical to scientific and engineering applications, such as numerical analysis and other computational approaches. In the first year, students choose three topics from six applied mathematics categories. At the end of the first year, students need to take a preliminary exam. 

Before the third year, students need to pass the general examination which culminates in a seminar on a research topic. The final requirement for this program is the doctoral dissertation which can be a mathematical contribution to one field of science or engineering.

Applied and Computational Math PhD Overview

• Tuition and Fees : $58,790/year 
• PhD Funding Opportunities: University, departmental, and program fellowships, research and teaching assistantships, external financial support

Applied and Computational Math PhD Admission Requirements

• $75 application fee
• Recommendation letters
• Fall semester grades (for students currently enrolled in master’s degree programs)
• English language tests (for international students)

Established in 1829, Rochester Institute of Technology is known for its excellent technology, engineering, and computing programs. It offers PhDs in Astrophysical Sciences and Technology, Biomedical and Chemical Engineering, Color Science, Imaging Science Sustainability, and many more disciplines.

Mathematical Modeling PhD

Students in this program plan their own trajectory based on required coursework, research, and their selected concentration. They can select from concentrations in the areas of applied inverse problems, biomedical, and discrete mathematics, among others. Some of the program’s courses cover subjects like numerical analysis and computing for mathematical modeling. 

Mathematical Modeling PhD Overview

• Acceptance Rate: 74%
• Tuition: $54,176/year 
• PhD Funding Opportunities: Federal loans, Outside Scholarships, Yellow Ribbon Program, employment through co-ops, internships, or assistantships

Mathematical Modeling PhD Admission Requirements

• Online graduate application
• Copies of official transcripts of all previously completed undergraduate and graduate courses
• Bachelor’s degree from an accredited university or college
• Current resume/CV
• Two letters of recommendation
• Personal statement of educational objectives
• TOEFL, IELTS, or PTE score (international students)
• $65 application fee

Stanford University is well-known for its enterprising student population and high return on investment. Founded in 1885 by California senator Leland Stanford, Stanford University is consistently ranking in the top 10 national universities. Stanford offers many PhD programs in its Graduate School of Engineering, including programs in engineering, medicine, and education.

Mathematics PhD Program

Students in this PhD program take courses in algebraic topology, differential topology, differential geometry, algebraic geometry, theory of probability, real analysis, modern algebra, and mathematical methods of imaging. In the first year, the coursework prepares the students for the qualifying examinations which test them on their skills in algebra and real-world analysis. 

Students must have a dissertation advisor and seek candidacy in their second year. Students will take their Area Examination during their third year, which must be completed by the conclusion of the Winter Quarter. In the fourth and fifth years, students are expected to finish their dissertation research.

Mathematics PhD Program Overview

• Acceptance Rate: 5.7%
• Tuition and Fees: $56,487
• PhD Funding Opportunities: Federal programs, university fellowships, research assistantships, teaching assistantships

Mathematics PhD Program Admission Requirements

• Bachelor’s degree from a regionally accredited US college or university 
• Unofficial transcripts/academic records
• Names and contact information of those who can provide between three and six letters of recommendation
• Online application
• $125 application fee or fee waiver

University of California, Berkeley was founded in 1868. UC Berkeley is known for strict academic criteria for its undergraduate programs. The school has more than 130 academic departments and 80 multidisciplinary research areas. UC Berkeley is among the top universities in the world.

PhD Program in Applied Mathematics

The applied mathematics program requires doctoral students with previous graduate study in mathematics must demonstrate exceptional mathematical research skills. Students in this course can learn about linear algebra and differential equations, multivariable calculus, and discrete mathematics. There are also major electives like mathematical biology and statistics.

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PhD Program in Applied Mathematics Overview

• Program Length: 5-6 years
• Acceptance Rate: 18%
• Tuition and Fees: $10,248.75/semester (resident); $17,799.75/semester (non-resident)
• PhD Funding Opportunities : Berkeley Fellowships, Chancellor’s Fellowships, Cota-Robles Fellowships, Mentored Research Awards, UC Dissertation-Year Fellowships, Conference Travel Grants, Parent and Caregiver Grants

PhD Program in Applied Mathematics Admission Requirements

• A bachelor’s degree or equivalent from an accredited institution
• A minimum GPA of 3.0
• Enough experience in your undergraduate program to complete the program
• At least three letters of recommendation
• $120 application fee (California residents); $140 application fee (all other applicants)

The University of Michigan - Ann Arbor is among the top 30 national universities. It was founded in 1817 in Ann Arbor. Some of the most popular majors from this university are business administration, experimental psychology, and economics. It also has good PhD level programs in education, electrical and computer engineering, and computer and information science.

Applied & Interdisciplinary Mathematics PhD

This PhD degree's purpose is to create graduates who are well-prepared to pursue successful careers in the mathematical sciences, whether in government or academia. Students in the applied and interdisciplinary mathematics course can learn about risk management and modeling of financial losses, mathematical theory of probability, and modern mathematics.

Applied & Interdisciplinary Mathematics PhD Overview

• Program Length: 5-6 years 
• Acceptance Rate: 26%
• Tuition and Fees: $25,230.38 (in state); $50,646.38 (out of state)
• PhD Funding Opportunities: Graduate Student Instructorship (GSI): Mathematics Graduate Student Instructor Teacher Training Program, and Graduate Student Research Assistantship (GSRA)

Applied & Interdisciplinary Mathematics PhD Admission Requirements

• Three letters of recommendation
• Personal Statement (500-word limit)
• TOEFL or IELTS Exam (non-native English speakers)
• Three semesters of calculus
• One or two semesters of differential equation courses
• One semester course in modern algebra, linear algebra, geometry or topology
• Advanced calculus of one and several variables 
• $75 application fee (US citizens); $90 application fee (international applicants)

Nelson Dewey, the first governor of Wisconsin, founded the University of Wisconsin in 1848. It has one of the best engineering and business majors. It also has several good graduate degrees from its Department of Mathematics like Doctor of Philosophy, Master of Arts – Foundations of Advanced Studies, and Master of Arts – Foundations for Research.

Mathematics, PhD

The University of Wisconsin’s Doctor of Philosophy program offers a foundation in fundamental and advanced graduate mathematics. Students in this PhD program can learn about linear algebra, abstract algebra, differential equations, knot theory, and real analysis.

Mathematics, PhD Overview

• Program Length: 5 years
• Acceptance Rate: 16%
• Tuition and Fees: $12,176/year (in state); $25,504/year (out of state)
• PhD Funding Opportunities: Graduate assistantships, fellowships, traineeships, student loans, student jobs, research and travel grants

Mathematics, PhD Admission Requirements

• Application to the UW–Madison Graduate School
• Master’s degree in mathematics or a related field
• Official transcripts
• Supplementary information form
• Three academic letters of recommendation and one additional letter
• Proof of English proficiency (for non-native speakers)
• One-page statement of purpose

Yale University was founded in 1701 in New Haven, Connecticut. It is among the top 10 national universities and is known for its Ivy League reputation . Yale has many PhD programs like anthropology, applied physics, architecture, biomedical engineering, chemical, and environmental engineering. 

PhD in Applied Mathematics

The applied mathematics program entails the study and application of mathematics to issues prompted by a diverse set of application fields. Students study and apply mathematics concepts to questions driven by a variety of applicable fields.

It covers topics such as discrete algorithms, linear and nonlinear partial differential equations, statistical theory and applications, information theory, econometrics, and classical and modern applied harmonic analysis. This program also covers topics like financial statistics, financial mathematics, econometrics probability theory, and information theory. 

PhD in Applied Mathematics Overview

• Acceptance Rate: 6.2%
• Tuition and Fees: $45,700/year
• PhD Funding Opportunities: Financial aid, university fellowships, dissertation fellowships, teaching fellowships

PhD in Applied Mathematics Admission Requirements

• A list of all the prior colleges or universities you’ve attended 
• Unofficial transcripts from each school
• TOEFL scores (non-native English-speaking applicants)

Can You Get a PhD in Mathematics Online?

Yes, you can get a PhD in Mathematics online. However, there aren’t many online PhD math programs out there because of two reasons. The first reason is that there isn’t a big demand for PhDs in Mathematics, so online universities are not inclined to find ways to make those programs more accessible.

The second reason is that in traditional doctoral studies graduate students in mathematics are used as research and teaching assistants. Usually, math doctoral studies take place within a community in which PhD students are apprentices learning alongside instructors and one another.

Best Online PhD Programs in Mathematics

How long does it take to get a phd in mathematics.

It takes about five years to get a PhD in Mathematics. Depending on the academic institution and the specifications of the program, it can take between three and six years. Some students prefer to have a master’s degree before entering a doctoral program in mathematics, but most programs do not require students to have more than a bachelor’s degree .

Some doctoral mathematics academic programs offer two degrees called dual degree programs. You can begin pursuing a Master’s Degree in Mathematics and begin earning credits for a PhD at the same time.

Is a PhD in Mathematics Hard?

Yes, a PhD in Mathematics is hard. Doctoral studies that require advanced knowledge of mathematics would be extremely difficult for most people. Whether you choose to pursue a PhD in Applied Mathematics or Pure Mathematics, you will encounter complex theoretical concepts.

How Much Does It Cost to Get a PhD in Mathematics?

On average, it costs $12,171 per year to get a PhD in Mathematics . This data is based on the National Center for Educational Statistics’s (NCES) report for the average costs of postgraduate studies at public institutions from 2018 to 2019. According to this data, it would cost a total of $48,684 for four years of studies. The cost to get a PhD in Mathematics has likely risen since 2019.

How to Pay for a PhD in Mathematics: PhD Funding Options

The PhD funding options that students can use to pay for a PhD in Mathematics include research fellowships, teaching assistantships, and tuition waivers. In the previous section, we mentioned the cost of doctoral degrees , but this number is with a deduction in the tuition.

These funding options enable students to lower the cost of tuition. For example, domestic students receive lower tuitions. Without funding options or deductions, a PhD can cost from $20,000 to $30,000 per year.

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What Is the Difference Between a Mathematics Master’s Degree and PhD?

The difference between a Master’s Degree in Mathematics and a PhD is the career opportunities they provide graduates. People who pursue a PhD in Mathematics usually seek jobs in research and academia, while people who pursue master’s degrees are more interested in advancing their professional careers.

A PhD in Math is more research-intensive and focused on historical and current perspectives within the mathematics field. Master’s degrees are a bit broader. Master’s degrees can take up to three years to complete, while PhD can take up to six years. Math PhD students also teach and engage in departmental activities, while master’s students do not.

Master’s vs PhD in Mathematics Job Outlook

It is usually required for postsecondary teachers to have PhDs. According to the Bureau of Labor Statistics (BLS), these professionals have a job outlook of 12 percent , four percent higher than the average for all occupations.

Mathematics and statisticians are usually required to hold a master’s degree and they have a job outlook of 33 percent , nearly three times the average for all occupations. According to these numbers, the job outlook for Master’s Degrees in Mathematics is higher than for PhDs in Mathematics. However, these are just two examples of careers available to master’s and PhD holders.

Difference in Salary for Mathematics Master’s vs PhD

According to PayScale, mathematicians with a master’s degree earn $85,000 per year, and mathematicians with a PhD degree earn $110,000 a year. This is a difference of $25,000 per year for mathematics master’s vs PhD holders.

Related Mathematics Degrees

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Why You Should Get a PhD in Mathematics

You should get a PhD in Mathematics because it expands your academic career opportunities and salary options. With a PhD in Math, you can work for private businesses or government agencies. Many PhD grads work in engineering, medicine, and software development, where they apply the advanced mathematical knowledge they acquired from their doctoral programs.

Reasons for Getting a PhD in Mathematics

• Specialized skills. A PhD in Mathematics equips you with highly valuable and marketable skills. Aside from subject-specific abilities, you can obtain several transferable skills that will be beneficial in practically any industry, such as exceptional numeracy, logical ability, holistic deduction, and reasoning skills.
• Higher salary. A PhD is a terminal graduate degree, meaning it is the highest level of academic certification you can achieve in this field. So with a PhD you will gain the highest skills and knowledge in mathematics. High-level skills and knowledge often translate to a high salary.
• Career opportunities. One of the best advantages math PhDs have is the option of following a variety of employment pathways. To some extent, all sciences are built on basic mathematical principles, so there are many career opportunities with a PhD in Mathematics. With this kind of degree, you can work in finance, academia, or IT.
• Research opportunities. Doctoral Degrees in Mathematics cover theoretical mathematics which is one of the best areas for research and research methodology. During your doctoral studies in math, you will study and research with some of the smartest people in the world.

Getting a PhD in Mathematics: Mathematics PhD Coursework

Getting a PhD in Mathematics is not an easy task. Students need to learn different math courses, some of which require advanced knowledge of mathematics. There is some standard mathematics PhD coursework, like classes in mathematical analysis or mathematical logic. Find out more details on common mathematics PhD courses below.

Mathematical Analysis

These kinds of courses explain the principles of mathematical analysis, including differentiability, continuity, sequence and series convergence, the Riemann integral, function sequences, and series. Students can also learn real applications of real-world analysis, including the estimation of indeterminate integrals, Laurent series, WKB theory, Cauchy theorem, and residual theorem.

Differential Topology

Courses in differential topology are a part of the theory of differential forms and De Rham’s theory. Somer differential topology courses teach about multilinear algebra, Stokes’ theorem, and an introduction to cohomology. But, in most cases, these courses teach about winding numbers, vector fields, index, smooth maps, transversality, and differential manifolds.

Discrete Mathematics

Courses in discrete mathematics usually cover finite geometry, finite groups, finite topology, finite fields, fundamental algebraic topology, combinatorics, and graph theory. Students can learn about the symmetry group of the regular icosahedron because it is a reoccurring subject throughout the course.

Theory of Probability

Theory of probability courses introduce students to discrete and continuous random variables, independent random variables, conditional probability, distribution functions, Poisson distributions, Bayes theorem, and joint distribution. Students in this course will learn how to use math principles in the analysis of random phenomena.

Algebraic Geometry

In most cases, algebraic geometry courses teach students about complex algebraic varieties, cohomology, affine, and projective algebraic geometry. They also cover the fundamental concepts of complex analytic varieties, Bezout’s theorem, algebraic curves, and surfaces, the language of schemes, and properties of morphisms.

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How to Get a PhD in Mathematics: Doctoral Program Requirements

Some of the most common doctoral program requirements to get a PhD in Mathematics are coursework, qualifying exams, research, classroom teaching, and thesis defense. Some doctoral program requirements may vary, but there are a few common steps for most math programs. Keep reading to find out more.

PhD students must finish eight or nine-term courses at the graduate level, in most cases maintaining at least a C average grade. The time for completion of a PhD math coursework is four to five years.

The residency requirement can vary from university to university. For most math PhD candidates, the usual residency requirement is around three years. The whole point of residency is to continue the education and training of students in a specialized field of mathematics.

A committee of three faculty members administers the Qualifying Examination, which is an oral test. Each student selects three qualifying test subjects and discusses them with appropriate examiners. Subjects must be in separate, reasonably broad mathematical fields. 

Typically, the primary topic is decided in cooperation with the potential thesis advisor. After completing the qualifying test, students must choose a thesis advisor, who is usually the primary topic examiner.

In academic positions, teaching is important, especially in mathematical doctoral studies. Many people in math PhD programs aspire to academic positions, so teaching is one of the best ways to learn about the mathematics academic profession. It is an excellent way to learn public presentation skills by completing at least one semester of classroom teaching.

Writing and defending a thesis for a PhD in Mathematics is the culmination of this graduate program. Usually, a thesis is anticipated to contain original research of a high enough quality to be published in a high-level research journal. The defense is usually held in the classroom of the Institute and is presented to the Thesis Examination Committee.

Potential Careers With a Mathematics Degree

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PhD in Mathematics Salary and Job Outlook

People with a PhD in Mathematics earn above the average annual salaries and have careers with above the average job growth. For example, actuaries have a 24 percent job growth till 2030. According to the Bureau of Labor Statistics, the average job growth rate for all occupations is 7.7 percent .

Earning a mathematics PhD is difficult but you will graduate with a wide range of job opportunities. People with a PhD in Math earn between $82,360 to $108,660 per year. Overall, people with PhDs in Mathematics have a good salary and job outlook.

What Can You Do With a PhD in Mathematics?

With a PhD in Mathematics, you can work in education or IT and for private, corporate, or government corporations. You can work as a postdoctoral research associate, a math professor in postsecondary or higher education, a director of analytics, a research scientist, a principal software engineer, or an asset manager.

Best Jobs with a PhD in Mathematics

• Data Scientist
• Mathematicians and Statistician
• Operations Research Analyst

What Is the Average Salary for a PhD in Mathematics?

The average salary for a PhD in Mathematics is $110,000 , according to PayScale. The salary range for doctorate-level degree mathematics jobs ranges from $48,690 to $167,040 per year. Lower-earning operations research analysts make an average of $48,690 , while top-earning data scientists earn $167,040 annually, according to the Bureau of Labor Statistics.

Highest-Paying Mathematics Jobs for PhD Grads

Best mathematics jobs with a doctorate.

There are many jobs you can find with a math PhD. The skills you will gain are applicable to almost any field. Find out more about the best PhD in Mathematics jobs below.

Data scientists are specialists who know how to extract and interpret data. They know how to use models and algorithms to analyze massive data repositories. They use various strategies to determine the optimal methods for developing a data model for a business. Data scientist is one of the most in-demand and highest-paid jobs in 2022. 

• Salary with a Mathematics PhD: $108,660
• Job Outlook: 22% job growth from 2020 to 2030
• Number of Jobs: 105,980
• Highest-Paying States: Washington, California, Delaware, New York, New Jersey

Actuaries are more concerned with the financial implications of risk and uncertainty. Actuaries are financial analysts who analyze past and present financial data and forecast future risks. They assess risk using financial theory, statistics, and math and develop plans that reduce the cost of risk.

• Salary with a Mathematics PhD: $105,900
• Job Outlook: 24% job growth from 2020 to 2030
• Number of Jobs: 27,700
• Highest-Paying States: Georgia, New Hampshire, New York, Connecticut, Vermont

Some of the main responsibilities economists have are to predict the customer demand for the firm's products and the firm's product sales. They deal extensively with microeconomic issues while researching trends, collecting, evaluating, and analyzing data. Economists also investigate how to efficiently allocate resources like raw materials, labor, land, or machines.

• Salary with a Mathematics PhD: $105,630
• Job Outlook: 13% job growth from 2020 to 2030
• Number of Jobs: 18,600
• Highest-Paying States: New York, District of Columbia California, New Hampshire, Illinois

In most cases mathematicians and statisticians work in academia and government. Usually, both of the positions require the use of analyzing data or applying statistical and mathematical techniques that help solve problems. 

Statisticians are increasingly involved in statistics and research methodology. Mathematicians are more involved in researching mathematical models or abstractly measuring values and creating results. Mathematics and statistics are the best duos that understand the principles of probability.

• Salary with a Mathematics PhD: $96,280
• Job Outlook: 33% job growth from 2020 to 2030
• Number of Jobs: 44,800
• Highest-Paying States: District of Columbia, New York, New Jersey, Virginia, Maryland

Operations research analysts utilize mathematics and logic to assist businesses in making informed decisions and solving complex problems. They are high-level problem solvers that use sophisticated approaches such as mathematical modeling and statistical analysis to help companies operate more economically.

• Salary with a Mathematics PhD: $82,360
• Job Outlook: 25% job growth from 2020 to 2030
• Number of Jobs: 104,100
• Highest-Paying States: Virginia, Alabama, Maryland, Hawaii, New York

Is a PhD in Mathematics Worth It?

Yes, a PhD in Mathematics is worth it if you are interested in solving math problems and analytical thinking. You also have to be willing to commit to the long educational journey and research required to achieve this level of degree.

Exploring the mathematical fields is challenging and demanding but with time it provides excellent career opportunities. Math positions are respected and well-paid with positive job outlooks. Other than its academic difficulty, there aren’t many downsides when it comes to doctoral math degrees because you can go on to work in many fields.

Additional Reading About Mathematics

[query_class_embed] https://careerkarma.com/blog/best-schools-for-mathematics/ https://careerkarma.com/blog/mathematics-for-computer-science/ https://careerkarma.com/blog/jobs-that-use-math/

PhD in Mathematics FAQ

Yes, a math PhD is useful because it will help you acquire some transferable skills that will be useful in virtually any industry. It is also useful for those interested in mathematics research or academia. If you wonder what you can do with a math degree , a PhD will equip you with many valuable skills.

Yes, a PhD in Mathematics is difficult to learn. It is difficult to be admitted to a doctoral program in mathematics, and it is also difficult and too abstract for most people to grasp advanced mathematical principles taught in PhD studies. Doctoral math studies are objectively difficult, even for those who are naturally strong at math and have excelled at it their whole life.

Mathematicians need several skills like active learning, complex problem-solving, critical thinking, and analytical skills. A good mathematician also needs information ordering and inductive and deductive reasoning abilities.

Mathematicians mainly explore mathematical principles or models and create their own mathematical theories and concepts. Most of the work they do is in the theoretical realm, that is why they mainly want to work in IT, social science, engineering, or financial fields.

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Dr. Tapomoy Guha Sarkar visited the Department of ...

10 June, 2013 | Pilani

Dr. Navin Singh has delivered an invited lecture o...

30 March, 2013 | Pilani

Dr. Anshuman Dalvi has delivered an invited lectu...

Research paper titled “conformational and or....

25 March, 2013 | Pilani

26 February, 2013 | Pilani

“Program on CP Violation in elementary parti...

19 February, 2013 | Pilani

Dr. Jayendra N Bandyopadhyay has delivered Colloqu...

Dr. jayendra n bandyopadhyay has delivered prestig....

18 February, 2013 | Pilani

Dr. Jayendra N Bandyopadhyay has delivered an ivi...

25 January, 2013 | Pilani

Dr. Madhukar Mishra has recently given a talk on &...

18 January, 2013 | Pilani

Dr. Jayendra N Bandyopadhyay has delivered an invi...

6 January, 2013 | Pilani

Dr Subhashis Gangopadhyay presented a paper title...

3 January, 2013 | Pilani

Dr. Tapomoy Guha Sarkar participated in the ASTRON...

19 December, 2012 | Pilani

Workshop on Introduction to Graphs and Geometric A...

22 January, 2009 | Pilani

K K Birla Goa, Hyderabad, Pilani

The main objective of the doctoral program is to create scholarly responsibility among researchers and provide them with an intellectually stimulating environment for focused research. The study phase of the research includes a solid foundation in advanced mathematics with a particular sequence of courses designed to enhance the professional ability of the candidates.

Research Areas

The department offers a full time Ph.D. programme in the following thrust areas of its research.

• Linear Algebra
• Numerical Analysis
• Graph Theory
• Control Theory
• Operator Theory
• Complex Analysis
• Topology & Geometry
• Functional Analysis
• Fracture Mechanics
• Harmonic & Wavelet Analysis
• Number Theory & Cryptography
• Financial Mathematics
• Theoretical and Computational Differential Equations
• Mathematical Modelling
• Discrete Mathematics
• M. Sc. In Mathematics or equivalent degree with 60% aggregate. Shortlisted candidates will have to appear for a written test and/or interview. Candidate having NET/GATE/M. Phil/JRF will be exempted from the written test
• M.Sc. or equivalent: ₹ 28,000 per month. After qualifying, the coursework through examinations, the scholarship will increase to ₹ 31,000 per month.
• M.Phil./M. Tech./JRF: ₹ 31,000 per month.
• The higher fellowship may be made available in subsequent years subject to the institute norms and performance of the candidates.

To learn more about the Ph.D admissions process at BITS Pilani, please visit the website https://www.bitsadmission.com

Associate Professor

Numerical Methods

Amiya Kumar Pani

Visiting Professor

Combinatorial Design & Testing

Numerical Analysis, Optimal Control Theory

Anirudh Singh Rana

Assistant Professor, Department of Mathematics, BITS Pilani, Pilani Campus

Fluid dynamics, Finite Volume methods, Non-equilibrium Thermodynamics, Micro/Nanofludics, Gas dynamics, Continuum Mechanics, Transport Phenomenon

Anupama Sharma

Assistant Professor, Gr-I

Dynamical Systems, Mathematical Biology, Mathematical modelling, Complex systems

Anushaya Mohapatra

Dynamical Systems, Mathematical Biology

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Master of Philosophy in Mathematics Doctor of Philosophy in Mathematics

MPhil(MATH) PhD(MATH)

Both full- and part-time

Full-time: 2 years Part-time: 4 years  

Full-time: 3 years (with a relevant research master’s degree), 4 years (without a relevant research master’s degree) Part-time: 6 years

Department of Mathematics

https://www.math.ust.hk/pg/

[email protected]

The Master of Philosophy (MPhil) Program seeks to strengthen students’ general background in mathematics and mathematical sciences, and to expose students to the environment and scope of mathematical research. A candidate for an MPhil degree is expected to demonstrate knowledge in the discipline and to synthesize and create new knowledge, making a contribution to the field. It can be a terminal degree or a preliminary degree leading to the PhD.  

The Doctor of Philosophy (PhD) Program aims to prepare students to become research scholars in an academic or industrial environment and enable students to do independent and original research. It provides a broad background in mathematics and mathematical sciences. Students can choose to focus their research in one of the three areas: Pure Mathematics, Applied Mathematics, and Probability and Statistics. A candidate for a PhD degree is expected to demonstrate mastery of knowledge in the chosen discipline and to synthesize and create new knowledge, making an original and substantial contribution to the discipline.

On successful completion of the MPhil program, graduates will be able to:

• Conduct research in mathematics;
• Demonstrate knowledge in the chosen discipline of mathematics;
• Be able to synthesize and create new knowledge and to making contributions to the discipline of mathematics; and
• Demonstrate communication skills in presenting reporting findings in mathematics.

On successful completion of the PhD program, graduates will be able to:

• Be able to conduct independent and original research in mathematics;
• Demonstrate mastery of knowledge in the chosen discipline of mathematics;
• Be able to synthesize and create new knowledge and to make original and substantial contributions to the discipline of mathematics;
• Demonstrate a broad knowledge in mathematics and mathematical sciences; and
• Demonstrate effective communication skills in presenting and publishing findings in mathematics.

The programs are offered by the Department of Mathematics with the following research foci and state-of-the-art facilities, strengthening students’ knowledge in mathematics and training them to carry out original research independently and innovatively.  

Research Foci

Algebra and Number Theory

The theory of Lie groups, Lie algebras and their representations is an important branch in modern mathematics and it has interactions with many other branches in mathematics and physics. Our research includes representation theory of reductive groups, quantum groups, infinite dimensional Lie groups and Lie algebras and applications to mathematical physics. Number theory has a long and distinguished history, and the concepts and problems relating to the theory have been instrumental in the foundation of a large part of mathematics. Number theory has flourished in recent years, as made evident by the proof of Fermat's Last Theorem. Our research specializes in automorphic forms and applications of number theory.  

Geometry and Topology

Geometry and topology provide an essential language describing all kinds of structures in nature. The subject has been vastly enriched by close interaction with other mathematical fields and with fields of science such as physics, astronomy and mechanics. The result has led to great advances in the subject, as highlighted by the proof of the Poincare conjecture. Active research areas in the department include algebraic geometry, differential geometry, low-dimensional topology, equivariant topology, combinatorics and combinatorial topology, and geometrical structures in mathematical physics.  

Analysis and Differential Equations

The analysis of real and complex functions plays a fundamental role in mathematics. This is a classical yet still vibrant subject that has a wide range of applications. Differential equations are used to describe many scientific, engineering and economic problems. The theoretical and numerical study of such equations is crucial in understanding and solving problems. Our research areas include complex analysis, exponential asymptotics, functional analysis, harmonic analysis, wavelets analysis, nonlinear equations and dynamical systems, integrable systems, fluid dynamics, and inverse problems.  

Applied and Computational Mathematics

The research focuses on development of mathematical models and advance numerical algorithms, as well as applications of mathematics to interdisciplinary science and engineering areas. The research areas include: modeling and simulation in fluid dynamics and materials science, computational fluid dynamics, multiscale modeling, kinetic theory, atomistic simulations, image processing, computational harmonic analysis, inverse problems and wave propagation, graph theory, optimization, evolutionary genetics, environmental science, numerical weather prediction, ocean and coastal modeling, numerical PDEs and numerical linear algebra, parallel algorithms, and numerical software.  

Financial Mathematics

Traditionally, financial mathematics includes three areas: asset allocation/management, derivative modeling, pricing and hedging, and division or firm-wise risk management. The mathematical tools include probability, stability, stochastic calculus and numerical analysis. The recent trend of financial mathematics research, however, has been data driven, parallel to the development in the financial market where both sell-side and buy-side firms are trying to adopt the methodologies of big data and machine learning, in order to gain advantages in proprietary trading, market making, hedging and investments. Some faculties of the department working in related fields have embraced the changes and adopted the new technologies in their teaching and research.  

Probability and Statistics

Statistics, the science of collecting, analyzing, interpreting, and presenting data, is an essential tool in a wide variety of academic disciplines as well as for business, government, medicine and industry. Our research is conducted in four categories. Time Series and Dependent Data: inference from nonstationarity, nonlinearity, long-memory behavior, and continuous time models. Resampling Methodology: block bootstrap, bootstrap for censored data, and Edgeworth and saddle point approximations. Stochastic Processes and Stochastic Analysis: filtering, diffusion and Markov processes, and stochastic approximation and control. Survival Analysis: survival function and errors in variables for general linear models. Probability current research includes limit theory.  

Data Science

Statistics and data science play a more and more important role in scientific research and applications, particularly in the era of big data. Some research interests include random matrix, survival analysis, financial statistics, econometrics, bioinformatics, statistical machine learning, statistical genetics and genomics, computational biology and bioinformatics, error in variables model, generalized linear models; biological and medical statistics, topological and geometric methods in data analysis.  

We also develop algorithms that provably and significantly improve the state-of-the-art for mathematical data analysis, in terms of accuracy, running time, memory efficiency, scalability, or other computational measures. This includes adapting existing algorithms or approaches to exploit contemporary computing architectures, new statistical paradigms, or heterogeneous information sources. We investigate the solution of specific practical data science problems that use novel methodology and are of broad interest. These problems include imaging, high dimensional data analysis, machine learning, social networks, genetics and genomics.  

The Department enjoys a range of up-to-date facilities and equipment for teaching and research purposes. It has two computer laboratories and a Math Support Center equipped with 100 desktop computers for undergraduate and postgraduate students. The Department also provides an electronic homework system and a storage cloud system to enhance teaching and learning.  

To assist computations that require a large amount of processing power, a High Performance Computing (HPC) laboratory was setup in 2000. The high-speed computers in this laboratory in which InfiniBand technology and GPU technology are applied are capable of delivering a processing power of 150 TFLOPS practically. By making use of these powerful computing facilities, our faculty and postgraduate students are able to solve computationally intensive problems in their innovative research projects so that they can stay at the forefront of their fields.

Minimum Credit Requirement

MPhil: 24 credits PhD: 36 credits  

Credit Transfer

PhD students who have obtained an MSc or MPhil degree from other institutions may be granted credit transfer of up to 18 credits, subject to departmental approval.  

Required Courses

MPhil: 24 credits in mathematics or related fields, normally including at least 18 credits of mathematics courses at postgraduate level. PhD: 36 credits in mathematics or related fields, including at least 24 credits of mathematics courses at postgraduate level. PhD students can choose to focus their research in one of the three areas:

Pure Mathematics;

Applied Mathematics; and

Probability and Statistics.  

Students with a first degree in an area other than mathematics may be required to take additional courses.  

Graduate Teaching Assistant Training

All full-time RPg students are required to complete PDEV 6800. The course is composed of a 10-hour training offered by the Center for Education Innovation (CEI), and session(s) of instructional delivery to be assigned by the respective departments. Upon satisfactory completion of the training conducted by CEI, MPhil students are required to give at least one 30-minute session of instructional delivery in front of a group of students for one term. PhD students are required to give at least one such session each in two different terms. The instructional delivery will be formally assessed.  

Professional Development Course Requirement

Students are required to complete PDEV 6770. The 1 credit earned from PDEV 6770 cannot be counted toward the credit requirements. PhD students who are HKUST MPhil graduates and have completed PDEV 6770 or other professional development courses offered by the University before may be exempted from taking PDEV 6770, subject to prior approval of the School.

Students are required to complete MATH 6771. The 1 credit earned from MATH 6771 cannot be counted toward the credit requirements. PhD students who are HKUST MPhil graduates in Mathematics and have completed MATH 6771 before may be exempted from taking MATH 6771, subject to prior approval of the Department Head or Department PG Coordinator.  

English Language Requirement

Full-time RPg students are required to take an English Language Proficiency Assessment (ELPA) Speaking Test administered by the Center for Language Education before the start of their first term of study. Students whose ELPA Speaking Test score is below Level 4, or who failed to take the test in their first term of study, are required to take LANG 5000 until they pass the course by attaining at least Level 4 in the ELPA Speaking Test before graduation. The 1 credit earned from LANG 5000 cannot be counted toward the credit requirements.  

Students must complete and pass LANG 5010, which should be taken in the first year of study. The 1 credit earned from LANG 5010 cannot be counted toward the credit requirements. PhD students who are HKUST MPhil graduates may be considered for exemption from this requirement, subject to prior approval from the Department Head and PG Coordinator.  

Postgraduate Seminar

Full-time students are required to complete MATH 6900 in their first four regular terms of study. The maximum number of credits to be earned from this course is 4.  

PhD Qualifying Examination

PhD students are required to pass a qualifying examination. Pure Mathematics To become PhD candidates, students must first pass a written qualifying examination (normally at the end of the first year of study) on two of the three subject areas: analysis, algebra, and geometry. At a later date (normally no later than the end of the second year of study) an oral examination on a major area excluding the two areas covered in the written examination. Applied Mathematics To become PhD candidates, students must first pass a written preliminary examination (normally before the end of the first year of study) on two subjects: advanced calculus and linear algebra. Students must also submit a thesis proposal, and pass an oral examination on the thesis proposal and two minor subjects. The oral examination should normally take place before the end of the second year of study. Probability and Statistics To become PhD candidates, students must pass an oral qualifying examination on one major subject and two minor subjects (normally no later than the end of the second year of study). For Probability students, the major subject is Probability while the two minor subjects would be Statistics and an area in Mathematics or an appropriate area outside Mathematics. For Statistics students, the major subject is Statistics while the two minor subjects would be Probability and an area in Mathematics or an appropriate area outside Mathematics.  

Thesis Research

Registration in MATH 6990; and

Presentation and oral defense of the MPhil thesis.

Registration in MATH 7990; and

Presentation and oral defense of the PhD thesis.

Last update: 31 July 2019

Scientific Computation Concentration

In addition to the existing program requirements, students who opt for the Scientific Computation concentration are required to:  

MPhil: Complete a minimum of 7 credits from the following course lists. PhD: Complete a minimum of 10 credits from the following course lists. The credits earned under the concentration will be counted toward the total credit requirements of the programs. Core Courses MPhil: at least 3 credits PhD: at least 6 credits All students must take MATH 6915 and MATH 6916. Credits earned from MATH 6915 can be repeated for up to 2 credits.  

Elective Courses

Conduct research in the area of scientific computation.

Last update: 29 August 2023

To qualify for admission, applicants must meet all of the following requirements. Admission is selective and meeting these minimum requirements does not guarantee admission.

Applicants seeking admission to a master's degree program should have obtained a bachelor’s degree from a recognized institution, or an approved equivalent qualification;

Applicants seeking admission to a doctoral degree program should have obtained a bachelor’s degree with a proven record of outstanding performance from a recognized institution; or presented evidence of satisfactory work at the postgraduate level on a full-time basis for at least one year, or on a part-time basis for at least two years.

Applicants have to fulfill English Language requirements with one of the following proficiency attainments:

TOEFL-iBT: 80*

TOEFL-pBT: 550

TOEFL-Revised paper-delivered test: 60 (total scores for Reading, Listening and Writing sections)

IELTS (Academic Module): Overall score: 6.5  and All sub-score: 5.5

* refers to the total score in one single attempt

Applicants are not required to present TOEFL or IELTS score if

their first language is English, or

they obtained the bachelor's degree (or equivalent) from an institution where the medium of instruction was English.

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