what does a math phd look like

• 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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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.  

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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.

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

Mathematics, PHD

On this page:, at a glance: program details.

• Location: Tempe campus
• Second Language Requirement: No

Program Description

Degree Awarded: PHD Mathematics

The PhD program in mathematics is intended for students with exceptional mathematical ability. The program emphasizes a solid mathematical foundation and promotes innovative scholarship in mathematics and its many related disciplines.

The School of Mathematical and Statistical Sciences has very active research groups in analysis, number theory, geometry and discrete mathematics.

Degree Requirements

84 credit hours, a written comprehensive exam, a prospectus and a dissertation

Required Core (3 credit hours) MAT 501 Geometry and Topology of Manifolds I (3) or MAT 516 Graph Theory I (3) or MAT 543 Abstract Algebra I (3) or MAT 570 Real Analysis I (3)

Other Requirements (3 credit hours) MAT 591 Seminar (3)

Electives (24-39 credit hours)

Research (27-42 credit hours) MAT 792 Research

Culminating Experience (12 credit hours) MAT 799 Dissertation (12)

Additional Curriculum Information Electives are to be chosen from math or related area courses approved by the student's supervisory committee.

Students must pass:

• two qualifying examinations
• a written comprehensive examination
• an oral dissertation prospectus defense

Students should see the department website for examination information.

Each student must write a dissertation and defend it orally in front of five dissertation committee members.

Admission Requirements

Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and Sciences.

Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics or a closely related area from a regionally accredited institution.

Applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in the last 60 hours of their first bachelor's degree program or a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in an applicable master's degree program.

All applicants must submit:

• graduate admission application and application fee
• official transcripts
• statement of education and career goals
• three letters of recommendation
• proof of English proficiency

Additional Application Information An applicant whose native language is not English must provide proof of English proficiency regardless of their current residency.

Additional eligibility requirements include competitiveness in an applicant pool as evidenced by coursework in linear algebra (equivalent to ASU course MAT 342 or MAT 343) and advanced calculus (equivalent to ASU course MAT 371), and it is desirable that applicants have scientific programming skills.

Next Steps to attend ASU

Learn about our programs, apply to a program, visit our campus, application deadlines, learning outcomes.

• Address an original research question in mathematics.
• Able to complete original research in theoretical mathematics.
• Apply advanced mathematical skills in coursework and research.

Career Opportunities

Graduates of the doctoral program in mathematics possess sophisticated mathematical skills required for careers in many different sectors, including education, industry and government. Potential career opportunities include:

• faculty-track academic
• finance and investment analyst
• mathematician
• mathematics professor, instructor or researcher
• operations research analyst
• statistician

Program Contact Information

If you have questions related to admission, please click here to request information and an admission specialist will reach out to you directly. For questions regarding faculty or courses, please use the contact information below.

• [email protected]
• 480/965-3951

Ph.D. Degree Programs

The UCSD Mathematics Department admits students into the following Ph.D. programs:

• Ph.D. in Mathematics -- Pure or Applied Mathematics.
• Ph.D. in Mathematics with a  Specialization in Computational Science .
• Ph.D. in Mathematics with a  Specialization in Statistics .

In addition, the department participates in the following Ph.D. programs:

• Ph.D. in  Bioinformatics .
• Ph.D. in  Mathematics and Science Education  (joint program between UCSD and SDSU).

For application information, go to  How to Apply (Graduate) .  

Ph.D. in Mathematics

The Ph.D. in Mathematics allows study in pure mathematics, applied mathematics and statistics. The mathematics department has over 60 faculty, approximately 100 Ph.D. students, and approximately 35 Masters students. A list of the UCSD mathematics faculty and their research interests can be found at  here . The Ph.D. in Mathematics program produces graduates with a preparation in teaching and a broad knowledge of mathematics. Our students go on to careers as university professors, as well as careers in industry or government.

In the first and second years of study, Ph.D. students take courses in preparation for three written qualifying examinations (quals). One qual must be taken in Algebra or Topology, and another in Real or Complex Analysis. A third qual may be taken in Numerical Analysis or Statistics or one of the remaining topics in the first two groups. All three quals must be passed by the start of the third year. After the qualifying exams are passed, the student is expected to choose an advisor and follow a course of study agreed on by the two of them. At this point, the student chooses a thesis topic, finds a doctoral committee and presents a talk on his or her proposed research topic. If the committee is satisfied with this talk, the student has "Advanced to Candidacy." The student will then pursue their research agenda with their advisor until they have solved an original problem. The student will submit a written dissertation and reconvene his or her committee for a Final Defense. At the Final Defense, the student gives a seminar talk that is very similar to a talk that he or she might give for a job interview.

Nearly every admitted Ph.D. student gets financial support. The financial support is most commonly in the form of a Teaching Assistantship, however, Research Assistantships and other fellowships are also available.

Because of the large faculty to student ratio, graduate students have many opportunities to interact with faculty in courses or smaller research seminars. The graduate students also run their own "Food for Thought" seminar for expository talks as well as a research seminar where they give talks about their research.

UCSD has excellent library facilities with strong collections in mathematics, science, and engineering. Ph.D. students are provided with access to computer facilities and office space.

Full-time students are required to register for a minimum of twelve (12) units every quarter, eight (8) of which must be graduate-level mathematics courses taken for a letter grade only. The remaining four (4) units can be approved upper-division or graduate-level courses in mathematics-related subjects (MATH 500 may not be used to satisfy any part of this requirement). After advancing to candidacy, Ph.D. candidates may take all course work on a Satisfactory/Unsatisfactory basis. Typically, students should not enroll in MATH 299 (Reading and Research) until they have passed at least two Qualifying Examinations at the PhD or Provisional PhD level, or obtained approval of their faculty advisor.  

Written Qualifying Examinations

Effective Fall Quarter 1998, the department made changes in their qualifying exam requirements with a view to:

• improving applied mathematics' access to students and the attractiveness of its program to applicants; and
• broadening the education of our doctoral students and leading more of them towards applied areas.

The department now offers written qualifying examinations in  SEVEN (7)  subjects. These are grouped into three areas as follows:  

• Three qualifying examinations must be passed. At least one must be passed at the Ph.D. level and a second must be passed at either the Ph.D. or Provisional Ph.D. level.
• Of the three qualifying exams, there must be at least one from each of Areas 1 and 2. 
• Students must pass at least two exams from distinct areas with a minimum grade of Provisional Ph.D. (For example, a Ph.D. pass in Real Analysis, Provisional Ph.D. pass in Complex Analysis, M.A. pass in Algebra would  NOT  satisfy this requirement, but a Ph.D. pass in Real Analysis, M.A. pass in Complex Analysis, Provisional Ph.D. pass in Algebra would, as would a Ph.D. pass in Numerical Analysis, Provisional Ph.D. pass in Applied Algebra, and M.A. pass in Real Analysis.) All exams must be passed by the September exam session prior to the beginning of the third year of graduate studies. (Thus, there is no limit on the number of attempts, encouraging new students to take exams when they arrive, without penalty.) Except for this deadline, there is no limit on the number of exams a student may attempt.

After qualifying exams are given, the faculty meet to discuss the results of the exams with the Qualifying Exam and Appeals Committee (QEAC). Exam grades are reported at one of four levels:  

Department policy stipulates that at least one of the exams must be completed with a Provisional Ph.D. pass or better by September following the end of the first year. Anyone unable to complete this schedule will be terminated from the doctoral program and transferred to one of our Master's programs. Any grievances about exams or other matters can be brought before the Qualifying Exam and Appeals Committee for consideration.

Exams are typically offered twice a year, one scheduled late in the Spring Quarter and again in early September (prior to the start of Fall Quarter). Copies of past exams are available on the  Math Graduate Student Handbook .

In choosing a program with an eye to future employment, students should seek the assistance of a faculty advisor and take a broad selection of courses including applied mathematics, such as those in Area 3.  

Master's Transferring to Ph.D.

Any student who wishes to transfer from masters to the Ph.D. program will submit their full admissions file as Ph.D. applicants by the regular closing date for all Ph.D. applicants (end of the fall quarter/beginning of winter quarter). It is the student's responsibility to submit their files in a timely fashion, no later than the closing date for Ph.D. applications at the end of the fall quarter of their second year of masters study, or earlier. The candidate is required to add any relevant materials to their original masters admissions file, such as most recent transcript showing performance in our graduate program. Letters of support from potential faculty advisors are encouraged. The admissions committee will either recommend the candidate for admission to the Ph.D. program, or decline admission. In the event of a positive recommendation, the Qualifying Exam Committee checks the qualifying exam results of candidates to determine whether they meet the appropriate Ph.D. program requirements, at the latest by the fall of the year in which the application is received. For students in the second year of the master's program, it is required that the student has secured a Ph.D. advisor before admission is finalized. An admitted student is supported in the same way as continuing Ph.D. students at the same level of advancement are supported. Transferring from the Master's program may require renewal of an I-20 for international students, and such students should make their financial plans accordingly. To be eligible for TA support, non-native English speakers must pass the English exam administered by the department in conjunction with the Teaching + Learning Commons.  

Foreign Language Requirement

There is no Foreign Language requirement for the Ph.D. in Mathematics.  

Advancement to Candidacy

It is expected that by the end of the third year (9 quarters), students should have a field of research chosen and a faculty member willing to direct and guide them. A student will advance to candidacy after successfully passing the oral qualifying examination, which deals primarily with the area of research proposed but may include the project itself. This examination is conducted by the student's appointed doctoral committee. Based on their recommendation, a student advances to candidacy and is awarded the C. Phil. degree.  

Dissertation and Final Defense

Submission of a written dissertation and a final examination in which the thesis is publicly defended are the last steps before the Ph.D. degree is awarded. When the dissertation is substantially completed, copies must be provided to all committee members at least four weeks in advance of the proposed defense date. Two weeks before the scheduled final defense, a copy of the dissertation must be made available in the Department for public inspection.  

Time Limits

The normative time for the Ph.D. in mathematics is five (5) years. Students must be advanced to candidacy by the end of eleven (11) quarters. Total university support cannot exceed six (6) years. Total registered time at UCSD cannot exceed seven (7) years.  

It may be useful to describe what the majority of students who have successfully completed their Ph.D. and obtained an academic job have done. In the past some students have waited until the last time limit before completing their qualifying exams, finding an advisor or advancing to candidacy. We strongly discourage this, because experience suggests that such students often do not complete the program. Although these are formal time limits, the general expectation is that students pass two qualifying exams, one at the Ph.D. level and one at the masters level by the beginning of their second year. (About half of our students accomplish this.) In the second year, a student begins taking reading courses so that they become familiar with the process of doing research and familiarize themselves with a number of faculty who may serve as their advisor. In surveying our students, on average, a student takes 4 to 5 reading courses before finding an advisor. Optimally, a student advances to candidacy sometime in their third year. This allows for the fourth and fifth year to concentrate on research and produce a thesis. In contrast to coursework, research is an unpredictable endeavor, so it is in the interest of the student to have as much time as possible to produce a thesis.

A student is also a teaching assistant in a variety of courses to strengthen their resume when they apply for a teaching job. Students who excel in their TA duties and who have advanced to candidacy are selected to teach a course of their own as an Associate Instructor. Because there are a limited number of openings to become an Associate Instructor, we highly recommend that you do an outstanding job of TAing in a large variety of courses and advance to candidacy as soon as possible to optimize your chances of getting an Associate Instructorship.

9500 Gilman Drive, La Jolla, CA 92093-0112

(858) 534-3590

To apply for admissions and financial aid, or for additional information on admissions requirements for the Ph.D. program in pure mathematics, please go to the appropriate Harvard Kenneth C. Griffin Graduate School of Arts and Sciences website listed below. All other inquiries may be directed to the Graduate Program Administrator of the Mathematics Department.

• Harvard Kenneth C. Griffin Graduate School of Arts and Sciences (Harvard Griffin GSAS)
• Mathematics Graduate Studies
• Financial Support

Graduate Program Administrator

The Department of Mathematics does not discriminate against applicants or students on the basis of race, color, national origin, ancestry or any other protected classification.

Preparing the Application The statement of purpose for graduate applications is carefully weighted by the admissions committee. The applicant’s statement should convince the committee that they are able to communicate effectively and with a deep understanding of mathematics. It is not intended to be a biographical sketch or a reflection on one’s decision to enter the field.

Three letters of recommendation are required. Letter writers should be faculty or others qualified to evaluate the applicant’s potential for graduate study in mathematics. The letters must be submitted online and by the application deadline.

Applicants should include any research papers, publications, and other original works they would like to have evaluated by the admissions committee.

The department requests that applicants submit GRE Mathematics Subject Test scores if practical. Applicants should check on the ETS website for test dates in their area to ensure the scores will be submitted before the application deadline. An official score report should be sent to Harvard Kenneth C. Griffin Graduate School of Arts and Sciences using code 3451.

While the admissions committee reviews all applications submitted before the deadline, missing math subject test scores provide one less data point available to evaluate the application. Depending on the strength of the application, the missing subject test scores may put the application at a disadvantage.

Applicants who are non-native English speakers and who do not hold an undergraduate degree from an institution at which English is the primary language of instruction must submit scores from the Internet Based Test (IBT) of the Test of English as a Foreign Language (TOEFL) or the International English Language Testing System (IELTS) Academic test.

Harvard Griffin GSAS requires applicants to upload an electronic copy of undergraduate transcripts. Hard copies of official transcripts are not required at the time of application.

Ph.D. Program in Pure Mathematics The department does not grant a terminal Master’s degree, but the Master’s can be obtained “on the way” to the Ph.D. by fulfilling certain course and language exam requirements.

In general, there is no transfer status application to the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences or to the Department of Mathematics. No formal credit is given for an MSc or MA earned elsewhere. All applicants are considered to be applying as first-year graduate students. The only difference Master’s study may make is to better prepare students for the Qualifying Exam.

All graduate students are admitted to begin their studies in the fall term. The department plans on an entering class of about twelve students. Since the admissions committee receives a few hundred applications, the competition is keen.

Funding Graduate Study Applicants are urged to apply for all funding available to them. If no outside funding is available to the applicant, financial aid in the form of scholarships, research assistantships, and teaching fellowships is available. In general, students who do not have outside support will get scholarship support in their first year, but students are required to act as a teaching fellow for one-half course (i.e. for a one-term course) in their second through fifth years.

The department strongly recommends applicants to seek out and apply for all sources of financing available to them for graduate study. Recommended sources for funding US graduate students are NSF Graduate Fellowships and NDSEG Fellowships . Applicants from the UK are urged to also apply for the Kennedy fellowships and applicants from UK, New Zealand, Canada and Australia for Knox fellowships . International students may apply for the Fullbright IIE or any home country fellowships available for study abroad.

Harvard John A. Paulson School of Engineering and Applied Sciences The Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) offers programs for both the Master’s degree and the Ph.D. degree in Applied Mathematics. Please visit the SEAS website for more information on degrees in applied mathematics at www.seas.harvard.edu

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.

New Graduate Students

We are thrilled that you will be joining the Berkeley Mathematics community this Fall! This page contains pertinent information that will help get you started as a graduate student at Berkeley. 

Complete New Student Onboarding 

You've accepted our offer; now help the Office of the Registrar prepare your student record by completing the "Scheduled Tasks" found on the "My Dashboard" tab of your  CalCentral  account. Scheduled tasks typically include completing the Statement of Legal Residence (SLR), immunizations form, and for international students, the Non-immigrant Information Form (NIF). In addition, you will be required to complete  two mandatory Sexual Violence/Sexual Harassment Prevention trainings (online and in-person) . You will receive more information on these requirements via CalCentral and from the department. Thus, it's important that you regularly check your CalCentral account for important campus notifications and tasks. 

For International Students - Obtaining Documents for your Visa Application via the NIF 

The Berkeley International Office (BIO) provides student advising to international students across campus and is responsible for issuing visa documents to all incoming international students. We highly encourage all international students to complete the NIF (Non-immigrant Information Form) as soon as possible in order to have plenty of time to gather the documents required to apply for a student visa. Your departmental offer letter (w/ signature) can serve as proof of financial support. If you need assistance locating your departmental offer letter please contact Christian Natividad at  [email protected] . We also recommend you visit  BIO's website  for new students as you will find a wealth of resources specific to international student arrival including information on visas, housing, money, enrollment, transportation, and health care. 

On this page:  Important Dates & Planning your Arrival  

Important Dates • Arrival • Housing Resources • Financial Matters • Medical Reminders • Student Groups

Orientation & Academics

Prelim Info • Math Orientation • Fall Enrollment 

Employment as a GSI or GSR  

GSI/GSR Overview • Requirements for 1st time GSIs • Language Requirements • Resources for GSIs

Campus Resources

Variety of useful Links 

Graduate Program Contacts  

Contact Info 

Important Dates & Planning your Arrival

Important dates: .

• Prelim Workshop  -  TBD (2024 schedule will be posted late June/early July).
• Enrollment Opens for New Graduate Students -   Friday, July 19, 2024
• Mathematics Graduate Student Orientation -  Wednesday, August 21, 2024, 1015 Evans - Full-day program
• Teaching Conference for first-time  international  GSIs  (REQUIRED)  - Thursday, August 22, 2024
• Teaching Conference for  all  first-time GSIs  (REQUIRED for domestic & international)  - Friday, August 23, 2024
• Fall 2024 Math Prelim Exam s  - Monday, August 26 & Tuesday, August 27. This exam will be held in person.
• Fall 2024 Graduate Division Graduate Student Orientation   - TBD.  Registration is required.
• First Day of Fall Semester  - Wednesday, August 21, 2024
• Practice Prelim  - TBD
• First Day of Fall Instruction  - Wednesday, August 28, 2024
• Grad-stravaganza  - Wednesday, September 4, 2024 from 4-6 p.m.

Planning your Arrival 

This summer, we recommend that you arrive in the Bay Area as early as possible in order to get settled for your first year of graduate study. An arrival in early August would be ideal so that you can participate in the prelim prep workshop and complete HR onboarding for new GSIs in advance of your appointment start date (8/1/24). We understand that international students may only enter the country within 30 days of their I-20 or DS-2019 start date - again we recommend arriving as early as your schedule allows. 

Our department orientation (8/21) is not required, but it is highly encouraged that you attend. Not only will you be able to meet your fellow classmates, but we'll also be reviewing campus and department resources, program requirements and expectations, and will host our own Sexual Violence Sexual Harrassment Training, which satisfies the university's in-person SVSH training requirement. We'll also end the day with the office draw!  

Please note that for those of you serving as a GSI this fall, you are required to attend the Teaching Conference for New GSIs and first-time international GSIs are required to attend the  Teaching in the U.S. Classroom Conference (tentative dates above).  

In summary, while it is encouraged that you arrive in Berkeley as early as possible (early/mid-July), the latest you can arrive and still attend all of the required trainings and take the fall prelim is mid-August.

Housing Resources in Berkeley   

University Campus Housing Website (for Ida Jackson, Manville, University Village)

CalRentals - University Listings for Off-Campus Housing and Summer Sublets

The Graduate Assembly's Housing Guide: Best Practices for Finding Housing

Berkeley International Office Housing Resources

SLMath Housing Links (Short-Term Housing)

Rent To Own Labs  (this site is unaffiliated with UC Berkeley)

*If you need a housing reference from the department please feel free to use Christian Natividad at  [email protected] . Please notify him in advance so he knows someone may be contacting him for more information. 

Financial Matters

Students in the Math Department are funded through a combination of sources including Graduate Student Instructor (GSI) or Graduate Student Researcher (GSR) salaries, fellowship stipends, university fellowships (Berkeley, Chancellor's, Regents, Ning, etc.) or external fellowships (NSF, NDSEG, NPSC, etc.). In order to receive fellowship payments from the university, students must be officially registered and in good standing. To be considered registered, you must be enrolled in at least one class, have had at least the first installment of fees paid, and have no registration blocks. Please note that full-time enrollment (12 units or a DSP approved reduced course load) is required to remain in compliance with fellowship policy. 

Here is an estimation of when you can expect to receive your first payments from the university (assuming you are a registered student at this time): 

• GSI or GSR Salary if HR onboarding completed by communicated deadlines: ~September 1, 2024 (for August work); if HR onboarding completed after the deadline: ~October 1, 2024 (for August & September work)  
• Department Relocation or recruitment stipends - last week of August 2024
• University Fellowship Awards (Berkeley, Chancellor's, Ning, etc.) - last week of August 2024
• External Fellowships - please refer to granting institution for pay dates  

 A delay in payment could be caused by not being considered a registered student, not having an up-to-date GLACIER record (for international students only), registration blocks, or department delays.  

To receive your fellowship payments via direct deposit please make sure to sign up for EFT (Electronic Funds Transfer). The EFT website can also be accessed in CalCentral.

To receive your GSI/GSR salary payment via direct deposit please sign up for EFT during your HR onboarding session with Berkeley Regional Services - ERSO. After your GSI/GSR appointment has been processed by ERSO, you can also sign up for direct deposit via  UC Path .  If you plan to receive paper checks instead of signing up for direct deposit, please make sure your "Local Address" is correct in CalCentral . Direct deposit is STRONGLY recommended to receive payments.  

Also, please keep in mind that receiving  financial assistance from the university may have tax implications  that you are not very familiar with. University and department staff are not able to give tax advice so we encourage you to consult the  IRS website  for more information, and or a personal/family certified tax accountant for assistance.  

International students can find more information on U.S. taxes via the  Berkeley International Office website on tax reporting.  

If you have any questions regarding financial matters please don't hesitate to contact us. 

Stay Healthy! Medical Checklist and Immunization Information 

Once enrolled for the fall term, your  Student Health Insurance Plan (SHIP)  coverage will start on August 1, 2024 and run through December 31, 2024. Your spring 2024 coverage, once enrolled, starts January 1, 2025 - July 31, 2025. If you have alternate health insurance coverage and do not want to enroll in SHIP please submit your waiver to the University Health Services (UHS) by the deadline in mid-July - see link below for more information and to submit the waiver. Before arriving, please review the new student medical checklist below so that you come prepared for any medical emergencies that may arise. Also, please make sure to review the UC Immunization Requirement policy either via the link below or through your CalCentral account.   

New Student Medical Checklist

UC Immunization Requirement

Information on Waiving SHIP - Deadline ~July 15, 202 4 

Connect with your New Berkeley Math Friends  

The Noetherian Ring - Women and Gender Minorities in the Department of Mathematics at UC Berkeley  

MGSA  &  MGSA Wiki

Unbounded Representation URep (website TBD) 

Berkeley GEMS - Gender Equity in Mathematical Studies 

More information on the Math Grad Life website

Orientation & Academics 

Prelim workshop & prelim exam .

(This prelim section is for Math and Applied Math graduate students only. Logic students can find more information on program requirements and the  Logic prelim exams here. ) 

Taking the prelim exam is the first requirement math graduate students will attempt in the program. All graduate students are required to pass the exam within their first three semesters of the program. It is held every semester (fall & spring) on the Monday and Tuesday mornings before instruction begins. This fall the exam will be held on  Monday, August 26 and Tuesday, August 27.  

The purpose of the prelim exam is to make sure that graduate students have sufficient working knowledge of foundational undergraduate material in the early stages of the PhD program, and to give early feedback on gaps in knowledge. Its intention is not to weed out students, but rather help strengthen one's understanding of core material. There is no penalty for having to retake the exam more than once so we recommend students take the exam in the fall. 

The Prelim Exam also gives you the chance to meet with your cohort and study! It's good to get in the habit of studying and working together - math doesn't have to be lonely in graduate school! We encourage you to attend the Prelim Workshop, which will be run by current graduate students. A practice prelim exam will be held the week before the exam. Topics will alternate between algebra and analysis. You can find the complete schedule on the workshop website (updates forthcoming).  

You can find more information on the  prelim as well as resources to help you prepare and pass exams here . 

Math Graduate Student Orientation (For all incoming Math, Applied Math, and Logic students)

The Mathematics Graduate Student Orientation is scheduled for  Wednesday, August 21st, 2024.  It will be an all day event, which will include a continental breakfast and lunch(RSVP form will be sent via email). More information including a full agenda will be sent closer to the date. You can see a  general agenda here . New students aren't required to attend, but it's highly recommended that students make every effort to be present. Curious about what to expect for your time here? Check out the  MGSA Wiki . 

Fall Enrollment 

Fall enrollment for new graduate students opens on Friday,   July 19th, 9:15am - 4:45 pm.  Registering on-time, and before August 1st, ensures that you will have timely access to health insurance coverage and access to campus resources. 

Course enrollment at Berkeley occurs in two phases (1 & 2) and is assigned based on one's standing (Grad, UG Freshman, Sophomore, etc.). For fall, new graduate students are able to sign up for classes in one phase in late July. All math graduate students must be enrolled full-time with a minimum 12-unit course load each semester. Students needing accommodations such as a reduced course load must be registered with the  Disabled Students' Program.  

In addition to being required to enroll in 12 units each semester, as a program requirement, all first years must enroll in at least four courses total across the fall and spring terms. At least two of these must be graduate courses in Mathematics. MPS 375 and 303 do not count towards the satisfaction of this requirement. MPS 375 is a pedagogy course that all first-time GSIs are required to take. For reference of the course offerings to expect, you can view our course offerings on the  Schedule of Classes  [use the lefthand bar to filter by term and course level (e.g. grad or undergrad)] or on our website  here . To view courses offered in other departments visit the  Berkeley Guide . I recommend using the top ribbon to search by subject rather than by keyword. 

Before finalizing your course schedule, please discuss your plans with your assigned first-year faculty adviser. You will be assigned a faculty adviser in early summer. If you have any general questions about enrollment and your options please feel free to contact us. 

Employment as a GSI or GSR 

The two primary forms of employment for graduate students in the Math Department are Graduate Student Instructor (GSI) and Graduate Student Researcher (GSR) appointments. GSRs are appointed directly by the supervising faculty member. The majority of first-year students work as GSIs for the department.

In late spring/early summer an application will be sent out to all students who will be working as GSIs this fall so that we can collect your teaching preferences. We will do our best to match you to your preferences, but please keep in mind that you may not be assigned your first choice. The majority of first-year students are assigned to our larger, lower-division courses as assignments are made based on seniority and prior GSI experience. When submitting your preferences please make sure that your own course schedule does not conflict with the course you may be assigned to as a GSI. Some instructors require attendance at lecture and you will always need to be available to proctor in-class midterms and final exams. The instructor teaching the course is your supervisor so please make sure to have your travel plans for winter and summer break approved by them in advance of making any travel arrangements. You may be needed several days after the final exam is given to complete final grading and administrative duties so it's important that you work with your supervising faculty member to identify when it will be okay for you to depart from campus for breaks.

Fall GSI appointments officially start on August 1st and end on December 31st. Thus, students are paid for the entire month of August and December even though GSI work typically doesn't begin until the start of the semester and usually ends before December 31st.  In order for students to be paid the correct amount it is important that they complete HR onboarding (official employment verification) with Berkeley Regional Services / ERSO by the stated deadlines. Students who do not complete onboarding by the stated deadline may not be paid their GSI salary until October 1st.  More information on onboarding sessions will be sent later this summer. 

Requirements for first-time GSIs: 

• Attend the daylong  Teaching Conference  sponsored by the  GSI Teaching & Resource Center . This conference is held each semester on the Friday before classes begin. Pre-registration is required. All first-time international GSIs must also complete the  Teaching Conference for International GSIs , which takes place in the fall semester the Thursday before classes begin. Pre-registration is also required for Thursday's offering.
• Successfully complete the  online course GSI Professional Standards and Ethics in Teaching   before  interacting with students (in person or online) as an instructor.
• Enroll in and complete the  300-level pedagogy course for first-time GSIs , MPS 375. All GSIs teaching for the first time on campus must take a 300-level pedagogy course, regardless of prior teaching experience or previous courses taken at other universities. If you have a course conflict with MPS 375, students may take a pedagogy course offered by another department as long as you have approval from the Vice Chair for Graduate Studies, Sug Woo Shin, and the alternate department. 
• All first-time GSIs must attend an hour-long  Academic Student Employee (ASE) orientation session . Advanced registration is NOT required.

Language Proficiency Requirements for International Students 

International students must satisfy an English language proficiency requirement in order to serve as a GSI. If English is your first language, if you attended a US institution for your undergraduate degree, or if you scored at least 26 on the speaking portion of the TOEFL IBT, then you are eligible to teach. However, you must still report your status through the  Language Proficiency Questionnaire . The deadline to complete the questionnaire for incoming students is June 1, 2024. 

All other students must pass an English language proficiency exam, and in some cases take a class before being eligible to teach. We will write further about the steps needed to satisfy this language requirement in advance, but please be aware that it is your responsibility to ensure that this requirement is satisfied, and that your offer of a GSI appointment is contingent upon satisfying this requirement.

Resources for GSIs:

GSI Teaching and Resource Center 

GSI, GSR, Reader and Tutor Guide

If you have any questions regarding GSI or GSR matters please don't hesitate to contact us. 

Campus Resources 

Logic Resources

Graduate Division Campus Resources Page  - Comprehensive website of all resources on campus

New Graduate Student Guide  - Graduate Division's most recent guide for new students

Guide to Graduate Policy  - Graduate Division Policies on Graduate Studies at Berkeley

Berkeley International Office (BIO)

Berkeley Parking & Transportation

CalDining - 2022-23 Meal Plan for Graduate Students  (on-campus dining commons)

Cal 1 Card  - Student ID Card 

Cal Rec Sports  - RSF, Campus Gym 

UC Berkeley Basic Needs Security

Food Security • CalFresh • Food Assistance Program • UCB Food Pantry

Campus Life  

Campus Safety

Gender Equity Resource Center (GenEq)

Graduate Assembly

Undocumented Student Program 

Math Stats Library

University Health Services (UHS) 

Counseling and Psychological Services (CAPS) at UHS

Graduate Program Contacts 

Graduate Advisor  Clay Calder |  [email protected]  or 510-642-0665

Graduate Advisor  Christian Natividad|  [email protected]  or 415-501-0125 

Director of Student Services  Vicky Lee |  [email protected]  or 510-644-4603

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PhD in Mathematics

Push the Limits of Knowledge to Find Solutions

We live in a world where math plays an increasingly important role in solving today's most pressing issues. The use of advanced algorithms and computational tools has expanded our understanding of math's vast applications and has also revealed how much we still need to discover.

The PhD in Mathematics at Clarkson University pushes forward innovative solutions by allowing you to acquire specialized knowledge, expand an applied perspective and conduct your own research. Tackle some of the most complex problems with the confidence and know-how of an expert.

Why Earn a PhD in Mathematics From Clarkson University?

Develop specialized knowledge using the resources of a technology-rich university. Through our interdisciplinary approach, collaborate across campus with different scholars and on different issues with an applied mathematical perspective. In addition to increasing your advanced fundamental knowledge of the field, complete your own research meant to bring forward innovative solutions.

We are also a small department with big opportunities. What does that mean? Our faculty are nationally recognized math researchers, conducting studies with academic, industry and government partners. They further serve as dedicated mentors, who guide you toward your individual goals. The Department's environment is collegial and collaborative: we see you as a valuable member of our team and include you to solve research problems together.

Along with research, PhD students gain valuable teaching experience by working with undergrads as teaching assistants or in other capacities.

The PhD in Mathematics consists of 90 credit hours of coursework, seminars and project work. The first two to three years are devoted to coursework, and the remainder of your time centers around research.

The coursework is distributed as follows:

• A minimum of 15 credit hours in a major area.
• A minimum of nine credit hours in a minor area.
• A minimum of six credit hours of work outside the department.

Students also take a General Comprehensive Exam at the end of the second semester. The topics cover Calculus, Differential Equations, Linear Algebra, Real Analysis, Probability and Statistics. By the fourth semester, you must pass two additional written Comprehensive Examinations from the following:

• Category I (Pure Math): Real Analysis, Complex Analysis, Sets and Topology and Numerical Analysis.
• Category II: Matrix Theory and Computations, Partial Differential Equations and Boundary Value Problems, Ordinary Differential Equations, Probability and Measure Theory or Statistics.

To complete the degree, prepare to defend a dissertation based on your research.

Review all requirements below. 

The PhD in Mathematics consists of 90 credit hours above a bachelor's degree. These credits are obtained through coursework, seminars and project work to fulfill PhD requirements. The program consists of extensive coursework completed during the first two to three years, followed by the development of an original research dissertation under the direction of one of our faculty members.

PhD candidates take at least 39 credit hours of approved coursework (30 of which may be taken for the MS degree). As required by University regulations, the coursework must contain a minimum of 15 hours in his/her major area, a minimum of nine hours in a minor area and a minimum of six hours of work outside the Department. Cross-registered graduate-level courses from other institutions are acceptable. The major area and minor area must be identified by the candidate's advisor and approved by the graduate committee.

During the program:

• PhD students must maintain an overall grade point average of at least 3.00 for their coursework.
• PhD students acquire at least six hours of seminar credit. A seminar is a course in which the student is expected to make presentations to the class. This is in addition to the minimum of 39 credit hours of approved coursework. One hour of seminar credit may be earned by either attending a regularly scheduled seminar and making one presentation or attending all colloquia for one semester and giving one presentation at a Department of Mathematics Seminar.
• PhD students make a formal presentation of a proposed thesis topic to their Thesis Committee within one year of passing their Comprehensive Exam. The topic must be acceptable to the committee.
• Candidates must write and defend (to their Thesis Committee) a dissertation that embodies the results of their original research. In association with this work, the student must obtain at least 21, but no more than 45, hours of thesis credit. The Thesis Committee consists of at least five Clarkson faculty members, including at least one from another department.
• Candidates complete a total of 90 hours of graduate credit. The Thesis Committee then certifies the satisfaction of these requirements.
• The typical duration of the PhD program is five to six years.

Our faculty make significant contributions to the world of mathematics and participate in interdisciplinary research with other academics. As a student, you benefit from their mentorship and work closely with them in a collaborative and encouraging environment. Learn more about their experience and areas of specialization.

Meet Our Faculty

The major areas of emphasis in our Department include:

• Applied math education
• Applied statistics
• Applied optimization
• Biomathematics
• Control theory
• Computational mathematics
• Data-driven science
• Dynamical and complex systems
• Image processing

As a student, you have access to Clarkson University centers like the Institute for a Sustainable Environment , the Institute for STEM Education and the Center for Complex Systems Science , where many of our faculty direct and oversee research projects. 

A complete application consists of the following:

• Online Application Form.
• Curriculum Vitae.
• Statement of purpose.
• Three letters of recommendation.
• Official transcripts.
• GRE test scores are optional for those with MS and BS degrees in mathematical sciences (mathematics, applied mathematics, and statistics); for others it is required, but waivers will be considered on a case-by-case basis.
• Minimum test score requirements: TOEFL (80) and TOEFL Essentials (8.5), IELTS (6.5), PTE (56) or Duolingo English Test (115).
• The English language-testing requirement is not waived based on language of instruction, nor do we accept university certificates. English testing is waived if an applicant has a degree from a country where English is the Native Language. Click here to see the list of these countries.

Prerequisites: Students must have a BS (or MS) or equivalent degree(s) in mathematics or a closely related field.

Criteria for Acceptance and Helpful Information

What makes a successful candidate.

Typical successful applicants have an undergraduate or a master’s degree in mathematics, applied mathematics, statistics or a related quantitative discipline. While the balance of topics in any student’s background may vary, we expect that applicants have an excellent foundation in undergraduate mathematics, including experience with proof-based courses, coding in at least one programming language such as MATLAB, Python, Julia, R, etc. and fluency in written and spoken English. Prior research experience is not a requirement, but it definitely helps in making a strong case for the student to be accepted.

Dates and Deadlines

Your application can be accepted at any time, although new students are accepted only for the Fall and Spring semesters. Please complete the application at least two months and up to 10 months in advance of the start of the semester. Once an application file is submitted, the faculty review takes at least one month to complete.

Teaching and Research Assistantship Positions

If you are additionally applying for a TA or RA position, please review all additional application deadlines in the Financial Assistance section below.

Reviews for admission and for a TA or RA position are separate, with the TA/RA process significantly more competitive due to a very limited number of available spots. Past successful applicants to TA/RA positions have had significant undergraduate research experience (e.g., participating in semester-long or year-long research projects or potentially co-authoring a publication), and some experience in teaching, whether as a tutor for college-level courses or an undergraduate teaching assistant. In most such cases, recommendation letters and statements of purpose give the committee a good understanding of the content of research/teaching experience and the degree to which students were involved.

Research Fit/Topics

It is critical that the evaluation committee understands how specifically a student may fit into the research program at Clarkson; statements that say “I will do whatever it takes/work on whatever topic” are typically not helpful — instead, we encourage applicants to describe a smaller set of research topics for which they believe they are well prepared and which may keep their interest for the next five years.

No Minimum GPA/GRE Score Required

There are no GPA, GRE or other scales/cutoffs that we use to accept/disqualify applicants; rather, the committee evaluates holistically the set of skills and experiences applicants bring with them.

Most current graduate students are supported by Teaching Assistantships or Research Assistantships. A full appointment covers 30 credit hours of tuition and provides a stipend that covers estimated living expenses.

Teaching Assistantships (TA)

Most PhD students are supported by a TA position. A full appointment requires the student to teach up to 20 hours/week (on average) during the Fall and Spring semesters. In return, a TA position covers the full tuition and pays a living stipend to the student that covers the estimated expenses of living in Potsdam for the full academic year (including during summer). Regularly, PhD students are prioritized for TA positions.

• To apply for a TA position, simply indicate your interest on the application form. Prior teaching experience is not necessary, but it is helpful. The number of TA positions available each year varies. Offers are issued during the Spring semester (for Fall admittance) after a competitive review of interested applicants by the admissions committee.
• All TA positions start in the Fall semester, and the contract runs for one year at a time. If you are looking to start in the Spring semester, you should expect to have external funds to cover the tuition and living expenses for your first semester here (Spring); you would then be competitively evaluated for any available TA positions open in the subsequent Fall.
• TAs can expect to be reappointed in subsequent years for up to five years total, under the condition that they maintain good academic progress (maintain a cumulative GPA above B/3.00 and complete examinations according to deadlines) and perform their teaching duties well.
• The tuition and stipend provided by the full-year TA contract can be used by international students to certify the level of income needed to apply for the F1 visa. The university does not cover the costs of applying for a visa or relocating to Potsdam, NY.
• All new TAs must complete a TA training during their first summer (“TA boot camp”) that involves a virtual/remote portion and an in-person portion during early August. During the TA boot camp, TAs get introduced to different teaching methods, teaching activities and typical U.S. university classroom dynamics. The TA boot camp is organized by the Institute for STEM Education, which additionally tracks the progress and development of TAs. We hope that through this continual teacher training, our PhD graduates have an edge in their first job search and can quickly translate this experience to all future academic positions.

Research Assistantships (RA)

Faculty members occasionally obtain a research grant that funds an RA position. An RA graduate student works closely with the faculty member to achieve research goals. In many cases, this research can form a significant part of the student’s PhD project; in other cases, it may be unrelated to the student’s research.

The RA position carries a similar hourly commitment to the TA position, with the difference that the commitment extends for the full duration of support (not simply Fall/Spring semesters) and with continued funding limited by performance, the need of the funded project and the duration of the grant. RA positions are often filled by second and more senior graduate students. When there are unfilled RA positions open to first-year students, the Department advertises them below.

The annual Mathematics Colloquium is Clarkson University's flagship series of talks in mathematics. Once a week during the Spring semester, we invite visiting speakers, faculty members and graduate students to present their most current research.

Mathematics Colloquium

The program is held on our main campus in Potsdam, New York. Many of our full-time, research-based master's and PhD programs are housed here, as well. Students have access to research facilities, onsite laboratories and other resources.

Potsdam Campus

Career Possibilities

Clarkson's PhD in Mathematics provides the rigorous training and research-heavy experience required to enter the world of academia. With your degree, apply to professorships and positions in prestigious research centers.

Your skills are also high in demand in today's job market. According to the U.S. Bureau of Labor Statistics, the number of available positions for mathematicians and statisticians is expected to grow 31 percent by 2031. Roles for mathematicians can be found in almost every industry that deals with science, engineering, data or technology.

Clarkson graduates may find themselves in the following roles:

• Algorithmic engineer
• Data scientist
• Entrepreneur
• Investment banking analyst
• Management consultant
• Operations research analyst
• Process engineer
• Project manager
• Quality assurance manager
• Software developer
• Statistician

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Department of Graduate Admissions Email: [email protected] Phone: 518-631-9831

Interested in learning more about the PhD in Mathematics? Contact the Office of Graduate Admissions today with your questions.

Find out more about the Department of Mathematics .

Super Power

When you need a mega-dose of computing power to solve complex problems, turn to ACRES — our high-performance computing cluster. Short for Accelerating Computational Research for Engineering and Science, ACRES delivers 160 teraflops of solving speed.

Learn More About ACRES

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Deepen your expertise.

Gain specialized knowledge, conduct original research and qualify for a wide range of academic and industry opportunities with a PhD in Mathematics from Clarkson.

PhD Qualifying Exams

The requirements for the PhD program in Mathematics have changed for students who enter the program starting in Autumn 2023 and later. 

Requirements for the Qualifying Exams

Students who entered the program prior to autumn 2023.

To qualify for the Ph.D. in Mathematics, students must pass two examinations: one in algebra and one in real analysis. 

Students who entered the program in Autumn 2023 or later

To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas: 

• real analysis
• geometry and topology
• applied mathematics

The exams each consist of two parts. Students are given three hours for each part.

Topics Covered on the Exams:

• Algebra Syllabus
• Real Analysis Syllabus
• Geometry and Topology Syllabus
• Applied Mathematics Syllabus

Check out some Past and Practice Qualifying Exams to assist your studying.

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take either or both of the exams in the autumn. If they pass either or both of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing either of the exams.

Students must pass both qualifying exams by the autumn of their second year. Ordinarily first-year students take courses in algebra and real analysis throughout the year to prepare them for the exams. The exams are then taken at the beginning of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Students who started in Autumn 2023 and later

Students must choose and pass two out of the four qualifying exams by the autumn of their second year. Students take courses in algebra, real analysis, geometry and topology, and applied math in the autumn and winter quarters of their first year to prepare them for the exams. The exams are taken during the first week of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Exam Schedule

Unless otherwise noted, the exams will be held each year according to the following schedule:

Autumn Quarter:  The exams are held during the week prior to the first week of the quarter. Spring Quarter:  The exams are held during the first week of the quarter.

The exams are held over two three-hour blocks. The morning block is 9:30am-12:30pm and the afternoon block is 2:00-5:00pm.

For the start date of the current or future years’ quarters please see the  Academic Calendar

Upcoming Exam Dates

Spring 2024.

The exams will be held on the following dates:

Monday, April 1st

Analysis Exam, Room 384H

Wednesday, April 3rd

Algebra Exam, Room 384I

Thursday, April 4th 

Geometry & Topology Exam, Room 384I

Friday, April 5th

Applied Math Exam, Room 384I

What’s the point of maths research? It’s the abstract nonsense behind tomorrow’s breakthroughs

Director of Research for Mathematics, University of Hull

Disclosure statement

Wolfram Bentz does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

University of Hull provides funding as a member of The Conversation UK.

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Whenever I tell people I’m a mathematical researcher, I’m usually met with some form of bewilderment. Occasionally that’s followed by the immediate end of the conversation. If there is a follow-up question, it’s usually not about the type of research I’m doing or how it’s funded but whether there’s anything left to discover in maths at all.

True, maths rarely makes the headlines and so most people probably don’t think of it as carrying out cutting-edge research. But neither does, say, geology and people don’t assume there’s nothing left to discover in that field. The difference is that everyone is familiar with maths from their schooldays in a way that contrasts vastly from the work of actual mathematicians. In school we learn formulas that are then used to calculate answers to specific problems. The right method correctly calculated will never fail.

Maths research, on the other hand, looks at the myriad problems for which we don’t have such a method. It’s about finding the tools and systems that other subject areas find so useful in formulating their own work. And sometimes it stumbles across facts about numbers that we have no conceivable use for at the moment but that one day could become vital to the world.

Any mathematical method used at school (or work or anywhere) was figured out at some point by a mathematician. Another mathematician may have proven that it always works. And another may have worked out how to use the method in the real world. Someone else might then have shown that it’s not a very efficient way to solve larger problems and developed a different approach instead.

The method may also have relied on several properties of the underlying number system discovered over a long stretch of time. Others before them will have accomplished the important but unglamorous task of precisely defining that number system, perhaps a very long time ago .

Prime purpose

Research mathematicians essentially still discover similar types of results today. We have simply moved on to different questions that have become important, to new methods for existing questions, to different systems that draw our attention, and to more advanced questions about things that have already been researched.

Here is an example of such a recent result. It deals with the distribution of prime numbers, like 7, 11, 23, or 37, which you cannot divide by another natural number other than 1 or themselves. We’ve found prime numbers as large as 22m digits long, and researchers are still looking .

If you look at a table of numbers, the prime numbers seem to be almost randomly mixed into the non-prime ones. For a long time, we have been able to describe the typical characteristics of prime numbers. As it turns out, prime numbers slowly but steadily appear less frequently – they “thin out” – among the larger numbers. What’s more, we can quantify this process in surprisingly precise terms.

With the average primes being further and further apart as we look at ever larger numbers, a typical question for maths researchers to ask is whether this process of thinning out also carries over to the smallest gaps between primes. In other words, will all large primes come at increasing distances from each other, or will we always find primes that are close to each other?

A breakthrough result in 2014 showed that no matter how high we go among the numbers, we will always find two primes that are closer to each other than some constant number. That number was initially a whopping 70m. This might not seem very close but the fact that we could identify a finite number was an important breakthrough. Other mathematicians then set out to reduce this value, and the best I am currently aware of is a much more manageable 246 .

Real applications – eventually

You might wonder how solving such abstract problems helps anyone outside of mathematics. First, there is a trickle-down effect. A fundamental result is useful in obtaining other pure mathematical results, which in turn are used to develop applied mathematics, which are then used by non-mathematicians. Second – and more importantly – mathematical theory is often ahead of its time, and the abstract nonsense of yesterday underpins the applied mathematics of today.

For example, number theory is the area that examines, among other items, questions like our prime number example. For many years this was considered the ultimate pure mathematics topic and completely unusable for any purpose other than satisfying human curiosity. The eminent early 20th-century number theorist and pacifist, G.H Hardy, was very proud to say : “No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” In other words, he was glad his work could not be used for military purposes.

Nowadays, the number theory results that seemed so useless less than a century ago are at the heart of the encryption algorithms that let us securely order a product or check our bank accounts online. In a way that would have horrified Hardy, British intelligence services had actually already discovered the same method in secret ahead of their civil colleagues.

When the next technological or scientific breakthrough requires a new type of mathematical model, it is likely that the subject already has the underlying theory in hand, waiting to be adapted to a new setting.

Underlying all of this is one of the fundamental truths about mathematical research. The applications of mathematics might change with scientific progress, making some mathematical topics more useful at times than others. But because mathematical results are based on logical deductions alone, they actually never become wrong, never get obsolete, and never truly get old. They are just waiting for the right application to arrive.

• Mathematics
• Universities
• Maths study
• Prime numbers

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New NASA Black Hole Visualization Takes Viewers Beyond the Brink

Ever wonder what happens when you fall into a black hole? Now, thanks to a new, immersive visualization produced on a NASA supercomputer, viewers can plunge into the event horizon, a black hole’s point of no return.

“People often ask about this, and simulating these difficult-to-imagine processes helps me connect the mathematics of relativity to actual consequences in the real universe,” said Jeremy Schnittman, an astrophysicist at NASA’s Goddard Space Flight Center in Greenbelt, Maryland, who created the visualizations. “So I simulated two different scenarios, one where a camera — a stand-in for a daring astronaut — just misses the event horizon and slingshots back out, and one where it crosses the boundary, sealing its fate.”

The visualizations are available in multiple forms. Explainer videos act as sightseeing guides, illuminating the bizarre effects of Einstein’s general theory of relativity. Versions rendered as 360-degree videos let viewers look all around during the trip, while others play as flat all-sky maps.

To create the visualizations, Schnittman teamed up with fellow Goddard scientist Brian Powell and used the Discover supercomputer at the NASA Center for Climate Simulation . The project generated about 10 terabytes of data — equivalent to roughly half of the estimated text content in the Library of Congress — and took about 5 days running on just 0.3% of Discover’s 129,000 processors. The same feat would take more than a decade on a typical laptop.

The destination is a supermassive black hole with 4.3 million times the mass of our Sun, equivalent to the monster located at the center of our Milky Way galaxy.

“If you have the choice, you want to fall into a supermassive black hole,” Schnittman explained. “Stellar-mass black holes, which contain up to about 30 solar masses,  possess much smaller event horizons and stronger tidal forces, which can rip apart approaching objects before they get to the horizon.”

This occurs because the gravitational pull on the end of an object nearer the black hole is much stronger than that on the other end. Infalling objects stretch out like noodles, a process astrophysicists call spaghettification .

The simulated black hole’s event horizon spans about 16 million miles (25 million kilometers), or about 17% of the distance from Earth to the Sun. A flat, swirling cloud of hot, glowing gas called an accretion disk surrounds it and serves as a visual reference during the fall. So do glowing structures called photon rings, which form closer to the black hole from light that has orbited it one or more times. A backdrop of the starry sky as seen from Earth completes the scene.

As the camera approaches the black hole, reaching speeds ever closer to that of light itself, the glow from the accretion disk and background stars becomes amplified in much the same way as the sound of an oncoming racecar rises in pitch. Their light appears brighter and whiter when looking into the direction of travel.

The movies begin with the camera located nearly 400 million miles (640 million kilometers) away, with the black hole quickly filling the view. Along the way, the black hole’s disk, photon rings, and the night sky become increasingly distorted — and even form multiple images as their light traverses the increasingly warped space-time.

In real time, the camera takes about 3 hours to fall to the event horizon, executing almost two complete 30-minute orbits along the way. But to anyone observing from afar, it would never quite get there. As space-time becomes ever more distorted closer to the horizon, the image of the camera would slow and then seem to freeze just shy of it. This is why astronomers originally referred to black holes as “frozen stars.”

At the event horizon, even space-time itself flows inward at the speed of light, the cosmic speed limit. Once inside it, both the camera and the space-time in which it's moving rush toward the black hole's center — a one-dimensional point called a singularity , where the laws of physics as we know them cease to operate.

“Once the camera crosses the horizon, its destruction by spaghettification is just 12.8 seconds away,” Schnittman said. From there, it’s only 79,500 miles (128,000 kilometers) to the singularity. This final leg of the voyage is over in the blink of an eye.

In the alternative scenario, the camera orbits close to the event horizon but it never crosses over and escapes to safety. If an astronaut flew a spacecraft on this 6-hour round trip while her colleagues on a mothership remained far from the black hole, she’d return 36 minutes younger than her colleagues. That’s because time passes more slowly near a strong gravitational source and when moving near the speed of light.

“This situation can be even more extreme,” Schnittman noted. “If the black hole were rapidly rotating, like the one shown in the 2014 movie ‘Interstellar,’ she would return many years younger than her shipmates.”

By Francis Reddy NASA’s Goddard Space Flight Center , Greenbelt, Md. Media Contact: Claire Andreoli 301-286-1940 [email protected] NASA’s Goddard Space Flight Center, Greenbelt, Md.

Related Terms

• Astrophysics
• Black Holes
• Galaxies, Stars, & Black Holes
• Galaxies, Stars, & Black Holes Research
• Goddard Space Flight Center
• Supermassive Black Holes
• The Universe

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How Biden Adopted Trump’s Trade War With China

The president has proposed new barriers to electric vehicles, steel and other goods..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Sabrina Tavernise, and this is “The Daily.”

[MUSIC PLAYING]

Donald Trump upended decades of American policy when he started a trade war with China. Many thought that President Biden would reverse those policies. Instead, he’s stepping them up. Today, my colleague, Jim Tankersley, explains.

It’s Monday, May 13.

Jim, it’s very nice to have you in the studio.

It’s so great to be here, Sabrina. Thank you so much.

So we are going to talk today about something I find very interesting and I know you’ve been following. We’re in the middle of a presidential campaign. You are an economics reporter looking at these two candidates, and you’ve been trying to understand how Trump and Biden are thinking about our number one economic rival, and that is China.

As we know, Trump has been very loud and very clear about his views on China. What about Biden?

Well, no one is going to accuse President Biden of being as loud as former President Trump. But I think he’s actually been fairly clear in a way that might surprise a lot of people about how he sees economic competition with China.

We’re going after China in the wrong way. China is stealing intellectual property. China is conditioning —

And Biden has, kind of surprisingly, sounded a lot, in his own Joe Biden way, like Trump.

They’re not competing. They’re cheating. They’re cheating. And we’ve seen the damage here in America.

He has been very clear that he thinks China is cheating in trade.

The bottom line is I want fair competition with China, not conflict. And we’re in a stronger position to win the economic competition of the 21st century against China or anyone else because we’re investing in America and American workers again. Finally.

And maybe the most surprising thing from a policy perspective is just how much Biden has built on top of the anti-China moves that Trump made and really is the verge of his own sort of trade war with China.

Interesting. So remind us, Jim, what did Trump do when he actually came into office? We, of course, remember Trump really talking about China and banging that drum hard during the campaign, but remind us what he actually did when he came into office.

Yeah, it’s really instructive to start with the campaign, because Trump is talking about China in some very specific ways.

We have a $500 billion deficit, trade deficit, with China. We’re going to turn it around. And we have the cards. Don’t forget —

They’re ripping us off. They’re stealing our jobs.

They’re using our country as a piggy bank to rebuild China, and many other countries are doing the same thing. So we’re losing our good jobs, so many.

The economic context here is the United States has lost a couple of million jobs in what was called the China shock of the early 2000s. And Trump is tapping into that.

But when the Chinese come in, and they want to make great trade deals — and they make the best trade deals, and not anymore. When I’m there, we turn it around, folks. We turn it around. We have —

And what he’s promising as president is that he’s going to bring those jobs back.

I’ll be the greatest jobs president that God ever created. I’ll take them back from China, from Japan.

And not just any jobs, good-paying manufacturing jobs, all of it — clothes, shoes, steel, all of these jobs that have been lost that American workers, particularly in the industrial Midwest, used to do. Trump’s going to bring them back with policy meant to rebalance the trade relationship with China to get a better deal with China.

So he’s saying China is eating our lunch and has been for decades. That’s the reason why factory workers in rural North Carolina don’t have work. It’s those guys. And I’m going to change that.

Right. And he likes to say it’s because our leaders didn’t cut the right deal with them, so I’m going to make a better deal. And to get a better deal, you need leverage. So a year into his presidency, he starts taking steps to amass leverage with China.

And so what does that look like?

Just an hour ago, surrounded by a hand-picked group of steelworkers, President Trump revealed he was not bluffing.

It starts with tariffs. Tariffs are taxes that the government imposes on imports.

Two key global imports into America now face a major new barrier.

Today, I’m defending America’s national security by placing tariffs on foreign imports of steel and aluminum.

And in this case, it’s imports from a lot of different countries, but particularly China.

Let’s take it straight to the White House. The president of the United States announcing new trade tariffs against China. Let’s listen in.

This has been long in the making. You’ve heard —

So Trump starts, in 2018, this series of tariffs that he’s imposing on all sorts of things — washing machines, solar panels, steel, aluminum. I went to Delaware to a lighting store at that time, I remember, where basically everything they sold came from China and was subject to the Trump tariffs, because that’s where lighting was made now.

Interesting.

Hundreds of billions of dollars of Chinese goods now start falling under these Trump tariffs. The Chinese, of course, don’t take this lying down.

China says it is not afraid of a trade war with the US, and it’s fighting back against President Trump with its own tariffs on US goods.

They do their own retaliatory tariffs. Now American exports to China cost more for Chinese consumers. And boom, all of a sudden, we are in the midst of a full-blown trade war between the United States and Beijing.

Right. And that trade war was kind of a shock because for decades, politicians had avoided that kind of policy. It was the consensus of the political class in the United States that there should not be tariffs like that. It should be free trade. And Trump just came in and blew up the consensus.

Yeah. And Sabrina, I may have mentioned this once or 700 times before on this program, but I talk to a lot of economists in my job.

Yeah, it’s weird. I talk to a lot of economists. And in 2018 when this started, there were very, very, very few economists of any political persuasion who thought that imposing all these tariffs were a good idea. Republican economists in particular, this is antithetical to how they think about the world, which is low taxes, free trade. And even Democratic economists who thought they had some problems with the way free trade had been conducted did not think that Trump’s “I’m going to get a better deal” approach was going to work. And so there was a lot of criticism at the time, and a lot of politicians really didn’t like it, a lot of Democrats, many Republicans. And it all added up to just a real, whoa, I don’t think this is going to work.

So that begs the question, did it?

Well, it depends on what you mean by work. Economically, it does not appear to have achieved what Trump wanted. There’s no evidence yet in the best economic research that’s been done on this that enormous amounts of manufacturing jobs came back to the United States because of Trump’s tariffs. There was research, for example, on the tariffs on washing machines. They appear to have helped a couple thousand jobs, manufacturing jobs be created in the United States, but they also raised the price of washing machines for everybody who bought them by enough that each additional job that was created by those tariffs effectively cost consumers, like, $800,000 per job.

There’s like lots of evidence that the sectors Trump was targeting to try to help here, he didn’t. There just wasn’t a lot of employment rebound to the United States. But politically, it really worked. The tariffs were very popular. They had this effect of showing voters in those hollowed-out manufacturing areas that Trump was on their team and that he was fighting for them. Even if they didn’t see the jobs coming back, they felt like he was standing up for them.

So the research suggests this was a savvy political move by Trump. And in the process, it sort of changes the political economic landscape in both parties in the United States.

Right. So Trump made these policies that seemed, for many, many years in the American political system, fringe, isolationist, economically bad, suddenly quite palatable and even desirable to mainstream policymakers.

Yeah. Suddenly getting tough on China is something everyone wants to do across both parties. And so from a political messaging standpoint, being tough on China is now where the mainstream is. But at the same time, there is still big disagreement over whether Trump is getting tough on China in the right way, whether he’s actually being effective at changing the trade relationship with China.

Remember that Trump was imposing these tariffs as a way to get leverage for a better deal with China. Well, he gets a deal of sorts, actually, with the Chinese government, which includes some things about tariffs, and also China agreeing to buy some products from the United States. Trump spins it as this huge win, but nobody else really, including Republicans, acts like Trump has solved the problem that Trump himself has identified. This deal is not enough to make everybody go, well, everything’s great with China now. We can move on to the next thing.

China remains this huge issue. And the question of what is the most effective way to deal with them is still an animating force in politics.

Got it. So politically, huge win, but policy-wise and economically, and fundamentally, the problem of China still very much unresolved.

Absolutely.

So then Biden comes in. What does Biden do? Does he keep the tariffs on?

Biden comes to office, and there remains this real pressure from economists to roll back what they consider to be the ineffective parts of Trump’s trade policy. That includes many of the tariffs. And it’s especially true at a time when almost immediately after Biden takes office, inflation spikes. And so Americans are paying a lot of money for products, and there’s this pressure on Biden, including from inside his administration, to roll back some of the China tariffs to give Americans some relief on prices.

And Biden considers this, but he doesn’t do it. He doesn’t reverse Trump’s tariff policy. In the end, he’s actually building on it.

We’ll be right back.

So Jim, you said that Biden is actually building on Trump’s anti-China policy. What exactly does that look like?

So Biden builds on the Trump China policy in three key ways, but he does it with a really specific goal that I just want you to keep in mind as we talk about all of this, which is that Biden isn’t just trying to beat China on everything. He’s not trying to cut a better deal. Biden is trying to beat China in a specific race to own the clean-energy future.

Clean energy.

Yeah. So keep that in mind, clean energy. And the animating force behind all of the things Biden does with China is that Biden wants to beat China on what he thinks are the jobs of the future, and that’s green technology.

Got it. OK. So what does he do first?

OK. Thing number one — let’s talk about the tariffs. He does not roll them back. And actually, he builds on them. For years, for the most part, he just lets the tariffs be. His administration reviews them. And it’s only now, this week, when his administration is going to actually act on the tariffs. And what they’re going to do is raise some of them. They’re going to raise them on strategic green tech things, like electric vehicles, in order to make them more expensive.

And I think it’s important to know the backdrop here, which is since Biden has taken office, China has started flooding global markets with really low-cost green technologies. Solar panels, electric vehicles are the two really big ones. And Biden’s aides are terrified that those imports are going to wash over the United States and basically wipe out American automakers, solar panel manufacturers, that essentially, if Americans can just buy super-cheap stuff from China, they’re not going to buy it from American factories. Those factories are going to go out of business.

So Biden’s goal of manufacturing jobs in clean energy, China is really threatening that by dumping all these products on the American market.

Exactly. And so what he wants to do is protect those factories with tariffs. And that means increasing the tariffs that Trump put on electric vehicles in hopes that American consumers will find them too expensive to buy.

But doesn’t that go against Biden’s goal of clean energy and things better for the environment? Lots of mass-market electric vehicles into the United States would seem to advance that goal. And here, he’s saying, no, you can’t come in.

Right, because Biden isn’t just trying to reduce emissions at all costs. He wants to reduce emissions while boosting American manufacturing jobs. He doesn’t want China to get a monopoly in these areas. And he’s also, in particular, worried about the politics of lost American manufacturing jobs. So Biden does not want to just let you buy cheaper Chinese technologies, even if that means reducing emissions.

He wants to boost American manufacturing of those things to compete with China, which brings us to our second thing that Biden has done to build on Trump’s China policy, which is that Biden has started to act like the Chinese government in particular areas by showering American manufacturers with subsidies.

I see. So dumping government money into American businesses.

Yes, tax incentives, direct grants. This is a way that China has, in the past decades, built its manufacturing dominance, is with state support for factories. Biden is trying to do that in particular targeted industries, including electric vehicles, solar power, wind power, semiconductors. Biden has passed a bunch of legislation that showers those sectors with incentives and government support in hopes of growing up much faster American industry.

Got it. So basically, Biden is trying to beat China at its own game.

Yeah, he’s essentially using tariffs to build a fortress around American industry so that he can train the troops to fight the clean energy battle with China.

And the troops being American companies.

Yes. It’s like, we’re going to give them protection — protectionist policy — in order to get up to size, get up to strength as an army in this battle for clean energy dominance against the Chinese.

Got it. So he’s trying to build up the fortress. What’s the third thing Biden does? You mentioned three things.

Biden does not want the United States going it alone against China. He’s trying to build an international coalition, wealthy countries and some other emerging countries that are going to take on China and try to stop the Chinese from using their trade playbook to take over all these new emerging industrial markets.

But, Jim, why? What does the US get from bringing our allies into this trade war? Why does the US want that?

Some of this really is about stopping China from gaining access to new markets. It’s like, if you put the low-cost Chinese exports on a boat, and it’s going around the world, looking for a dock to stop and offload the stuff and sell it, Biden wants barriers up at every possible port. And he wants factories in those places that are competing with the Chinese.

And a crucial fact to know here is that the United States and Europe, they are behind China when it comes to clean-energy technology. The Chinese government has invested a lot more than America and Europe in building up its industrial capacity for clean energy. So America and its allies want to deny China dominance of those markets and to build up their own access to them.

And they’re behind, so they’ve got to get going. It’s like they’re in a race, and they’re trailing.

Yeah, it’s an economic race to own these industries, and it’s that global emissions race. They also want to be bringing down fossil-fuel emissions faster than they currently are, and this is their plan.

So I guess, Jim, the question in my mind is, Trump effectively broke the seal, right? He started all of these tariffs. He started this trade war with China. But he did it in this kind of jackhammer, non-targeted way, and it didn’t really work economically. Now Biden is taking it a step further. But the question is, is his effort here going to work?

The answer to whether it’s going to work really depends on what your goals are. And Biden and Trump have very different goals. If Trump wins the White House back, he has made very clear that his goal is to try to rip the United States trade relationship with China even more than he already has. He just wants less trade with China and more stuff of all types made in the United States that used to be made in China. That’s a very difficult goal, but it’s not Biden’s goal.

Biden’s goal is that he wants America to make more stuff in these targeted industries. And there is real skepticism from free-market economists that his industrial policies will work on that, but there’s a lot of enthusiasm for it from a new strain of Democratic economists, in particular, who believe that the only chance Biden has to make that work is by pulling all of these levers, by doing the big subsidies and by putting up the tariffs, that you have to have both the troops training and the wall around them. And if it’s going to work, he has to build on the Trump policies. And so I guess you’re asking, will it work? It may be dependent upon just how far he’s willing to go on the subsidies and the barriers.

There’s a chance of it.

So, Jim, at the highest level, whatever the economic outcome here, it strikes me that these moves by Biden are pretty remarkably different from the policies of the Democratic Party over the decades, really going in the opposite direction. I’m thinking of Bill Clinton and NAFTA in the 1990s. Free trade was the real central mantra of the Democratic Party, really of both parties.

Yeah, and Biden is a real break from Clinton. And Clinton was the one who actually signed the law that really opened up trade with China, and Biden’s a break from that. He’s a break from even President Obama when he was vice president. Biden is doing something different. He’s breaking from that Democratic tradition, and he’s building on what Trump did, but with some throwback elements to it from the Roosevelt administration and the Eisenhower administration. This is this grand American tradition of industrial policy that gave us the space race and the interstate highway system. It’s the idea of using the power of the federal government to build up specific industrial capacities. It was in vogue for a time. It fell out of fashion and was replaced by this idea that the government should get out of the way, and you let the free market drive innovation. And now that industrial policy idea is back in vogue, and Biden is doing it.

So it isn’t just a shift or an evolution. It’s actually a return to big government spending of the ‘30s and the ‘40s and the ‘50s of American industrialism of that era. So what goes around comes around.

Yeah, and it’s a return to that older economic theory with new elements. And it’s in part because of the almost jealousy that American policymakers have of China and the success that it’s had building up its own industrial base. But it also has this political element to it. It’s, in part, animated by the success that Trump had making China an issue with working-class American voters.

You didn’t have to lose your job to China to feel like China was a stand-in for the forces that have taken away good-paying middle-class jobs from American workers who expected those jobs to be there. And so Trump tapped into that. And Biden is trying to tap into that. And the political incentives are pushing every future American president to do more of that. So I think we are going to see even more of this going forward, and that’s why we’re in such an interesting moment right now.

So we’re going to see more fortresses.

More fortresses, more troops, more money.

Jim, thank you.

You’re welcome.

Here’s what else you should know today. Intense fighting between Hamas fighters and Israeli troops raged in parts of Northern Gaza over the weekend, an area where Israel had declared Hamas defeated earlier in the war, only to see the group reconstitute in the power vacuum that was left behind. The persistent lawlessness raised concerns about the future of Gaza among American officials. Secretary of State Antony Blinken said on “Face the Nation” on Sunday that the return of Hamas to the North left him concerned that Israeli victories there would be, quote, “not sustainable,” and said that Israel had not presented the United States with any plan for when the war ends.

And the United Nations aid agency in Gaza said early on Sunday that about 300,000 people had fled from Rafah over the past week, the city in the enclave’s southernmost tip where more than a million displaced Gazans had sought shelter from Israeli bombardments elsewhere. The UN made the announcement hours after the Israeli government issued new evacuation orders in Rafah, deepening fears that the Israeli military was preparing to invade the city despite international warnings.

Today’s episode was produced by Nina Feldman, Carlos Prieto, Sidney Harper, and Luke Vander Ploeg. It was edited by M.J. Davis Lin, Brendan Klinkenberg, and Lisa Chow. Contains original music by Diane Wong, Marion Lozano, and Dan Powell, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Sabrina Tavernise. See you tomorrow.

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Hosted by Sabrina Tavernise

Produced by Nina Feldman ,  Carlos Prieto ,  Sydney Harper and Luke Vander Ploeg

Edited by M.J. Davis Lin ,  Brendan Klinkenberg and Lisa Chow

Original music by Diane Wong ,  Marion Lozano and Dan Powell

Engineered by Alyssa Moxley

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Donald Trump upended decades of American policy when he started a trade war with China. Many thought that President Biden would reverse those policies. Instead, he’s stepping them up.

Jim Tankersley, who covers economic policy at the White House, explains.

On today’s episode

Jim Tankersley , who covers economic policy at the White House for The New York Times.

Background reading

Mr. Biden, competing with Mr. Trump to be tough on China , called for steel tariffs last month.

The Biden administration may raise tariffs on electric vehicles from China to 100 percent .

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