phd maths subjects

• 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.

Browse PhDs Now

Join thousands of students.

Join thousands of other students and stay up to date with the latest PhD programmes, funding opportunities and advice.

PhD in Mathematics

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

Degree Requirements

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

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

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

Financial Aid

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

More Information

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

Ph.D. in Mathematics

General info.

• Faculty working with students: 37
• Students: 52
• Students receiving Financial Aid: 100%
• Part time study available: No
• Application terms: Fall
• Application deadline: December 13

Mark Haskins Director of Graduate Studies Department of Mathematics Duke University Box 90320 Durham, NC 27708-0320

Phone: (919) 660-2800

Email:  [email protected]

Website:  http://www.math.duke.edu/graduate/

Program Description

The graduate program in Mathematics at Duke offers research training in a wide variety of topics in mathematics.  Major areas of research specialization include algebra, number theory, algebraic geometry, analysis, differential geometry and physics, topology, dynamical systems, partial differential equations, scientific computing, and stochastic processes.  There are strong groups in mathematical biology, and in the analysis of high-dimensional data using ideas from topology and harmonic analysis.  These and other research efforts often include collaborations with researchers in the medical school and other departments.  These opportunities for interdisciplinary research and the breadth of research areas in the department make the Duke math department an ideal place to prepare for a job inside or outside of academia.

• Mathematics: PhD Admissions and Enrollment Statistics
• Mathematics: PhD Completion Rate Statistics
• Mathematics: PhD Time to Degree Statistics
• Mathematics: PhD Career Outcomes Statistics

Application Information

Application Terms Available:  Fall

Application Deadline:  December 13

Graduate School Application Requirements See the Application Instructions page for important details about each Graduate School requirement.

• Transcripts: Unofficial transcripts required with application submission; official transcripts required upon admission
• Letters of Recommendation: 3 Required
• Statement of Purpose: Required (see department guidance below)
• Résumé: Required
• GRE General (Optional)
• GRE Subject: Mathematics (Optional)
• English Language Exam: TOEFL, IELTS, or Duolingo English Test required* for applicants whose first language is not English *test waiver may apply for some applicants
• GPA: Undergraduate GPA calculated on 4.0 scale required

Department-Specific Application Requirements (submitted through online application)

Statement of Purpose Guidelines Summarize your mathematical work/background and current interests.  (If your principal academic focus has been outside mathematics, tell us about that work and what brings you to mathematics now.)  Explain why you chose to apply to Duke Math (what makes the department a good fit for you?).  It is generally beneficial to keep the statement under two pages.

Writing Sample None required

We strongly encourage you to review additional department-specific application guidance from the program to which you are applying: Departmental Application Guidance

List of Graduate School Programs and Degrees

Overview of the PhD Program

For specific information on the Applied Mathematics PhD program, see the navigation links to the right. 

What follows on this page is an overview of all Ph.D. programs at the School; additional information and guidance can be found on the  Graduate Policies  pages. 

General Ph.D. Requirements

• 10 semester-long graduate courses, including at least 8 disciplinary.   At least 5 of the 10 should be graduate-level SEAS "technical" courses (or FAS graduate-level technical courses taught by SEAS faculty), not including seminar/reading/project courses.  Undergraduate-level courses cannot be used.  For details on course requirements, see the school's overall PhD course requirements  and the individual program pages linked therein.
• Program Plan (i.e., the set of courses to be used towards the degree) approval by the  Committee on Higher Degrees  (CHD).
• Minimum full-time academic residency of two years .
• Serve as a Teaching Fellow (TF) in one semester of the second year.
• Oral Qualifying Examination Preparation in the major field is evaluated in an oral examination by a qualifying committee. The examination has the dual purpose of verifying the adequacy of the student's preparation for undertaking research in a chosen field and of assessing the student's ability to synthesize knowledge already acquired. For details on arranging your Qualifying Exam, see the exam policies and the individual program pages linked therein.
• Committee Meetings : PhD students' research committees meet according to the guidelines in each area's "Committee Meetings" listing.  For details see the "G3+ Committee Meetings" section of the Policies of the CHD  and the individual program pages linked therein.
• Final Oral Examination (Defense) This public examination devoted to the field of the dissertation is conducted by the student's research committee. It includes, but is not restricted to, a defense of the dissertation itself.  For details of arranging your final oral exam see the  Ph.D. Timeline  page.
• Dissertation Upon successful completion of the qualifying examination, a committee chaired by the research supervisor is constituted to oversee the dissertation research. The dissertation must, in the judgment of the research committee, meet the standards of significant and original research.

Optional additions to the Ph.D. program

Harvard PhD students may choose to pursue these additional aspects:

• a Secondary Field (which is similar to a "minor" subject area).  SEAS offers PhD Secondary Field programs in  Data Science and in  Computational Science and Engineering .   GSAS  lists  secondary fields offered by other programs.
• a Master of Science (S.M.) degree conferred  en route to the Ph.D in one of several of SEAS's subject areas.  For details see here .
• a Teaching Certificate awarded by the Derek Bok Center for Teaching and Learning .

SEAS PhD students may apply to participate in the  Health Sciences and Technology graduate program  with Harvard Medical School and MIT.  Please check with the HST program for details on eligibility (e.g., only students in their G1 year may apply) and the application process.

In Applied Mathematics

• First-Year Exploration
• Areas of Application
• AM & Economics
• How to Declare
• Who are my Advisors?
• Secondary Field
• Senior Thesis
• Research for Course Credit (AM 91R & AM 99R)
• AB/SM Information
• Peer Concentration Advisors (PCA) Program
• Student Organizations
• How to Apply
• PhD Timeline
• PhD Model Program (Course Guidelines)
• Oral Qualifying Examination
• Committee Meetings
• Committee on Higher Degrees
• Research Interest Comparison
• Collaborations
• Cross-Harvard Engagement
• Clubs & Organizations
• Centers & Initiatives
• Alumni Stories

Mathematics

Share this page.

This program is designed for students looking to conduct original mathematical research with the aim of becoming a research mathematician . Students will be located in Cambridge, Massachusetts, one of the most active centers of mathematics in the world. Other universities in the area include Boston College, Boston University, Brandeis University, MIT, and Northeastern University.

Students will have access to a wide range of resources including the Center of Mathematical Sciences and Applications, which brings together researchers from an extensive variety of disciplines and institutions and hosts conferences, seminars, and workshops.

Most graduates of the program have been very successful at securing postdoctoral fellowships in academia. A number of recent graduates have won prestigious fellowships including the Clay Fellowship, the Simons Fellowship, and NSF Graduate Research Fellowship. Others now have jobs in industry. 

Additional information on the graduate program is available from the Department of Mathematics and requirements for the degree are detailed in Policies .

Admissions Requirements

Please review admissions requirements and other information before applying. You can find degree program-specific admissions requirements below and access additional guidance on applying from the Department of Mathematics .

Statement of Purpose

The statement of purpose should convince the admissions committee that the applicant is 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.

Standardized Tests

GRE General: Not Accepted GRE Subject: Required

Theses & Dissertations

Theses & Dissertations for Mathematics

See list of Mathematics faculty

APPLICATION DEADLINE

Questions about the program.

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

Mathematics phd program.

The Ph.D. program in the Department of Mathematics provides students with in-depth knowledge and rigorous training in all the subject areas of mathematics. A core feature is the first-year program, which helps bring students to the forefront of modern mathematics. Students work closely with faculty and each other and participate fully in both research and student-run seminars.

Questions? Email [email protected]

• The firm deadline for applications for Autumn 2024, is December 4, 2023.
• The (general and advanced) GRE tests are no longer accepted. Please do not submit these scores.

Mathematics, PhD

Zanvyl krieger school of arts and sciences.

The goal of our PhD program is to train graduate students to become research mathematicians. Each year, an average of five students complete their theses and  go on to exciting careers  in mathematics both inside and outside of academia.

Faculty research interests  in the Johns Hopkins University Department of Mathematics are concentrated in several areas of pure mathematics, including analysis and geometric analysis, algebraic geometry and number theory, differential geometry, algebraic topology, category theory, and mathematical physics. The department also has an active group in data science, in collaboration with the  Applied Math Department .

The Department values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

A brief overview of our graduate program is below. For more detailed information, please see the links at the right.

Program Overview

All students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Teaching assistant salaries for the 2022-2023 academic year are $33,000, and exceptional applicants are also considered for supplementary fellowships. Students making satisfactory progress can expect to be supported for six years.

PhD candidates take two or three courses per semester over the first several years of the program. These are a mix of required and intermediate-level graduate courses, independent studies, and special topics classes offered by our faculty.

By the beginning of their second year, students are asked to demonstrate competency in algebra and in analysis by passing written qualifying exams in these two broad areas. Students are then expected to choose an advisor, who will supervise their dissertation and also administer an oral qualifying exam to be taken in the second or third year. More specifics about all these requirements are described on the  requirements page .

All graduate students are invited to attend  weekly research seminars in a variety of topic areas  as well as regular department teas and a weekly wine and cheese gathering attended by many junior and senior members of the department. A graduate student lunch seminar series provides an opportunity for our students to practice their presentation skills to a general audience.

PhD students will gain teaching experience as a teaching assistant for undergraduate courses. Most of our students lead two TA sections per week, under the supervision of both the faculty member teaching the course and the director of undergraduate studies. Students wanting more classroom experience (or extra pay) can teach their own sections of summer courses. First-year students are given a reduced TA workload in the spring semester, in preparation for the qualifying exams.

In addition to their stipend, each student is awarded an annual travel allowance to enable them to attend conferences for which limited funding is available or visit researchers at other institutions.

Financial Aid

Students admitted to the Ph.D. program receive teaching assistantships and full tuition fellowships. Exceptional applicants become candidates for one of the university's George E. Owen Fellowships.

William Kelso Morrill Award

The William Kelso Morrill Award for excellence in the teaching of mathematics is awarded every spring to the graduate student who best exemplifies the traits of Kelso Morrill: a love of mathematics, a love of teaching, and a concern for students.

Excellence in Teaching Awards

Three awards are given each year to a junior faculty member and graduate student teaching assistants who have demonstrated exceptional ability and commitment to undergraduate education.

Admission Requirements

Admission to the PhD program is based on primarily on academic records, letters of recommendation, and a personal statement. The Department of Mathematics values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

Via the online application , applicants should submit:

• A Statement of Purpose
• An optional Personal Statement
• Transcripts from all institutions attended
• Three letters of recommendation
• Official GRE scores for both the general and the subject test
• Official TOEFL scores (if English is not your first language)

The required Statement of Purpose discusses your academic interests, objectives, and preparation. The optional Personal Statement describes your personal background, and helps us create a more holistic understanding of you as an applicant. If you wish you may also discuss your personal background in the Statement of Purpose (e.g. if you have already written a single essay addressing both topics), instead of submitting separate statements.

Application fee waivers are available based on financial need and/or participation in certain programs .

Many frequently asked questions about the graduate admission process are answered here .

No application materials should be mailed to the department. All application materials are processed by the Graduate Admissions Office .

Undergraduate Background

The following is an example of what the math department would consider a good background for a student coming out of a four-year undergraduate program at a college or university in the U.S. (assuming a semester system):

• Calculus in one variable (two semesters, or AP credits)
• Multivariable Calculus (one semester)
• Linear Algebra (one semester)
• Complex analysis (one semester)
• Real analysis (two semesters)
• Abstract algebra (two semesters)
• Point-set topology (one semester)

Many admitted students have taken upper-level undergraduate mathematics courses or graduate courses. Nevertheless, the department does admit very promising students whose preparation falls a little short of the above model. In such cases, we strongly recommend that the student start to close the gap over the summer, before arriving for the start of the fall semester.

Financial Support   

Students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Teaching assistant salaries for the 2022–2023 academic year are $33,000. Students making satisfactory progress can expect to be supported for six years. Exceptional applicants are considered for supplementary fellowships of $6,000 each year for three years.

Students from underrepresented groups may be eligible for other university-wide supplemental fellowships. Summer teaching is available for students seeking extra income.

Additional Information for International Students

Student Visa Information:  The Office of International Services at Homewood  will assist admitted international students in obtaining a student visa.

English Proficiency: Johns Hopkins University requires students to have adequate English proficiency for their course of study. Students must be able to read, speak, and write English fluently upon their arrival at the university. Applicants whose native language is not English must submit proof of their proficiency in English before they can be offered admission and before a visa certificate can be issued. Proficiency can be demonstrated by submitting results from either the Test of English as a Foreign Language (TOEFL) or the IELTS . Johns Hopkins prefers a minimum score of 100 on the TOEFL or a Band Score of 7 on the IELTS. Results should be sent to Johns Hopkins directly by TOEFL or IELTS. Applicants taking the IELTS must additionally upload a copy of their score through the application system. However, do not send the student copy or a photocopy of the TOEFL.

Program Requirements

Course requirements.

Mathematics PhD candidates must show satisfactory work in Algebra (110.601-602), Real Variables (110.605), Complex Variables (110.607), and one additional non-seminar mathematics graduate course in their first year. The first-year algebra and analysis requirement can be satisfied by passing the corresponding written qualifying exam in September of the first year; these students must complete at least two courses each semester. In addition, PhD candidates must take Algebraic Topology (110.615) and Riemannian Geometry (110.645) by their second year. Students having sufficient background can substitute an advanced topology course for 110.615, or an advanced geometry course for 110.645 with the permission of the instructor.

Candidates must show satisfactory work in at least two mathematics graduate courses each semester of their second year, and if they have not passed their oral qualifying exam, in the first semester of their third year.

Qualifying Exams

Candidates must pass written qualifying exams by the beginning of their second year in Analysis (Real & Complex) and in Algebra. Exams are scheduled for September and May of each academic year, and the dates are announced well in advance.

Candidates must pass an oral qualifying examination in the student’s chosen area of research by April 10 of the third year. The topics of the exam are chosen in consultation with the faculty member who has agreed (provisionally) to be the student’s thesis advisor, who will also be involved in administering the exam.

PhD Dissertation

Candidates must produce a written dissertation based upon independent and original research. After completion of the thesis research, the student will defend the dissertation by means of the  Graduate Board Oral exam . The exam must be held at least three weeks before the Graduate Board deadline the candidate wishes to meet.

Our PhD program does not have a foreign language requirement.

Skip to content

Georgia Institute of Technology College of Sciences

Search form.

• You are here:
• Graduate Programs
• Doctoral Programs

PhD in Mathematics

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

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

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

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

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

The Coursework

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

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

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

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

Breadth requirements.  The students must demonstrate:

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

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

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

Special Topics and Reading Courses.

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

Credit Transfers

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

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

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

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

Comprehensive Examinations

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

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

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

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

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

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

Dissertation and Defense

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

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

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.  

Publications — Over 100 years of publishing excellence

• Book Author Resources
• Submit a Book Proposal
• AMS Rights, Licensing, and Permissions
• Open Math Notes
• Frequently asked questions
• Member Journals
• Research Journals
• Translation Journals
• Distributed Journals
• Open Access Journals
• Guidelines and Policies
• Journal Author Resources

Librarian Resources

• eBook Collections
• COUNTER Usage Statistics
• My Subscriptions
• Subscription Information
• Licensing Information

Mathematical Reviews/MathSciNet®

• MathSciNet ®
• Reviewer Home
• MathSciNet ® Subscriptions

Membership — Welcome to your membership center

Join the ams, renew your membership, give a membership, individual membership.

• Member Benefits
• Member Directory
• Reciprocating Societies
• Members in Developing Countries

Institutional Membership

• Domestic Institutions
• International Institutions
• Two-Year Institutions
• Graduate Student Chapter Program

Other Member Types

• Corporate Memberships
• Associate Memberships

Meetings & Conferences — Engage with colleagues and the latest research

National meetings.

• Joint Mathematics Meetings
• Upcoming JMMs
• Previous JMMs
• Special Lectures
• Professional Enhancement Programs (PEPs)

Sectional Meetings

• Upcoming Sectionals
• Previous Sectionals
• Presenting Papers
• Hosting Sectionals

Other Meetings, Conferences & Workshops

• Mathematics Research Communities
• Education Mini-conference
• International Meetings
• Mathematics Calendar
• Short Courses
• Workshop for Department Chairs and Leaders

Meetings Resources

• Suggest a Speaker
• AMS Meetings Grants
• Submitting Abstracts
• Welcoming Environment Policy
• MathSafe – supporting safe meetings

News & Outreach — Explore news, images, posters, and mathematical essays

News from the ams.

• AMS News Releases
• Feature Stories
• Information for Journalists
• In Memory Of

Math Voices

• Feature Column
• Math in the Media
• Column on Teaching and Learning

Explorations

• Recognizing Diverse Mathematicians
• AMS Posters
• Mathematics & Music
• Mathematical Imagery
• Mathematical Moments

Professional Programs — Resources and opportunities to further your mathematical pursuits

Professional development.

• Employment Services
• Mathjobs.org
• BEGIN Career Initiative
• Mathprograms.org
• Mathematical Opportunities Database
• Research Seminars

Institutional Information and Data

• Annual Survey of the Mathematical and Statistical Sciences
• CBMS Survey
• Other Sources of Data
• Directory of Institutions in the Mathematical Sciences
• Professional Directory

Grants & Support

• AMS-Simons Grants for PUI Faculty
• Travel Grants
• Fellowships & Scholarships
• Epsilon Fund
• Child Care Grants

Awards & Recognition

• AMS Prizes & Awards
• Fellows of the AMS

Education — Resources to support advanced mathematics teaching and learning

For students.

• Information for Undergraduate and High School Students
• Research Experiences for Undergraduates (REUs)
• Considering Grad School
• Find Grad Programs
• Applying to Grad School
• What do Mathematicians Do?

For Teachers

• Teaching Online
• Teaching Resources
• Inclusive Classrooms
• Assessing Student Learning
• Education Webinars

For Department Leaders & Mentors

• Information for Department Leaders
• paraDIGMS (Diversity in Graduate Mathematical Sciences)

Government Relations — Advocating for the mathematical sciences

Elevating mathematics in congress.

• Our Mission
• Letters, Statements, & Legislation
• Congressional Briefings

Legislative Priorities

• Federal Issues of Concern
• Federal Budget Process

Get Involved

• Advocacy Resources
• Take Action

DC-Based Fellowships

• Congressional Fellowship
• Mass Media Fellowship
• Catalyzing Advocacy in Science & Engineering (CASE) Fellowship

Giving to the AMS — Your gifts make great things happen for mathematics   Make a Gift

What you can support.

• The 2020 Fund
• Next Generation Fund
• Birman Fellowship for Women Scholars
• JMM Child Care Grants
• MathSciNet for Developing Countries

Create a Legacy

• Make a Tribute Gift
• Create a Permanent Fund
• Establish a Prize, Award or Fellowship
• Bequests and Charitable Estate Planning

Honoring Your Gift

• Donor Stories
• Donor Wall of Honor
• Thomas S. Fiske Society
• AMS Contributors Society
• AMS Gardens

Giving Resources

• AMS Development Committee
• AMS Gift Acceptance Policy

About the AMS — Advancing research. Connecting the mathematics community.

Our organization.

• Executive Staff
• Equity, Diversity, & Inclusion
• Jobs at AMS
• Customer Service

Our Governance

• Board of Trustees
• Executive Committee

Governance Operations

• Calendar of Meetings
• Policy Statements & Guidelines

On March 21 st , the AMS website will be down for regularly scheduled maintenance from 5:00am–8:00am

Institution name

Program type, masters programs (check all that apply), phd specialties (check all that apply), financial support available, gre required, online options available, number of phds awarded in the last year, enrollments, canadian province, list or edit your graduate program in the mathematical sciences.

• Add new listing
• Edit existing listing

Search results

Click the school name or up/down arrow symbol to view the complete program listing.

Records with are incomplete.

Air Force Institute of Technology Dept of Math & Stat

Bridge/postbaccalaureate.

• Certificate

Institution overview

Masters programs, phd programs, certificate programs, alabama a & m university dept of physics chemistry and mathematics , arizona state university s a levin math, comp & modeling sciences ctr , arizona state university school of math & stat sci , auburn univ, auburn dept of math , auburn university dept of math & stat, augusta university dept of population health sciences , baylor university department of mathematics, baylor university dept of statistical science, baylor university statistical science, binghamton university department of mathematical sciences, boston college boston college department of mathematics, boston college department of mathematics, boston university dept of math & stat, boston university school of public health biostatistics, bowling green state university department of mathematics and statistics, brigham young university dept of math, brown university department of mathematics, brown university div of applied math, bryn mawr college dept of math clark sci ctr rm 332 , california institute of technology dept of computing & math sci , california institute of technology dept of math , carleton university sch of math & stat , carnegie mellon university department of mathematical sciences, carnegie mellon university dept of stat , case western reserve university mathematics, applied mathematics and statistics, case western reserve university population & quantitative health sciences , centenary university education and mathematics department , central michigan university department of mathematics, chapman university faculty of mathematics, physics & comptation , claremont graduate university institute of mathematical sciences , clarkson university department of mathematics, clarkson university dept of mathematics , college of charleston department of mathematics, college of william & mary applied sci dept , colorado school of mines department of applied mathematics and statistics, colorado state university department of mathematics , colorado state university dept of stat , columbia university department of mathematics, columbia university department of statistics, columbia university dept of appl phys & appl math , columbia university, school of public health dept of biostatistics , columbia university, teachers college math educ dept , concordia university department of mathematics & statistics, cornell university bowers cis statistics and data science, cornell university center for applied mathematics, cornell university computational biology, cornell university dept of math , cornell university operations res & inform engr , cuny graduate school & university center phd program in math, dalhousie univ engineering math dept , dalhousie university department of engineering mathematics and internetworking, dalhousie university department of mathematics & statistics, dana farber cancer inst statistics , dartmouth college department of mathematics , delaware state university dept of mathematical sciences, department of statistics, columbia university dept of statistics, drexel university department of mathematics, duke university dept of stat sci , emory university dept of math , emory university dept. of biostatistics & bioinformatics , florida atlantic university department of mathematical sciences, florida institute of technology dept of math sci, florida institute of technology mathematical sciences , florida international university dept of math & stat , florida international university dept of statistics , florida state university 214 oceanography/statistics building , florida state university department of mathematics, george mason university department of statistics , george mason university mathematical sciences, george washington university dept of eng mgmnt & sys eng , george washington university dept of stat , georgetown university department of mathematics and statistics, georgia institute of technology school of indus & sys engr , georgia institute of technology school of mathematics, georgia southern university, jiann-ping-hsu college of public health hiann ping-shu coll of public health , georgia state university department of mathematics and statistics, harvard univ, school of public health biostatistics dept , harvard university department of statistics , harvard university school of engineering & applied sciences , harvard university math dept department of mathematics, howard university dept of math , icahn school of medicine at mount sinai biomath sci dept , idaho state university dept of mathematics and statistics , illinois institute of technology appl math dept , illinois state university mathematics department, indiana university bloomington dept of math , indiana university, bloomington dept of stat , indiana university-purdue university indianapolis dept of mathematical sciences , iowa state university department of mathematics, iowa state university department of statistics, johns hopkins bloomberg school of public health biostatistics, johns hopkins univ, bloomberg school of public health dept of biostatistics, johns hopkins university applied mathematics and statistics, johns hopkins university department of mathematics, johns hopkins university, baltimore dept of applied mathematics and statistics , johns hopkins university, baltimore elec & comp eng dept , kansas state university dept of math, kansas state university dept of statistics , kennesaw state university school of data science and analytics, kent state university department of mathematical sciences, kent state university, geauga and twinsburg academic center mathematical sciences, lehigh university department of mathematics, lehigh university div of appl math & stat , louisiana state university, baton rouge dept of math , louisiana tech university comp analysis & modeling prog, louisiana tech university prog of math & stat , lsu health sciences center, new orleans dept of biostats, marquette university mathematical and statistical sciences, massachusetts institute of technology center for computational science and engineering, massachusetts institute of technology department of mathematics, massachusetts institute of technology oper res ctr , massachusetts institute of technology or/stat grp, mgmt sci , mcgill university dept of math & stat , mcmaster university dept of math & stat , medical university of south carolina department of public health sciences, medical university of south carolina dept of public health sciences, memorial univ of newfoundland math & stat dept , michigan state university dept of math, michigan state university dept of stat & probability, michigan tech university department of mathematical sciences , middle tennessee state university dept of math sci , mississippi state university dept of math & stat , missouri university of science & technology math & stat dept , montana state university dept of math sci, montclair state university department of mathematics, morgan state university dept of math , naval postgraduate school department of applied mathematics, new jersey institute of technology department of mathematical sciences, new mexico institute of mining & technology dept of math, new mexico state university, las cruces dept of mathematical sciences, new york university dept of math , new york university, courant institute department of mathematics , new york university, stern school of business ioms-statistics grp , north carolina state university department of mathematics , north carolina state university deptartment of statistics, north dakota state university, fargo department of statistics , north dakota state university, fargo mathematics department, northeastern university department of mathematics, northern illinois university dept of math sci , northwestern university department of statistics , northwestern university engineering sciences & applied mathematics dept , northwestern university industrial engineering & management sciences, oakland university dept of mathematics and statistics , ohio st univ, columbus math biosciences inst , ohio state university, columbus dept of statistics, ohio state university, columbus integrated systems engineering , ohio university department of mathematics, oklahoma state university department of mathematics, oklahoma state university dept of stat , old dominion university dept of math & stat, old dominion university mathematics and statistics, oregon state university department of mathematics, oregon state university dept of statistics , penn state university, university park department of mathematics, pennsylvania state university, monto alto dept of math , pennsylvania state university, university park dept of statistics , portland state university fariborz maseeh dept of math & stat, prairie view a & m university mathematics department , princeton university dept of mathematics , princeton university mathematics, princeton university prog in applied & comp math, purdue university dept of math , purdue university dept of statistics, purdue university school of industrial engineering, rensselaer polytechnic institute dept of math sci , rice university computational & applied mathematics (caam) , rice university mathematics, rochester institute of technology school of mathematics and statistics, royal military coll of canada math & comp sci dept , rutgers school of public health biostatistics and epidemiology, rutgers school of public health dept of biostats, rutgers the state university of new jersey new brunswick dept of stat & biostat , rutgers the state university of new jersey new brunswick rutcor-rutgers ctr for oper res , rutgers university-newark dept of math & comp sci., rutgers, the state university of new jersey mathematics department, ryerson university department of mathematics, saint louis university department of mathematics and statistics, saint louis university college for public health & social justice dept of epid & biostats , san diego state university dept of math & stat , simon fraser university department of statistics & actuarial science , simon fraser university dept of math , south dakota state university department of mathematics and statistics, southern illinois university carbondale school of mathematical and statistical sciences, southern methodist university department of mathematics, southern methodist university stat sci dept , stanford university department of mathematics, stanford university dept of management sci & engr , stanford university dept of statistics , stanford university statistics, stevens institute of technology dept of math sci , stony brook university applied mathematics and statistics, stony brook university dept of math , stony brook university institute for mathematical sciences , stony brook university mathematics, syracuse university dept of math , temple university department of mathematics, texas a & m university dept of math, texas a&m university dept of statistics, texas christian university dept of math, texas state university department of mathematics, texas tech university dept of math and stat , the medical college of wisconsin, inc. division of biostatistics, the ohio state university department of mathematics, the ohio state university statistics, the university of chicago phd program in mathematics, the university of texas at arlington department of mathematics, the university of texas at austin oden institute for computational engineering and sciences, tufts university department of mathematics, tulane university biostat & data science , tulane university department of mathematics , union college graduate management inst , univ of british columbia department of statistics , univ of british columbia okanagan math & stat dept , univ of calgary dept of math & stats , univ of calgary div of appl math , univ of calgary div of pure math , univ of calif, santa cruz department of statistics , univ of kansas medical center department of biostatistics , univ of kentucky math dept, univ of manitoba dept of statistics , univ of manitoba warren ctr for act stud & res , univ of missouri-columbia math education , univ of pennsylvania applied math & computational sci , univ of pennsylvania epidemiology, biostatistics and informatics , univ of toronto statistical sciences , univ of waterloo dept of appl math , univ of western ontario dept of appl math , univ of western ontario dept of stat & actuarial sci , université du québec à montréal department of mathematics , university at albany, state university of new york department of mathematics & statistics, university at buffalo-suny dept of biostatistics , university at buffalo-suny dept of mathematics , university laval département de mathématiques et de statistique , university of alabama appl stat program , university of alabama department of mathematics, university of alabama at birmingham department of mathematics, university of alabama at birmingham dept of biostat , university of alabama-huntsville dept of math sci, university of alabama-tuscaloosa information systems, statistics, and management science, university of alaska fairbanks department of mathematics and statistics, university of albany, suny dept of epidemiology & biostat , university of alberta dept of math & stat sci, university of arizona department of mathematics, university of arizona program in applied mathematics gidp , university of arizona statistics gidp , university of arizona, mel & enid zuckerman college of public health dept of biostats, coll of public health , university of arkansas at fayetteville dept of math sci, university of arkansas at little rock department of mathematics and statistics, university of british columbia dept of math, university of calgary mathematics and statistics, university of california santa cruz applied mathematics, university of california santa cruz mathematics, university of california, berkeley biostatistics , university of california, berkeley department of statistics, university of california, berkeley dept of math , university of california, berkeley group in logic and the methodology of science, university of california, davis department of statistics, university of california, davis dept of mathematics & graduate group in applied mathematics, university of california, irvine department of statistics , university of california, irvine dept of math , university of california, los angeles biomathematics dept , university of california, los angeles department of statistics and data science, university of california, los angeles dept of biostatistics , university of california, merced applied mathematics department, university of california, riverside department of mathematics, university of california, riverside dept of statistics , university of california, san diego dept of mathematics , university of california, santa barbara dept of mathematics , university of california, santa barbara dept of stat & appl probability , university of california, santa cruz applied mathematics , university of california, santa cruz dept of math , university of central florida department of mathematics, university of central florida department of statistics & data science , university of chicago department of statistics, university of chicago dept of math, university of cincinnati department of mathematical sciences, university of cincinnati mathematical sciences, university of cincinnati operations, business analytics & is , university of cincinnati, medical college division of biostatistics & bioinformatics , university of colorado, boulder dept of appl math , university of colorado, boulder dept of math , university of colorado, boulder mathematics, university of colorado, colorado springs dept of mathematics, university of colorado, denver biostat & informatics dept , university of colorado, denver department of mathematical and statistical sciences, university of connecticut statistics department, university of connecticut, storrs department of mathematics, university of connecticut, storrs department of statistics, university of delaware department of mathematical sciences, university of delaware dept of mathematical sciences, university of denver department of mathematics , university of florida department of statistics, university of florida dept of math , university of florida college of public health dept of biostatistics, university of georgia dept of stat , university of georgia mathematics department, university of guelph dept of math & stat , university of hawai'i at manoa department of mathematics, university of houston dept of math, university of illinois at chicago epid & biostat div , university of illinois at chicago math, stat & comp sci dept, university of illinois at urbana-champaign department of mathematics, university of illinois at urbana-champaign dept of mech sci & engr , university of iowa applied mathematical and computational sciences program, university of iowa department of statistics and actuarial science, university of iowa dept of biostatistics, university of iowa dept of math, university of kansas dept of mathematics, university of kentucky dept of mathematics, university of kentucky dr. bing zhang department of statistics, university of kentucky college of public health college of public health, university of kentucky college of public health* dept of biostatistics, university of lethbridge department of mathematics and computer science , university of louisiana at lafayette dept of math , university of louisville department of mathematics, university of louisville dept of bioinform & biostats , university of manitoba dept of math , university of manitoba mathematics, university of maryland decision, operations & information technologies, university of maryland baltimore county mathematics and statistics, university of maryland college park applied mathematics and statistics, & scientific computation program, university of maryland, college park department of mathematics, university of maryland, college park measure stat & evaluation program , university of massachusetts amherst dept of mathematics & statistics, university of massachusetts, amherst biostatistics & epidemiology department, university of memphis department of mathematical sciences, university of memphis dept of math sci , university of miami department of mathematics, university of miami dept of math , university of miami mgmnt sci dept , university of michigan department of statistics, university of michigan dept of biostatistics, university of michigan dept of math, university of minnesota div of biostat, sph, university of minnesota school of mathematics, university of minnesota school of statistics, university of mississippi dept of math , university of missouri department of mathematics, university of missouri-columbia dept of stat , university of missouri-kansas city dept of math & stat, university of missouri-st louis supply chain & analytics, university of montana-missoula department of mathematical sciences, university of montreal dept of math & stat, university of nebraska - lincoln mathematics, university of nebraska medical center department of biostatistics, university of nebraska-lincoln department of mathematics, university of nebraska-lincoln dept of stat , university of nevada, las vegas dept of math sciences , university of nevada, reno dept of math & stat, university of new brunswick, fredericton math & stat dept , university of new brunswick, saint john mathematics and statistics, university of new hampshire dept of math & stat , university of new mexico dept of math & stat, university of new mexico mathematics and statistics, university of new orleans mathematics, university of north carolina at chapel hill dept of math , university of north carolina at chapel hill dept of stat & oper res, university of north carolina at charlotte department of mathematics & statistics, university of north carolina at charlotte department of mathematics and statistics, university of north carolina at charlotte mathematics and statistics, university of north carolina at greensboro dept of math & stat , university of north texas department of mathematics , university of north texas school of public health dept of biostats , university of northern colorado school of math sci , university of notre dame appl and comp math & stat , university of notre dame department of mathematics, university of oklahoma deptartment of mathematics, university of oklahoma, health science center biostat & epidemiology dept , university of oregon department of mathematics, university of ottawa dept of math & stat , university of pennsylvania dept of mathematics , university of pennsylvania dept of stat , university of pittsburgh department of biostatistics, university of pittsburgh department of statistics, university of pittsburgh dept of mathematics, university of regina department of mathematics and statistics, university of rhode island dept of mathematics, university of rochester department of biostatistics and computational biology, university of rochester dept of math , university of saskatchewan department of mathematics & statistics, university of sherbrooke math dept , university of south carolina department of mathematics, university of south carolina, columbia dept of epidemiology & biostatistics, university of south carolina, columbia dept of statistics , university of south florida dept of epidem & biostats , university of south florida dept of math & stat , university of southern california dept of mathematics , university of southern mississippi school of mathematics and natural sciences, university of tennessee dept of business analytics and statistics, university of tennessee mgmnt sci program , university of tennessee at chattanooga department of mathematics, university of tennessee, knoxville dept of math, university of texas at arlington dept of math , university of texas at austin computational science, engineering, and mathematics, university of texas at austin dept of math , university of texas at dallas department of mathematical sciences, university of texas at el paso dept of math sci , university of texas rio grande valley school of mathematical & statistical sciences, university of texas rio grande valley school of mathematical and statistical sciences, university of texas-school of public health department of biostatistics , university of toledo dept of mathematics & statistics, university of toronto department of mathematics, university of utah dept of mathematics, university of vermont math & stat dept , university of victoria dept of mathematics & statistics, university of virginia department of statistics, university of virginia dept of math, university of virginia mathematics, university of washington biostatistics dept , university of washington department of applied mathematics, university of washington department of mathematics, university of washington department of statistics, university of waterloo applied mathematics, university of waterloo combinatorics & optimization, university of waterloo department of pure mathematics, university of waterloo dept of statistics & actuarial science, university of western ontario department of mathematics , university of windsor department of mathematics & statistics, university of wisconsin milwaukee mathematical sciences, university of wisconsin, madison department of mathematics, university of wisconsin, madison dept of industrial & systems engineering, university of wisconsin, madison statistics dept , university of wisconsin, milwaukee mathematical sciences, university of wyoming dept of math & statistics , university puerto rico, rio piedras department of mathematics, utah state university dept of mathematics & statistics, vanderbilt univerity department of mathematics, vanderbilt university department of mathematics, vanderbilt university, school of medicine dept biostat , virginia commonwealth university department of mathematics & applied mathematics, virginia commonwealth university statistical sciences & operations research, virginia commonwealth university, medical center dept of biostatistics , virginia polytechnic institute and state university dept of math , virginia polytechnic institute and state university dept of statistics , washington state university dept of math & statistics , washington state university dept of mathematics and statistics, washington university dept of elect & sys engr , washington university in st. louis dept of mathematics and statistics, wayne state university department of mathematics, wesleyan university dept of math & comp sci, west virginia university lane dept csee , west virginia university school of mathematical and data sciences , west virginia university school of mathematical and data sciences, western michigan university department of statistics, western michigan university dept of math , wichita state university math, stats & physics dept, wilfrid laurier university department of mathematics, worcester polytechnic institute department of mathematical sciences, wright state university, dayton mathematics & statistics , yale university biostatistics department, yale university dept of math , yale university dept of statistics and data science , yeshiva university department of mathematical sciences, york univ math & stat dept , york university dept of math & stat .

PhD Qualifying Exams

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

Requirement for students starting in Autumn 2023 and later:  To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas: (i) algebra, (ii) real analysis, (iii) geometry and topology, (iv) applied mathematics. 

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

Topics Covered on the Exams:

Algebra Syllabus

Applied Mathematics Syllabus

 Geometry and Topology Syllabus

Real Analysis Syllabus

Past and Practice Qualifying Exams

Timeline for Completion:

Current Requirement: 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. 

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.

Requirement for students starting 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. 

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

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, Room 384H

Wednesday, April 3rd: Algebra, Room 384I

Thursday, April 4th: Geometry & Topology, Room 384I

Friday, April 5th: Applied Math, Room 384I

© Stanford University . Stanford , California 94305 .

Department of Mathematics

• Prospective Graduate Student FAQ
• Graduate Board Oral Exam
• Graduate Courses
• Qualifying Exams
• Recent PhD Theses
• Graduate Awards

The goal of our PhD program is to train graduate students to become research mathematicians. Each year, an average of five students complete their theses and go on to exciting careers in mathematics both inside and outside of academia.

Faculty research interests in the Johns Hopkins University Department of Mathematics are concentrated in several areas of pure mathematics, including analysis and geometric analysis, algebraic geometry and number theory, differential geometry, algebraic topology, category theory, and mathematical physics. The department also has an active group in data science, in collaboration with the Applied Math Department .

The Department values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

A brief overview of our graduate program is below. For more detailed information, please see the links at the right.

Program Overview

All students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Students making satisfactory progress are guaranteed support for five years. A sixth year is generally possible for students who are on track to complete their Ph.D. and would benefit from the additional year.

PhD candidates take two or three courses per semester over the first several years of the program. These are a mix of required and intermediate-level graduate courses, independent studies, and special topics classes offered by our faculty.

By the beginning of their second year, students are asked to demonstrate competency in algebra and in analysis by passing written qualifying exams in these two broad areas. Students are then expected to choose an advisor, who will supervise their dissertation and also administer an oral qualifying exam to be taken in the second or third year. More specifics about all these requirements are described on the requirements page .

All graduate students are invited to attend weekly research seminars in a variety of topic areas as well as regular department teas and a weekly wine and cheese gathering attended by many junior and senior members of the department. A graduate student lunch seminar series provides an opportunity for our students to practice their presentation skills to a general audience.

PhD students will gain teaching experience as a teaching assistant for undergraduate courses. Most of our students lead two TA sections per week, under the supervision of both the faculty member teaching the course and the director of undergraduate studies. Students wanting more classroom experience (or extra pay) can teach their own sections of summer courses. First-year students are given a reduced TA workload in the spring semester, in preparation for the qualifying exams.

In addition to their stipend, each student is awarded an annual travel allowance to enable them to attend conferences for which limited funding is available or visit researchers at other institutions.

UCI Mathematics

Ph.d program, doctor of philosophy (phd) in mathematics.

To earn a PhD in Mathematics one must satisfy the following requirements:

• Completion of all required coursework
• Completion of required written examinations
• Completion of Advancement to Candidacy Oral Examination & Graduate Division paperwork
• Completion of Teaching Experience
• Submission of Doctoral Dissertation & Graduate Division paperwork

When accepted into the doctoral program, the student embarks on a program of formal courses, seminars, and individual study courses to prepare for the Ph.D. written examinations, advancement to candidacy oral examination, and dissertation.

Upon entering the program, students are expected to take Math 210 (Real Analysis), Math 220 (Complex Analysis) and Math 230 (Algebra), which must be passed with a grade of B or better.  Students must complete these sequences by the end of the second year.

By the start of the second year , students must achieve at least two passes at the M.S. level among four exams in Real Analysis, Complex Analysis, Algebra and Applied Mathematics.  

By the start of the third year , students must achieve at least two passes at the Ph.D. level among four exams in Real Analysis, Complex Analysis, Algebra and Applied Mathematics.

To satisfy the exam requirements, students may take the Comprehensive Exam (offered in the Spring of every year) or the Qualifying Exams (offered before the start of the fall quarter) in these areas. Students may not attempt to take an exam in a particular subject area more than 3 times .  A student who passes a Qualifying examination prior to taking the corresponding course will be exempted from taking the course.

Please Note: Corresponding qualifying exam coursework, MATH 210,220, & 230 cannot be used to satisfy both exam and coursework requirements (i.e. you can’t ‘double dip’).

Some students may require additional background prior to entering Math 210.  This will be determined by assessment prior to the start of the students’ first year by the Vice Chair for Graduate Studies, upon consultation with the graduate studies committee.  Such students will be directed into Math 205 during their first year.  These students may pass one Comprehensive Exam in the area of Analysis in lieu of achieving a M.S. pass on the Qualifying Exam, which must be satisfied prior to the start of the students’ second year. The Comprehensive Exam in Analysis will be offered once per year in the Spring quarter.

By the end of the second year, students must declare a major specialization from the following areas:

• Applied & Computational Mathematics
• Geometry & Topology
• Probability

Students are required to take two series of courses from their chosen area (students who later decide to change their area must also take two series of courses from the new area).  Additionally, all students must take two series of courses outside their declared major area of specialization.  Special topics courses within certain areas of specialization and courses counted toward the M.S. degree, (other than MATH 205), will count toward the fulfillment of the major specialization requirement.

By the beginning of their third year, students must have an advisor specializing in their major area.  With the advisor's aid, one should begin to form a committee for the Advancement to Candidacy PhD oral examination.  This committee will be approved by the Department on behalf of the Dean of Graduate Studies and the Graduate Council and will have five faculty members.  At least one (and at most two), of the committee members must be faculty from outside the Department.  Before the end of the third year, students must have a written proposal, approved by their committee, for the Advancement to Candidacy oral examination.  The proposal should explain the role of at least two series of courses from the student's major area of specialization that will be used to satisfy the Advancement to Candidacy requirements.  The proposal should also explain the role of additional research reading material as well as providing a plan for investigating specific topics under the direction of the student's advisor(s).  Only one of the core courses, MATH 210ABC, 220ABC, and 230ABC may count for the course requirement for Advancement to Candidacy Examinations.

After one meets these requirements, the Graduate Studies Committee recommends to the Dean of Graduate Studies the advancement to candidacy for the PhD. degree.  Students should advance to candidacy by the beginning of their fourth year .  After advancing to candidacy, a student is expected to be fully involved in research toward writing his or her PhD dissertation.  Ideally, a student should keep in steady contact/interaction with their doctoral committee.  Teaching experience and training is an integral part of the PhD program.  All doctoral students are expected to participate in the Department's teaching program, unless otherwise communicated during the admissions process.

The candidate must demonstrate independent, creative research in Mathematics by writing and defending a dissertation that makes a new and valuable contribution to mathematics in the candidate's area of concentration.  Upon advancement to candidacy a student must form a thesis committee, a subcommittee of the advancement examination committee, consisting of at least three total faculty members, chaired by the student's advisor.  The committee guides and supervises the candidate's research, study, and writing of the dissertation; participates in or attends the oral defense of the dissertation; and recommends that the PhD be conferred upon approval of the doctoral dissertation.

The normal time for completion of the PhD is six years , and the maximum time permitted is seven years (please note the department may only provide financial support for a maximum of six years ). 

Completion of the PhD degree must occur within 9 quarters of Advancement to PhD candidacy.

Areas of Specialization and Their Corresponding Advancement to Candidacy Courses

PhD students will choose one specialization from the following six areas, as offered by the Mathematics Department, which determines coursework requirements.  Each area of specialization will have a core course, which the Department will do its best to offer each year.  The department will offer other courses every other year, or more frequently depending on student demands and other department priorities.

Algebra : Math 230ABC (core), Math 232ABC, Math 233ABC, 234ABC, 235ABC, 239ABC

Analysis : Math 210ABC (core), Math 220ABC (core), Math 211ABC, Math 260ABC, Math 295ABC, Math 296

Applied & Computational Mathematics: Math 290ABC (core), Math 225ABC, Math 226ABC, Math 227AB, Math 291ABC, Math 295ABC

Geometry & Topology: Math 218ABC (core), Math 222ABC, Math 240ABC, Math 245ABC, Math 250ABC

Logic : Math 280ABC (core), Math 281ABC, Math 282ABC, Math 285ABC

Probability : Math 210ABC, Math 211ABC, Math 270ABC, Math 271ABC, Math 272ABC, Math 274

*PhD Requirements Summarized*

By the beginning of the 2nd year: Pass at the MS level two exams in real analysis, complex analysis, algebra or applied math.

By end of the 2nd year: (1) Declare a major specialization; (2) complete the course series 210ABC, 220ABC, 230ABC.

By the beginning of the 3rd year : (1) Pass at the PhD level two qualifying exams in real analysis, complex analysis, algebra or applied math; (2) Select an advisor specialist in the major area and form a committee for the Advancement to Candidacy oral exam.

Before the end of the 3rd year: (1) Have a written proposal, approved by the committee, for the PhD Advancement to Candidacy examination.

By the beginning of the 4th year: (1) Advanced to Candidacy at the PhD level; (2) form a thesis committee (that is, a subcommittee of the advancement examination committee)

Completion of the PhD:  Average completion time is 5.8 years ; maximum time permitted is seven years . The Department will not financially support students past their sixth year in the PhD program.  Completion of the PhD degree must occur within 9 quarters (three years) of advancement to PhD candidacy.

Graduate Program in Mathematical and Computation Biology (MCSB)

The graduate program in Mathematical, Computational Systems Biology (MCSB) is designed to meet to meet the interdisciplinary training challenges of modern biology and function in concert with selected department programs, including the Ph.D. in Mathematics.

http://mcsb.uci.edu/

• Admission Policies
• Financial Support
• Ph.D. in Atmosphere Ocean Science
• M.S. at Graduate School of Arts & Science
• M.S. at Tandon School of Engineering
• Current Students

Ph.D. in Mathematics, Specializing in Applied Math

Table of contents, overview of applied mathematics at the courant institute.

• PhD Study in Applied Mathematics
• Applied math courses

Applied mathematics has long had a central role at the Courant Institute, and roughly half of all our PhD's in Mathematics are in some applied field. There are a large number of applied fields that are the subject of research. These include:

• Atmosphere and Ocean Science
• Biology, including biophysics, biological fluid dynamics, theoretical neuroscience, physiology, cellular biomechanics
• Computational Science, including computational fluid dynamics, adaptive mesh algorithms, analysis-based fast methods, computational electromagnetics, optimization, methods for stochastic systems.
• Data Science
• Financial Mathematics
• Fluid Dynamics, including geophysical flows, biophysical flows, fluid-structure interactions, complex fluids.
• Materials Science, including micromagnetics, surface growth, variational methods,
• Stochastic Processes, including statistical mechanics, Monte-Carlo methods, rare events, molecular dynamics

PhD study in Applied Mathematics

PhD training in applied mathematics at Courant focuses on a broad and deep mathematical background, techniques of applied mathematics, computational methods, and specific application areas. Descriptions of several applied-math graduate courses are given below.

Numerical analysis is the foundation of applied mathematics, and all PhD students in the field should take the Numerical Methods I and II classes in their first year, unless they have taken an equivalent two-semester PhD-level graduate course in numerical computing/analysis at another institution. Afterwards, students can take a number of more advanced and specialized courses, some of which are detailed below. Important theoretical foundations for applied math are covered in the following courses: (1) Linear Algebra I and II, (2) Intro to PDEs, (3) Methods of Applied Math, and (4) Applied Stochastic Analysis. It is advised that students take these courses in their first year or two.

A list of the current research interests of individual faculty is available on the Math research page.

Courses in Applied Mathematics

The following list is for AY 2023/2024:

--------------------------------------

(MATH-GA.2701) Methods Of Applied Math

Fall 2023, Oliver Buhler

Description:  This is a first-year course for all incoming PhD and Masters students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations.

Prerequisites : Elementary linear algebra, ordinary differential equations; at least an undergraduate course on partial differential equations is strongly recommended.

(MATH-GA.2704) Applied Stochastic Analysis

Spring 2024, Jonathan Weare

This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective.  Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.

Prerequisites: Basic Probability (or equivalent masters-level probability course), Linear Algebra (graduate course), and (beginning graduate-level) knowledge of ODEs, PDEs, and analysis.

(MATH-GA.2010/ CSCI-GA.2420) Numerical Methods I

• Fall 2023, Benjamin Peherstorfer

Description:   This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, and nonlinear optimization, and iterative methods. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

Prerequisites:   A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language).

(MATH-GA.2020 / CSCI-GA.2421) Numerical Methods II

Spring 2024, Aleksandar Donev

This course (3pts) will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments in MATLAB/Python will form an essential part of the course. The course will introduce students to numerical methods for (approximately in this order):

• The Fast Fourier Transform and pseudo-spectral methods for PDEs in periodic domains
• Ordinary differential equations, explicit and implicit Runge-Kutta and multistep methods, IMEX methods, exponential integrators, convergence and stability
• Finite difference/element, spectral, and integral equation methods for elliptic BVPs (Poisson)
• Finite difference/element methods for parabolic (diffusion/heat eq.) PDEs (diffusion/heat)
• Finite difference/volume methods for hyperbolic (advection and wave eqs.) PDEs (advection, wave if time permits).

Prerequisites

This course requires Numerical Methods I or equivalent graduate course in numerical analysis (as approved by instructor), preferably with a grade of B+ or higher.

( MATH-GA.2011 / CSCI-GA 2945) Computational Methods For PDE

Fall 2023, Aleksandar Donev & Georg Stadler

This course follows on Numerical Methods II and covers theoretical and practical aspects of advanced computational methods for the numerical solution of partial differential equations. The first part will focus on finite element methods (FEMs), and the second part on finite volume methods (FVMs) including discontinuous Galerkin (FE+FV) methods. In addition to setting up the numerical and functional analysis theory behind these methods, the course will also illustrate how these methods can be implemented and used in practice for solving partial differential equations in two and three dimensions. Example PDEs will include the Poisson equation, linear elasticity, advection-diffusion(-reaction) equations, the shallow-water equations, the incompressible Navier-Stokes equation, and others if time permits. Students will complete a final project that includes using, developing, and/or implementing state-of-the-art solvers.

In the Fall of 2023, Georg Stadler will teach the first half of this course and cover FEMs, and Aleks Donev will teach in the second half of the course and cover FVMs.

A graduate-level PDE course, Numerical Methods II (or equivalent, with approval of syllabus by instructor(s)), and programming experience.

• Elman, Silvester, and Wathen: Finite Elements and Fast Iterative Solvers , Oxford University Press, 2014.
• Farrell: Finite Element Methods for PDEs , lecture notes, 2021.
• Hundsdorfer & Verwer: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations , Springer-Verlag, 2003.
• Leveque: Finite Volume Methods for Hyperbolic Problems , Cambridge Press, 2002.

-------------------------------------

( MATH-GA.2012 ) Immersed Boundary Method For Fluid-Structure Interaction

Not offered AY 23/24.

The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry.  It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, swimming of creatures large and small, and the flight of insects.  Non-biological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems. Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables, with interaction equations involving the Dirac delta function; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite-difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; a simple but effective method for adding mass to an immersed boundary; numerical simulation of rigid immersed structures or immersed structures with rigid parts; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells.  Some recent developments to be discussed include stability analysis of the IB method and a Fourier-Spectral IB method with improved boundary resolution.

Course requirements include homework assignments and a computing project, but no exam.  Students may collaborate on the homework and on the computing project, and are encouraged to present the results of their computing projects to the class.

Prerequisite:   Familiarity with numerical methods and fluid dynamics.

(MATH-GA.2012 / CSCI-GA.2945) :  High Performance Computing

Not offered AY 23/24

This class will be an introduction to the fundamentals of parallel scientific computing. We will establish a basic understanding of modern computer architectures (CPUs and accelerators, memory hierarchies, interconnects) and of parallel approaches to programming these machines (distributed vs. shared memory parallelism: MPI, OpenMP, OpenCL/CUDA). Issues such as load balancing, communication, and synchronization will be covered and illustrated in the context of parallel numerical algorithms. Since a prerequisite for good parallel performance is good serial performance, this aspect will also be addressed. Along the way you will be exposed to important tools for high performance computing such as debuggers, schedulers, visualization, and version control systems. This will be a hands-on class, with several parallel (and serial) computing assignments, in which you will explore material by yourself and try things out. There will be a larger final project at the end. You will learn some Unix in this course, if you don't know it already.

Prerequisites for the course are (serial) programming experience with C/C++ (I will use C in class) or Fortran, and some familiarity with numerical methods.

(MATH-GA.2011) Monte Carlo Methods

Fall 2023, Jonathan Weare and Jonathan Goodman

Topics : The theory and practice of Monte Carlo methods. Random number generators and direct sampling methods, visualization and error bars. Variance reduction methods, including multi-level methods and importance sampling. Markov chain Monte Carlo (MCMC), detailed balance, non-degeneracy and convergence theorems. Advanced MCMC, including Langevin and MALA, Hamiltonian, and affine invariant ensemble samplers. Theory and estimation of auto-correlation functions for MCMC error bars. Rare event methods including nested sampling, milestoning, and transition path sampling. Multi-step methods for integration including Wang Landau and related thermodynamic integration methods. Application to sampling problems in physical chemistry and statistical physics and to Bayesian statistics.

Required prerequisites:

• A good probability course at the level of Theory of Probability (undergrad) or Fundamentals of Probability (masters)
• Linear algebra: Factorizations (especially Cholesky), subspaces, solvability conditions, symmetric and non-symmetric eigenvalue problem and applications
• Working knowledge of a programming language such as Python, Matlab, C++, Fortran, etc.
• Familiarity with numerical computing at the level of Scientific Computing (masters)

Desirable/suggested prerequisites:

• Numerical methods for ODE
• Applied Stochastic Analysis
• Familiarity with an application area, either basic statistical mechanics (Gibbs Boltzmann distribution), or Bayesian statistics

(MATH-GA.2012 / CSCI-GA.2945) Convex & Non Smooth Optimization

Spring 2024, Michael Overton

Convex optimization problems have many important properties, including a powerful duality theory and the property that any local minimum is also a global minimum. Nonsmooth optimization refers to minimization of functions that are not necessarily convex, usually locally Lipschitz, and typically not differentiable at their minimizers. Topics in convex optimization that will be covered include duality, CVX ("disciplined convex programming"), gradient and Newton methods, Nesterov's optimal gradient method, the alternating direction method of multipliers, the primal barrier method, primal-dual interior-point methods for linear and semidefinite programs. Topics in nonsmooth optimization that will be covered include subgradients and subdifferentials, Clarke regularity, and algorithms, including gradient sampling and BFGS, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational. Students may submit a final project on a pre-approved topic or take a written final exam.

Prerequisites: Undergraduate linear algebra and multivariable calculus

Q1: What is the difference between the Scientific Computing class and the Numerical Methods two-semester sequence?

The Scientific Computing class (MATH-GA.2043, fall) is a one-semester masters-level graduate class meant for graduate or advanced undergraduate students that wish to learn the basics of computational mathematics. This class requires a working knowledge of (abstract) linear algebra (at least at the masters level), some prior programming experience in Matlab, python+numpy, Julia, or a compiled programming language such as C++ or Fortran, and working knowledge of ODEs (e.g., an undergrad class in ODEs). It only briefly mentions numerical methods for PDEs at the very end, if time allows.

The Numerical Methods I (fall) and Numerical Methods II (spring) two-semester sequence is a Ph.D.-level advanced class on numerical methods, meant for PhD students in the field of applied math, masters students in the SciComp program , or other masters or advanced undergraduate students that have already taken at least one class in numerical analysis/methods. It is intended that these two courses be taken one after the other, not in isolation . While it is possible to take just Numerical Methods I, it is instead strongly recommended to take the Scientific Computing class (fall) instead. Numerical Methods II requires part I, and at least an undergraduate class in ODEs, and also in PDEs. Students without a background in PDEs should not take Numerical Methods II; for exceptions contact Aleks Donev with a detailed justification.

The advanced topics class on Computational Methods for PDEs follows on and requires having taken NumMeth II or an equivalent graduate-level course at another institution (contact Aleks Donev with a syllabus from that course for an evaluation), and can be thought of as Numerical Methods III.

Q2: How should I choose a first graduate course in numerical analysis/methods?

• If you are an undergraduate student interested in applied math graduate classes, you should take the undergraduate Numerical Analysis course (MATH-UA.0252) first, or email the syllabus for the equivalent of a full-semester equivalent class taken elsewhere to Aleks Donev for an evaluation.
• Take the Scientific Computing class (fall), or
• Take both Numerical Methods I (fall) and II (spring), see Q1 for details. This is required of masters students in the SciComp program .

• Top Colleges
• Top Courses
• Entrance Exams
• Admission 2024
• Study Abroad
• Study in Canada
• Study in UK
• Study in USA
• Study in Australia
• Study in Germany
• IELTS Material
• Scholarships
• Sarkari Exam
• Visual Stories
• College Compare
• Write a review
• Login/ Register
• Login / Register

Ph.D Mathematics Syllabus and Subjects

Updated on - Jan 4, 2023

PhD Mathematics is a three-five-year doctoral program that focuses on familiarizing students with the research to find solutions to mathematical problems. They prioritize practical experience and skills. The PhD Mathematics syllabus is intended to provide students with all of the information they require to meet the demands of the industry. The PhD Mathematics syllabus teaches students about Mathematical Analysis, Finding Statics, Research Methodology, Data and Dynamics, and many more.

PhD Mathematics Semester Wise Syllabus

The PhD Mathematics course syllabus is designed to provide students with an understanding of mathematical advances in research and training. The PhD Mathematics course curriculum is intended to provide an in-depth examination of mathematical patterns in various career opportunities such as Science, Geography, Oceanography, Data Interpretation, and so on. The PhD Mathematics subject list syllabus is divided below into semesters:

PhD Mathematics First Year Syllabus

The table below contains the subjects from the PhD Mathematics first-year syllabus:

PhD Mathematics Second Year Syllabus

The table below contains the subjects from the PhD Mathematics second-year syllabus:

PhD Mathematics Third Year Syllabus

The table below contains the subjects from the PhD Mathematics third-year syllabus:

PhD Mathematics Subjects

The PhD Mathematics course is a two-year study period on the student's chosen specialization in mathematical patterns. The following are the subjects in PhD Mathematics:

• Differential Equation
• Mathematical Finance
• Differential Geometry Mechanics
• Discrete Mathematics
• Metric Space
• Computational Techniques
• English Literature
• Number Theory
• Computer Science
• Linear Programming
• Probability Theory

PhD Mathematics Course Structure

The PhD Mathematics course subject and syllabus cover fourteen topics. The theoretical component of the course focuses on the principles and values of mathematical patterns and mechanics, English Literature, and computers. The course structure initially focuses on familiarizing students with advanced mathematics and training them on the fundamentals of problem-solving patterns. The following topics are covered in the PhD Mathematics course:

• Three years
• Fourteen Subjects
• Projects and Dissertation
• Research Methodology

PhD Mathematics Teaching Methodology and Techniques

The theoretical component of the PhD. Mathematics course subjects and syllabus focuses on the principles and values of mathematical patterns and Mechanics, English Literature, and computers. The course structure is designed to familiarize students with the fundamentals of mathematical patterns through hands-on experience.

• Case Studies
• Paper Presentation

PhD Mathematics Project

The PhD Mathematics program combines theory with project work. The project's goal is to ensure that students are familiar with finding, reasoning, and obtaining solutions to existing mathematical problems. Some of the PhD Mathematics Project topics are as follows:

• Game Theory and Algebra
• Dynamic System and Ergodic Theory
• Geometrics Flows in Hermitian Geometry
• Projects in Computational Topology

PhD Mathematics Reference Books

Various books touch on different topics in PhD Mathematics. These books provide guidelines and basic information on research and its techniques. Listed below are some PhD Mathematics books for reference:

Get Free Scholarship worth 25000 INR

How do birds flock? Researchers do the math to reveal previously unknown aerodynamic phenomenon

Findings have potential applications for transportation and energy.

In looking up at the sky during these early weeks of spring, you may very well see a flock of birds moving in unison as they migrate north. But how do these creatures fly in such a coordinated and seemingly effortless fashion?

Part of the answer lies in precise, and previously unknown, aerodynamic interactions, reports a team of mathematicians in a newly published study. Its breakthrough broadens our understanding of wildlife, including fish, who move in schools, and could have applications in transportation and energy.

"This area of research is important since animals are known to take advantage of the flows, such as of air or water, left by other members of a group to save on the energy needed to move or to reduce drag or resistance," explains Leif Ristroph, an associate professor at New York University's Courant Institute of Mathematical Sciences and the senior author of the paper, which appears in the journal Nature Communications . "Our work may also have applications in transportation -- like efficient propulsion through air or water -- and energy, such as more effectively harvesting power from wind, water currents, or waves."

The team's results show that the impact of aerodynamics depends on the size of the flying group -- benefiting small groups and disrupting large ones.

"The aerodynamic interactions in small bird flocks help each member to hold a certain special position relative to their leading neighbor, but larger groups are disrupted by an effect that dislodges members from these positions and may cause collisions," notes Sophie Ramananarivo, an assistant professor at École Polytechnique Paris and one of the paper's authors.

Previously, Ristroph and his colleagues uncovered how birds move in groups -- but these findings were drawn from experiments mimicking the interactions of two birds. The new Nature Communications research expanded the inquiry to account for many flyers.

To replicate the columnar formations of birds, in which they line up one directly behind the other, the researchers created mechanized flappers that act like birds' wings. The wings were 3D-printed from plastic and driven by motors to flap in water, which replicated how air flows around bird wings during flight. This "mock flock" propelled through water and could freely arrange itself within a line or queue, as seen in a video of the experiment.

The flows affected group organization in different ways -- depending on the size of the group.

For small groups of up to about four flyers, the researchers discovered an effect by which each member gets help from the aerodynamic interactions in holding its position relative to its neighbors.

"If a flyer is displaced from its position, the vortices or swirls of flow left by the leading neighbor help to push the follower back into place and hold it there," explains Ristroph, director of NYU's Applied Mathematics Laboratory, where the experiments were conducted. "This means the flyers can assemble into an orderly queue of regular spacing automatically and with no extra effort, since the physics does all the work.

"For larger groups, however, these flow interactions cause later members to be jostled around and thrown out of position, typically causing a breakdown of the flock due to collisions among members. This means that the very long groups seen in some types of birds are not at all easy to form, and the later members likely have to constantly work to hold their positions and avoid crashing into their neighbors."

The authors then deployed mathematical modeling to better understand the underlying forces driving the experimental results.

Here, they concluded that flow-mediated interactions between neighbors are, in effect, spring-like forces that hold each member in place -- just as if the cars of a train were connected by springs.

However, these "springs" act in only one direction -- a lead bird can exert force on its follower, but not vice versa -- and this non-reciprocal interaction means that later members tend to resonate or oscillate wildly.

"The oscillations look like waves that jiggle the members forwards and backwards and which travel down the group and increase in intensity, causing later members to crash together," explains Joel Newbolt, who was an NYU graduate student in physics at the time of research.

The team named these new types of waves "flonons," which is based on the similar concept of phonons that refer to vibrational waves in systems of masses linked by springs and which are used to model the motions of atoms or molecules in crystals or other materials.

"Our findings therefore raise some interesting connections to material physics in which birds in an orderly flock are analogous to atoms in a regular crystal," Newbolt adds.

The study's other authors included the Courant Institute's Nickolas Lewis, Mathilde Bleu, Jiajie Wu, and Christiana Mavroyiakoumou.

The work was supported by grants from the National Science Foundation (DMS-1847955, DMS-1646339).

• New Species
• Bird Flu Research
• Environmental Issues
• Environmental Science
• Wildlife gardening
• Molecular biology
• May 2003 Tornado Outbreak Sequence
• Overfishing

Story Source:

Materials provided by New York University . Note: Content may be edited for style and length.

Related Multimedia :

• YouTube Video: Flocking Birds Experiment

Journal Reference :

• Joel W. Newbolt, Nickolas Lewis, Mathilde Bleu, Jiajie Wu, Christiana Mavroyiakoumou, Sophie Ramananarivo, Leif Ristroph. Flow interactions lead to self-organized flight formations disrupted by self-amplifying waves . Nature Communications , 2024; 15 (1) DOI: 10.1038/s41467-024-47525-9

Cite This Page :

Explore More

• Food in Sight? The Liver Is Ready!
• Acid Reflux Drugs and Risk of Migraine
• Do Cells Have a Hidden Communication System?
• Mice Given Mouse-Rat Brains Can Smell Again
• How Do Birds Flock? New Aerodynamics
• Cancer: Epigenetic Origin Without DNA Mutation
• Climate Change Driving Biodiversity Loss
• Why Can't Robots Outrun Animals?
• Evolution of Gliding in Marsupials
• Novel One-Dimensional Superconductor

Trending Topics

Strange & offbeat.