phd maths subjects

Ph.D. Degree Programs

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

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

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

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

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

Ph.D. in Mathematics

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

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

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

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

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

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

Written Qualifying Examinations

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

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

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

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

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

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

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

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

Master's Transferring to Ph.D.

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

Foreign Language Requirement

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

Advancement to Candidacy

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

Dissertation and Final Defense

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

Time Limits

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

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

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

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PhD in Mathematics

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

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

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

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

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

The Coursework

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

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

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

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

Breadth requirements.  The students must demonstrate:

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

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

• Area I subjects:​

• Area II subjects:​

Special Topics and Reading Courses.

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

Credit Transfers

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

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

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

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

Comprehensive Examinations

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

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

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

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

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

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

Dissertation and Defense

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

PhD Qualifying Exams

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

Requirements for the Qualifying Exams

Students who entered the program prior to autumn 2023.

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

Students who entered the program in Autumn 2023 or later

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

• real analysis
• geometry and topology
• applied mathematics

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

Topics Covered on the Exams:

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

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

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

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

Students who started in Autumn 2023 and later

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

Exam Schedule

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

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

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

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

Upcoming Exam Dates

Autumn 2024.

The exams will be held during the week of September 16th. The date for each exam will be posted in mid-August. 

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Research Programmes

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The Faculty of Mathematics offers three doctoral (PhD) and one MPhil research programmes.

Select a course below to visit the University’s Course Directory where you can read about the structure of the programmes, fees and maintenance costs, entry requirements and key deadlines.

Research Areas and Potential Supervisors

Determining whether your interests and ambitions align with our research and expertise is a vital part of the application and admissions process. When we receive your formal application, we will consider the information you provide on your research interests carefully, alongside other factors such as your academic suitability and potential, how you compare to other applicants in the field, and whether we have a suitable academic supervisor with the capacity to take on new students.

We are committed to widening participation in mathematical research at Cambridge. We welcome and encourage applications from people from groups underrepresented in postgraduate study.

Before making an application to study with us we recommend you:

• Investigate our areas of research and consider how they fit with your interests and ambitions.

A list of broad research areas is provided below, together with links to further information. Your interests may span more than one area. On your application form you will be asked to indicate at least one broad area of interest. This is to help us direct your application to the most suitable group of people to review it.

• Identify 2 or 3 appropriate supervisor(s) with whom you might work.

The information linked below will take you to lists of supervisors working in each broad research area, with an indication of their availability. You are encouraged to make informal contact with potential supervisors prior to making an application. Initial contact should be made by email. In your email we recommend you provide a concise explanation of your areas of interest, how your research interests align with the supervisor(s) research, and that you highlight any relevant work you have done in this area. We recommend that you attach an up-to-date CV. The purpose of this contact is to enquire on supervisor capacity and willingness to supervise, and to see if there is a good fit between your interests and theirs.

If you haven’t had a response to an informal enquiry, you are still welcome to apply and list the individual concerned on your application form, although you may also wish to consider other options.

• Give some thought to your intended research and why you want to study with us.

On your application form you will be asked to submit a short research summary, details of your research experience and your reasons for applying to undertake a PhD/MPhil with us. Whilst you are not expected to submit a detailed research proposal at any stage of the process, we do want to know that you have considered the areas of research that you wish to pursue.

Research areas

Click on a research area to find out more about available supervisors and their research:

Please note that a  large majority of the successful applicants for PhD studentships with  the High Energy Physics, and General Relativity & Cosmology (GR) groups   will have taken Part III of the Mathematical Tripos.

Funding Opportunities

Each Department works hard to secure funding for as many offer holders as possible, either from within its own funds, in collaboration with funding partners, or via the University Postgraduate Funding Competition. However, funding is not guaranteed via these routes, and you should investigate funding opportunities early in the process to be sure that you can meet advertised deadlines.

All application deadlines are 23:59pm (midnight) UK time on the stated date. So that your application can be given full consideration please apply by the following deadlines:

Note for PhD applicants:

We will accept applications for an October start up until the general University deadline in May, but your chances of obtaining funding are significantly reduced. In addition, space limitations may mean that late applications cannot be considered (i.e., the most appropriate supervisor may already have committed to taking other students).

Only in exceptional circumstances will we consider admission to a later start date in the academic year (i.e., January or April). If you intend to apply for a later start date please contact us at [email protected] so we can advise you on the feasibility of your plan.

Note for MPhil applicants:

We will accept applications until the general University deadline in February, but you will not be considered for funding. In addition, space limitations may mean that late applications cannot be considered (i.e., the most appropriate supervisor may already have committed to taking other students).

Most interviews are expected to take place in the second half of January.

The purpose of the interview is to try to ascertain the extent of the applicant's relevant knowledge and experience, and to gauge whether their interests and abilities align with the research of the potential supervisor and/or research group. It will most likely consist of a discussion of your background and motivations for applying to the course, as well as some questions on relevant topics.

Not all applicants will be selected for interview.

If you are selected for interview, you will be contacted by email at the address you provided on your application. The email should confirm:

• the location of the interview (it may be in-person or on-line dependent upon interviewer availability, your distance from Cambridge, as well as individual preferences),
• the interview format and whether you should prepare anything specific in advance,
• the approximate duration of the interview,
• who you will be meeting.

Prior to interview you may declare a disability, serious health problem or caring responsibility which may require reasonable adjustments for the interview to be made.

Due to interviewer availability and the tight admissions timetable, we can usually only rearrange the time and date of your interview under exceptional circumstances.

Decision timeline

Both DAMTP and DPMMS make most of their PhD/MPhil admissions decisions for October entry in January and early February, and you should not expect to receive a decision on your application before mid-February (even if you apply much earlier). We expect to have made decisions on all applications by mid-July. The Department makes every effort to take decisions on applications at the earliest opportunity. In some cases, however, it may take some time for a decision to be made. Applications may need to be viewed by several potential supervisors before a final decision can be reached.

To consider your application formally we must receive a complete application form, together with all supporting documents, by the deadline.

Communication of outcomes

You will be notified of the formal outcome of your application via the Applicant Portal.

Following an interview, you can normally expect to receive notification of the outcome within a week or two.

If you are successful, the University’s Postgraduate Admissions Office will issue a formal offer of admission which will outline all your conditions. As processing times can vary, we may also contact you informally to notify you of our decision.

We do not provide formal feedback to applicants who are unsuccessful at either the application or interview stage.

Take a look at our frequently asked questions for PhD applicants.

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Ph.D. Program in Mathematics

Degree requirements.

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

NYU Shanghai Ph.D. Track

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

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

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

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

Probability

PDE/Analysis

The Written Comprehensive Examination

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

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

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

The Oral Preliminary Examination

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

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

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

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

The Dissertation Defense

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

Summer Internships and Employment

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

Mentoring and Grievance Policy

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

Visiting Doctoral Students

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

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Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2023   2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Reham Alahmadi - Asymptotic Study of Toeplitz Determinants with Fisher-Hartwig Symbols and Their Double-Scaling Limits

Anne Sophie Rojahn –  Localised adaptive Particle Filters for large scale operational NWP model

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

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PhD in Mathematics

• Updated on  
• Apr 24, 2023

Do you love calculations and solving big mathematical problems? Dating back to ancient times, Mathematics as a field has witnessed immense changes and growth with technological advancements. Carl Friedrich Gauss, a famous mathematician, called this field the ‘queen of sciences’ because the mathematical principles and theories are used in multifarious disciplines like Sciences , Finance , Engineering , Medicine and Social Sciences . From calculating and measuring to the development of a multitude of theories, laws, and patterns, pursuing a career after BSc Maths can be highly rewarding. In this blog, we will shed light on various elements of the PhD in Mathematics program and will provide insightful knowledge on the same. 

This Blog Includes:

What is phd in mathematics, why study phd in mathematics, top phd in mathematics specializations, phd in mathematics syllabus, eligibility criteria for phd in mathematics, top universities offering phd in mathematics , top colleges in india , career opportunities and salaries , phd in mathematics vs phd in economics.

Relying heavily on the practical side, the students in the doctoral program are familiarised with mathematical logic and analysis, statistical, topology and stochastic processes. Running for a duration of 3-5 years, the doctorate program imparts advanced knowledge in the field of Mathematics and equips students with skills that can be used to apply and solve complex real-life problems. Not just in the education sector, but a PhD in Mathematics opens the door to a multitude of career opportunities in the corporate and other sectors of the economy. 

Many opportunities in research institutes and universities are available for candidates who are interested in a research career. And for those who want to teach, there are lots of well-paying teaching opportunities available in private engineering institutions. Many international businesses research laboratories, financial services companies, and others are aggressively hiring Indian mathematicians. Some of the most common reasons why a PhD in Mathematics is a popular choice among students are:

• This curriculum trains students to keep up with the growth frontiers of knowledge and provides research skills relevant to the country’s current social and economic objectives.
• It learns how to create an effective research report and how to show facts graphically.
• Accountancy and commercial services, finance, investing and insurance, and government and public administration are additional options.

Ranging from Computational Sciences and Statistics to Natural Sciences, PhD in Mathematics offers an array of career opportunities in the research field. Being the heart and soul of modern scientific questions, the doctorate program helps contemporary inventions in today’s generation. Here are some of the most popular specialization programs that you can opt for: 

• PhD in Mathematical and Computer Sciences
• PhD in Mathematical Sciences
• PhD in Mathematical Education
• PhD in Mathematical Statistics
• PhD in Computational Mathematics and Decision Sciences
• PhD in Mathematical Modeling
• PhD in Natural and Mathematical Sciences
• PhD in Applied Mathematics
• PhD in Statistics

Although the PhD in Mathematics course curriculum differs per college, it mostly comprises certain common core courses from which students can choose based on their interests. The following is a list of common subjects and subjects covered in the syllabus:

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

To take admission in the choice of course, the students have to fulfil certain eligibility criteria as mentioned by the university. Although different educational institutes have their own set of prerequisites, here are some of the most common parameters that you must satisfy in order to get enrolled in a PhD in Mathematics course:

• A 3 years undergraduate degree in a field related to mathematics followed by a postgraduate degree like MSc Mathematics or a 4-year undergraduate honours degree in the field which provides relevant quantitative training to the students such as Mathematics, Engineering, Computer Science, Statistics, Physics, etc.
• A valid English language proficiency test scorecard like IELTS , TOEFL or PTE . 
• GRE scorecard, if needed by the university.
• A passing score in the entrance exam conducted by the university, if any.

Apart from the certificate documents of the aforementioned criteria, the applicants also have to submit university transcripts, Letter of Recommendation (LOR), a Statement of Purpose ( SOP ), Curriculum Vitae or a Resume , and other documents as mentioned by the university. 

Providing the best in class infrastructure, a highly qualified faculty and industrial exposure essential to building a successful career ahead, the universities mentioned below are popular choices when it comes to pursuing PhD in Mathematics: 

Also Read: PhD Scholarships in UK for Indian Students

The table below lists the top PhD in Mathematics colleges and universities that offer the given programme full-time:

Must Read: IIT Delhi And Queensland University’s Joint PhD Program

PhD in Mathematics is one of the most popular professional options among students. Mathematical graduates have several career prospects both overseas and in India. PhD graduates can work in a variety of mathematical fields, such as Numerical Analysis, Computational Complex Analysis Group, Biomathematics Group, Complexity and Networks, Dynamical Systems, Fluid Dynamics, Mathematical Physics, and so on. Graduates with customer service skills and a basic understanding of the business can work in both private and public sector banks . They can also look for work in market research, public accounting companies, government and private banks, government and private financial sectors, budget planning, consultancies, and businesses, among other places. Some of the most sought-after job prospects for PhD in Mathematics graduates are mentioned below: 

Note: Mathematicians’ employment in India is anticipated to rise by 23-30% due to a surge in demand for knowledge and experience in private sector analytics businesses. The private sector provides more compensation and more opportunities. If they include sophisticated computer abilities and statistical tools in their profile, the package will be increased.

PhD in Mathematics and PhD in Economics both have a promising future in the field of study in a variety of areas. Many colleges in India and abroad choose both courses as part of their academic framework. The table below represents the differences between both the options:

PhD in Mathematics takes around 3-5 years to complete.

Doing PhD in Mathematics can open doors for a lot of career options for example – Mathematician, demographer, professor, economist, researcher, etc.

Candidates applying for PhD in Mathematics must have scored at least 60% marks in their class 12th, undergraduate and postgraduate program. Also, if the university conducts an entrance examination then the candidate must score passing marks in that.

Hopefully, you have got an insight into various aspects pertaining to PhD in Mathematics. Are you also looking for opportunities to study abroad ? If the answer is yes, the experts at Leverage Edu can make your journey easier as they will be guiding you throughout the process. To take help from the experts simply register on our website or call us at 1800-572-000.

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I want to become a good professor in maths….i m of 35 years old now…but just going to complete my pg

Please explain the difference between phd from a govt institute and phd from a private institute with pros n cons

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Ph.D Mathematics Syllabus and Subjects

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:

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State Education Department Releases Preliminary Data on English Language Arts, Mathematics and Science State Exams

Testing data to inform instructional decisions and individualized student learning plans for the 2024-25 school year .

The State Education Department today announced that preliminary data on the Grades 3-8 English Language Arts (ELA) and mathematics (Math) assessments, and the Grades 5 and 8 Science assessments, have been released to schools and school districts to provide parents and families with their students’ assessment results and inform instructional decisions and individualized learning plans for students during the 2024-25 school year. The data are considered preliminary until they undergo the local district review and verification process, which will close on September 4, 2024. 

The percentage of students scoring proficient by grade in each subject is shown below. Overall, the 2023-24 state assessment data from public and charter schools show that proficiency rates of students in Grades 3-8 on the ELA and Math assessments are 46% and 52%, respectively; the proficiency rate of students in both Grades 5 and 8 on the Science exam is 35%. Data are subject to change after the local review and verification period has closed and statewide quality checks have been completed.  

These annual tests, required by the federal Every Student Succeeds Act (ESSA), are designed to measure how well students are mastering the learning standards that guide classroom instruction and are a valuable tool to help ensure students have the supports needed to succeed. The annual tests are intended to be one measure used to assess student learning and one tool of many used by educators to ensure students are getting the services and supports they need to prepare for career, college, and civic readiness.   

Following the verification period, NYSED will engage in the labor-intensive process of updating assessment reports based on the data certified by schools and districts. Our goal is to provide teachers, administrators, and parents with as much information as possible about their students’ performance and make it available as quickly as possible to improve classroom instruction. The public release of all final state assessment data is anticipated by November. 

Additional information about the tests is available on the Department’s State Assessment website . 

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Dissertations

Most Harvard PhD dissertations from 2012 forward are available online in DASH , Harvard’s central open-access repository and are linked below. Many older dissertations can be found on ProQuest Dissertation and Theses Search which many university libraries subscribe to.

GCSE results 2024: how did each subject perform?

GCSE results released today show that overall grades were broadly similar to 2023 , including when broken down by subject.

This year’s exams maintained standards from last year, which was the final step in Ofqual’s return to normal exams and grading after the Covid-19 pandemic.

Examiners were asked to ensure the standard of work this year was “broadly similar” to 2023, though to “bear in mind any residual impact of disruption on performance”.

While almost all mitigations have been removed, GCSE students in some subjects were still provided with formulae sheets this year.

Top grades fell between 2022 and 2023 as grading was brought in line with pre-pandemic standards. Some variation in results in subjects between years is always expected.

Here are the headline takeaways from today’s GCSE results, broken down by subject (scroll down or click the links below to go straight to each subject):

• Arts and D&T

The proportion of overall entries receiving a pass or higher (grade 4/C) in maths this year was down slightly from 61 per cent last year to 59.6 per cent in 2024.

This year was the same as the 59.6 per cent who got a grade 4/C or above in 2019.

Meanwhile, 16.7 per cent received a grade 7/A or higher in maths GCSE this year - a decrease from 17.2 per cent who got the top grades in 2023 but above 15.9 per cent in 2019.

The 2023 GCSE results saw an uptick in the number of students failing to achieve grade 4 for English and maths and therefore an increase in those having to resit in November.

However, less than a quarter of the students who took GCSE maths in November 2023 passed - meaning the majority failed their resits .

Only 17.4 per cent of entries from students aged 17 or over achieved a grade 4/C in maths in England this year - slightly up from 16.4 per cent in 2023.  

• GCSE results: English and maths pass rate down
• Why grades 1-3 are not a fail
• How much does attendance affect exam results?
• GCSE resits: everything you need to know  

GCSE English language and literature

Like in maths, achieving that grade 4/C needed to pass English language is key for GCSE students.

This year, 61.6 per cent of entries managed to achieve a passing grade or higher in English language, a fall from last year when 64.2 per cent were awarded grade 4/C or above.

The English language overall pass rate was also slightly below 2019, when 61.8 per cent of entries passed English language.

However, much of this is driven by entries from students aged 17 or over who are likely resitting. In England, 20.9 per cent of entries from these students were awarded a grade 4/C - a drop from 25.9 per cent last year.

Speaking in a Joint Council for Qualifications briefing this morning, Claire Thomson, AQA’s director of regulation and compliance, said the drop in pass rates was “largely around the 17-year-olds and over who are skewing the distributions. If you look at just the 16-year-olds, they are very stable with minimal movement over the years”.

“The 17 and over cohort has grown and come back over pre-pandemic levels, which is altering the results,” she added.

For English literature, 73.7 per cent of overall entries achieved a grade 4/C or higher - down slightly from 73.9 per cent in 2023 but up slightly from 73.4 per cent in 2019.

In literature, 20.1 per cent achieved a grade 7/A or above, and 15.6 per cent in language. This compares with the 20.6 per cent who achieved grade 7/A in literature in 2023 and 16 per cent in language.

This year, teachers with students doing the Romeo and Juliet option in AQA’s English literature paper warned the extract students received was “difficult” and could leave some young people disadvantaged.

GCSE double science, biology, chemistry and physics

In the double science GCSE, the proportion of entries getting top grades (7/A and above) rose to 8.8 per cent from 8.5 per cent in 2023.

Meanwhile, 57.1 per cent of entries received a pass or higher in double science, compared with 56.6 per cent last year.

In the three sciences, the proportion getting the top grades increased slightly for physics and chemistry and remained the same for biology.

Along with GCSE maths, physics and combined science students were allowed to have formulae and equation sheets for another year.

This was to recognise the cohort had “experienced two years of national closures during secondary school”.

In biology, 42 per cent achieved the top grades - compared with 42 per cent in 2023 and 42.3 per cent in 2019.

Meanwhile in physics, 44 per cent achieved the top grades - higher than the 43 per cent in 2023 and 43.8 per cent in 2019.

In chemistry, 44.7 per cent achieved the top grades - higher than the 43.6 per cent in 2023 and 43.9 per cent in 2019.

GCSE Spanish, French and German

The proportion of entries receiving top grades in modern foreign languages rose this year.

In Spanish, 26.7 per cent received a grade 7/A or higher and 69.8 per cent passed with a grade 4/C or higher. These are compared with 26.1 per cent that got the top grades in 2023 and 69.2 per cent that passed.

Adjustments were made to grading standards for French and German GCSEs again this year to better align results with Spanish.

Last year, exam boards were required by Ofqual to award more generously at grades 9, 7 and 4. This year, exam boards were asked to make further positive adjustments at the same grades for GCSE German, and grades 7 and 4 for French.

Considering this, 32.1 per cent achieved a grade 7/A above in German, compared with 27.6 per cent last year, and 28 per cent in French, up from 26 per cent in 2023.

The pass rate also increased. In French, 71.2 per cent were awarded a grade 4/C or higher - up from 70.5 per cent last year and 69.7 in 2019.

In German, 77.5 per cent passed - up from 76.9 per cent in 2023 and 75.8 per cent in 2019.

Just 9.3 per cent achieved a grade 9 in German GCSE last year, though this was up from 5.8 per cent in 2019. This has continued to rise to 10.4 per cent in 2024.

GCSE history and geography

This year, 25.8 per cent of entries scored the top grades (grade 7/A or above) in GCSE history.

This is an increase of 0.5 percentage points from 25.3 per cent in 2023.

In geography, a similar proportion of entries achieved a grade 7/A or above at 24.5 per cent, the same as 24.5 per cent in both 2023 and 2019.

The pass rate in both GCSEs increased very slightly to 63.9 per cent in history and 65 per cent in geography.

GCSE computing

Computing GCSE saw an increase in both the proportion of top grades and the pass rate for 2024.

This year, 28.3 per cent of entries were awarded a grade 7/A or above, and 68.3 per cent a grade 4/C or above.

This is compared with last year when 24.4 per cent of entries received a grade 7/A or higher in computing, and 64.6 per cent made grade 4/C or higher.

These were still slightly above 2019 levels when 21.6 per cent got at least a grade 7/A and 62.6 per cent achieved a grade 4/C or above.

After research into grading standards, Ofqual asked exam boards to award more generously at grades 9, 7 and 4 for computer science this year.

GCSE art and design, drama, music and design and technology

There was a mixed picture for the arts in terms of increases and decreases for 2024 compared with last year.

For art and design, 23.6 per cent got a grade 7/A or above and 76.3 per cent a grade 4/C or above. In 2023, 23.9 per cent of entries achieved a grade 7/A or above and 76.1 per cent got at least a grade 4/C.

Last year in design and technology, 64.8 per cent achieved a grade 4/C or higher, and 21 per cent got the top grades. This year saw some increases, with 22.4 per cent getting the top grades and 65.6 per cent achieving a pass of grade 4/C or better.

Meanwhile, music top grades decreased slightly from 33.3 per cent in 2023 to 32.9 per cent this year. Last year, 76.4 per cent of students received a grade 4/C or higher, and this only decreased slightly to 75.9 per cent this year.

Similarly, in drama for 2024, 76.7 per cent achieved a passing grade. In 2023, 75.9 per cent achieved a grade 4/C and up.

In the top grades, 26.9 per cent of students got at least a grade 7/A this year, a small increase on 2023 when the figure was 25.8 per cent.

GCSE PE saw 23 per cent of entries awarded a grade 7/A or above and 72.1 per cent getting a grade 4/C or above.

This is relatively similar compared to 2023, when 22.7 per cent of entries received the top grades and 72.4 per cent achieved a grade 4/C or higher.

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Toward a code-breaking quantum computer

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The most recent email you sent was likely encrypted using a tried-and-true method that relies on the idea that even the fastest computer would be unable to efficiently break a gigantic number into factors.

Quantum computers, on the other hand, promise to rapidly crack complex cryptographic systems that a classical computer might never be able to unravel. This promise is based on a quantum factoring algorithm proposed in 1994 by Peter Shor , who is now a professor at MIT.

But while researchers have taken great strides in the last 30 years, scientists have yet to build a quantum computer powerful enough to run Shor’s algorithm.

As some researchers work to build larger quantum computers, others have been trying to improve Shor’s algorithm so it could run on a smaller quantum circuit. About a year ago, New York University computer scientist Oded Regev proposed a  major theoretical improvement . His algorithm could run faster, but the circuit would require more memory.

Building off those results, MIT researchers have proposed a best-of-both-worlds approach that combines the speed of Regev’s algorithm with the memory-efficiency of Shor’s. This new algorithm is as fast as Regev’s, requires fewer quantum building blocks known as qubits, and has a higher tolerance to quantum noise, which could make it more feasible to implement in practice.

In the long run, this new algorithm could inform the development of novel encryption methods that can withstand the code-breaking power of quantum computers.

“If large-scale quantum computers ever get built, then factoring is toast and we have to find something else to use for cryptography. But how real is this threat? Can we make quantum factoring practical? Our work could potentially bring us one step closer to a practical implementation,” says Vinod Vaikuntanathan, the Ford Foundation Professor of Engineering, a member of the Computer Science and Artificial Intelligence Laboratory (CSAIL), and senior author of a paper describing the algorithm .

The paper’s lead author is Seyoon Ragavan, a graduate student in the MIT Department of Electrical Engineering and Computer Science. The research will be presented at the 2024 International Cryptology Conference.

Cracking cryptography

To securely transmit messages over the internet, service providers like email clients and messaging apps typically rely on RSA, an  encryption scheme invented by MIT researchers Ron Rivest, Adi Shamir, and Leonard Adleman in the 1970s (hence the name “RSA”). The system is based on the idea that factoring a 2,048-bit integer (a number with 617 digits) is too hard for a computer to do in a reasonable amount of time.

That idea was flipped on its head in 1994 when Shor, then working at Bell Labs, introduced an algorithm which proved that a quantum computer could factor quickly enough to break RSA cryptography.

“That was a turning point. But in 1994, nobody knew how to build a large enough quantum computer. And we’re still pretty far from there. Some people wonder if they will ever be built,” says Vaikuntanathan.

It is estimated that a quantum computer would need about 20 million qubits to run Shor’s algorithm. Right now, the largest quantum computers have around 1,100 qubits.

A quantum computer performs computations using quantum circuits, just like a classical computer uses classical circuits. Each quantum circuit is composed of a series of operations known as quantum gates. These quantum gates utilize qubits, which are the smallest building blocks of a quantum computer, to perform calculations.

But quantum gates introduce noise, so having fewer gates would improve a machine’s performance. Researchers have been striving to enhance Shor’s algorithm so it could be run on a smaller circuit with fewer quantum gates.

That is precisely what Regev did with the circuit he proposed a year ago.

“That was big news because it was the first real improvement to Shor’s circuit from 1994,” Vaikuntanathan says.

The quantum circuit Shor proposed has a size proportional to the square of the number being factored. That means if one were to factor a 2,048-bit integer, the circuit would need millions of gates.

Regev’s circuit requires significantly fewer quantum gates, but it needs many more qubits to provide enough memory. This presents a new problem.

“In a sense, some types of qubits are like apples or oranges. If you keep them around, they decay over time. You want to minimize the number of qubits you need to keep around,” explains Vaikuntanathan.

He heard Regev speak about his results at a workshop last August. At the end of his talk, Regev posed a question: Could someone improve his circuit so it needs fewer qubits? Vaikuntanathan and Ragavan took up that question.

Quantum ping-pong

To factor a very large number, a quantum circuit would need to run many times, performing operations that involve computing powers, like 2 to the power of 100.

But computing such large powers is costly and difficult to perform on a quantum computer, since quantum computers can only perform reversible operations. Squaring a number is not a reversible operation, so each time a number is squared, more quantum memory must be added to compute the next square.

The MIT researchers found a clever way to compute exponents using a series of  Fibonacci numbers that requires simple multiplication, which is reversible, rather than squaring. Their method needs just two quantum memory units to compute any exponent.

“It is kind of like a ping-pong game, where we start with a number and then bounce back and forth, multiplying between two quantum memory registers,” Vaikuntanathan adds.

They also tackled the challenge of error correction. The circuits proposed by Shor and Regev require every quantum operation to be correct for their algorithm to work, Vaikuntanathan says. But error-free quantum gates would be infeasible on a real machine.

They overcame this problem using a technique to filter out corrupt results and only process the right ones.

The end-result is a circuit that is significantly more memory-efficient. Plus, their error correction technique would make the algorithm more practical to deploy.

“The authors resolve the two most important bottlenecks in the earlier quantum factoring algorithm. Although still not immediately practical, their work brings quantum factoring algorithms closer to reality,” adds Regev.

In the future, the researchers hope to make their algorithm even more efficient and, someday, use it to test factoring on a real quantum circuit.

“The elephant-in-the-room question after this work is: Does it actually bring us closer to breaking RSA cryptography? That is not clear just yet; these improvements currently only kick in when the integers are much larger than 2,048 bits. Can we push this algorithm and make it more feasible than Shor’s even for 2,048-bit integers?” says Ragavan.

This work is funded by an Akamai Presidential Fellowship, the U.S. Defense Advanced Research Projects Agency, the National Science Foundation, the MIT-IBM Watson AI Lab, a Thornton Family Faculty Research Innovation Fellowship, and a Simons Investigator Award.

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• Vinod Vaikuntanathan
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• Cryptography
• Cybersecurity
• Computer science and technology
• Computer Science and Artificial Intelligence Laboratory (CSAIL)
• Electrical Engineering & Computer Science (eecs)
• Defense Advanced Research Projects Agency (DARPA)
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Maths and English GCSE failure rates near 40%

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BU Team Wins Major National Science Foundation Grant to Help PhD Students Attack Climate Change

$3 million five-year award will encourage multidisciplinary approaches to converting and storing sustainable energy.

Intense flooding in Connecticut (left) and wildfires in Colorado (right) are recent examples of extreme weather events that researchers will use to explore solutions to climate change. Photos via AP/Daniel Brown/Sipa USA and via AP/Arnold Gold/Hearst Connecticut Media

$3 million, five-year award will encourage multidisciplinary approaches to converting and storing sustainable energy

If recent weather patterns and disasters, like extreme heat and droughts, catastrophic flooding, and devastating wildfires, have proven anything, it’s that better solutions for sustainable energy are needed to help combat the lasting effects of climate change.

Culminating an up-and-down two-year journey, Malika Jeffries-EL, a Boston University College of Arts & Sciences professor of chemistry, along with a team of BU researchers, has been awarded a five-year, $3 million Research Traineeship grant (NRT) from the National Science Foundation to help PhD students collaborate across disciplines to develop new ideas to convert and store sustainable energy sources. The award will create a new training program that unifies resources in engineering, chemistry, computer science, and data sciences to provide participating students with a broad exposure to energy-related issues.

“I was just really proud,” says Jeffries-EL, who is also the associate dean of the Graduate School of Arts & Sciences. “I was proud of myself and my team for doing their part and working so hard to get things through at the last minute. My point of pride is that I fought for this proposal. This was a two-year struggle to get this approved. It’s a very large grant and I’m so excited about the work ahead.”

Along with Jeffries-EL, the project’s co–principal investigators are Emily Ryan , a College of Engineering associate professor of mechanical engineering and of materials science and engineering, James Chapman , an ENG assistant professor of mechanical engineering,  David Coker , a CAS professor of chemistry and of computing and data sciences, and Brian Kulis, an ENG associate professor of electrical and computer engineering.

The NRT Research Traineeship grant is a collaborative award between BU’s Institute for Global Sustainability and the Rafik B. Hariri Center for Computing and Computational Science & Engineering . (NRT grants are intended to assist graduate students in developing the skills, knowledge, and competencies needed to pursue a range of STEM careers.)

“Computation and data science are playing key roles in designing and discovering new materials to address society’s clean renewable energy needs,” says Coker, director of the Hariri Institute’s Center for Computational Science. “This NRT grant will fund the development of new graduate training programs that synergistically bring together data and computational scientists and materials fabrication, synthesis, and characterization experts to guide and educate a new generation of researchers capable of working at the intersections of these fields and pushing forward this critical frontier research.”

Jeffries-EL, who joined the CAS faculty in 2016, says that most scientists, herself included, begin their careers so focused on their own disciplines that they struggle to think and look outside their research silos. Climate change, she says, is simply too big, too broad, too daunting to continue approaching it in isolation. It is expected that over the NRT program’s five years, 100 to 125 BU PhD students will participate in the training.

“We are at a point where we need to be intentional with problems we are tackling,” Jeffries-EL says. “It’s all interconnected. These are complicated problems, and it requires an interdisciplinary approach and interdisciplinary science.”

And getting PhD students involved in the work is critical.

The grant, she says, is so important because it will allow PhD students, just starting out in their research, to immediately learn to think outside of their field. “We are going to catch them at the beginning. This is an intricate part of their training. If they make it all the way through graduate school [without doing interdisciplinary research], they won’t think they need to think this way. We will breed them to think about interdisciplinary research from the start.”

If chemists and data scientists and engineers and biologists all start approaching the problem of climate change and sustainable energy by thinking together, rather than as individuals, Jeffries-EL says, the possibilities are endless.

“We are teaching students to think bigger to take on bigger challenges,” she says. “We have to encourage students to be bold and think big, and that you might fail, but that’s OK, because you will learn as you go.”

We are teaching students to think bigger, to take on bigger challenges. We have to encourage students to be bold, and think big, and that you might fail, but that’s OK, because you will learn as you go. Malika Jeffries-EL

Ryan, an associate director of the Institute for Global Sustainability (IGS), says society needs cleaner energy storage and generation technologies to reduce greenhouse gas emissions and overcome the challenges brought about by climate change. 

“The next generation of scientists and engineers will require a multidisciplinary background and perspective to develop the energy systems of tomorrow,” Ryan says. “We will partner with BU’s Institute for Global Sustainability to provide students a broader education in energy and climate change that includes not only the technical aspects, but also incorporates health, justice, policy, and more to positively impact society. The cutting-edge technology research at BU along with the diverse expertise of IGS will provide students a unique education that would not be possible elsewhere.”

As an example of how an interdisciplinary approach could work, she points to BU’s new Faculty for Computing & Data Sciences.

“Everybody is benefiting from data science,” Jeffries-EL says. “Some are more attentive to what it might do for them. If we can think about how we can leverage data science in many different fields, that will help people have the right mindset around this. It’s all about getting people to think differently from the start.”

The reason sustainable energy research is so important, Jeffries-EL says, is that with so much attention focused on creating and growing new energy sources, from wind to solar, there needs to be an equal amount of time spent on finding ways to harness those new sources of renewable energies.

“We need to be more mindful about how we use energy, and how we store energy. I was fortunate to find like-minded people with the same interest. We can create all the energy in the world, but if we can’t store it in batteries or in some other way, it’s useless. How do you harness the energy, and store and transport it?”

One lesson learned from the work of the BU Institute for Global Sustainability, Jeffries-EL says, “is that we can come up with the coolest technology in the world, but if no one trusts it, it will sit on the shelf.”

The new grant is the second time in two years that BU researchers have been awarded an NSF Research Traineeship grant. In 2023, a $3 million grant went to “A Convergent Training Program on Biological Control,” codirected by Elise Morgan , ENG dean ad interim, and Mary Dunlop , an ENG associate professor. Their work is aimed at training a diverse group of PhD students for the workforce in biotech, synthetic biology, manufacturing, robotics, sustainability, and other sectors.

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