phd in mathematics requirements

Department of Mathematics

Requirements for the ph.d. degree.

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

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

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

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

The dissertation prospectus should be submitted during the third year. 

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

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

Master’s Degrees :

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

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

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

Department of Mathematics

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PhD 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.

Teaching Seminar

Candidates must take, attend, participate in, and pass the course 110.771 (GTA Teaching Seminar). The seminar is an important part of the preparation for classroom teaching, and thus an essential part of mathematics graduate education. The seminar is generally required in a student’s first year at JHU. A student supported by an external fellowship may delay taking the seminar until the spring before they are required to TA (but may not postpone the seminar entirely).

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. More information as well as old exams and syllabi can be found on the Qualifying Exams page .

Candidates must pass an oral qualifying examination in the student’s chosen area of research by April 10th 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.

The MA Degree

Although the Department of Mathematics does not admit students seeking a terminal MA degree, students in the PhD program may earn an MA degree.

MA candidates must complete:

• Four graduate courses given by the Hopkins Department of Mathematics
• Two additional courses at the graduate or 400 level, other than 110.401, 110.405 and 110.415, given by the Hopkins Department of Mathematics, or with the permission of the graduate program director, graduate mathematics courses given by other departments or universities.

All courses used to satisfy the requirements must be completed with a grade of B- or better. (Advanced graduate courses completed with a grade of P can also be used to satisfy the requirements.)

Ph.D. Program Overview

Description.

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

Core Courses

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

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

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

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

The six core courses are:

MATH 6110, Real Analysis

MATH 6120, Complex Analysis

MATH 6310, Algebra 1

MATH 6320, Algebra 2

MATH 6510, Introductory Algebraic Topology

MATH 6520, Differentiable Manifolds.

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

"What is...?" Seminar

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

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

Special Committee

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

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

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

Current Advising Committee

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

Admission to Candidacy

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

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

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

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

Masters Degree in the Minor Field

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

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

Graduate Student Funding

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

Special Requests

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

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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.

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Ph.D. Requirements

The Ph.D. program has two tracks: pure and applied. The course requirements differ somewhat between the two tracks, although the number of credit hours is the same. Both pure and applied tracks have the following requirements.

• Pass two qualifying exams: one in Algebra or Analysis, and one in Numerical Analysis or Probability/Statistics
• Pass a preliminary exam in your area of research
• Demonstrate knowledge of adequate computer skills
• Complete the Research Skills and Responsible Scholarship training sequence offered by the department
• Pass an oral comprehensive exam
• Write and defend a Ph.D. dissertation containing original research

After graduation, our Ph.D. alumni typically obtain positions either in academia, as postdoctoral fellows or tenure-track faculty, or in the private sector. 

Long Tradition of Excellence

We have a long tradition of excellence. The first Ph.D. granted at KU was in Mathematics, to Arnold Emch in the year 1895.

Since then, our graduate program has been a central part of the research and teaching mission of our department, and is an important component of our long term planning.

Required Enrollment Hours

GTA/GRA must be enrolled in at least six hours in fall and spring. Non GTA/GRA international students must be enrolled in nine hours.

Pure Track Requirements

Qual Courses Required:  

• 727 or Qual in Probability/Statistics
• 765 or Qual in Analysis
• 781 or Qual in Numerical Analysis
• 791 or Qual in Algebra
• (with permission advanced students may substitute corresponding 800-level courses)

Courses Required (9 total):

• 800, 810, 830
• One of the following: 840, 910, or 920
• Four additional courses at 800 level or above.

The requirements for exams, research skills, enrollment hours and research component are the same for both the pure and applied tracks.

Applied Track Requirements

• One 2-course sequence: 850/851 or 865/866 or 881/882
• One of the following: 840, 850 or 950

Requirements for Both Pure and Applied Tracks

• Pass two qualifying exams, one in algebra or analysis and one in numerical analysis or probability and statistics by the beginning of the fifth semester.
• Preliminary by the beginning of the eighth semester.
• Comprehensive Oral
• Final Thesis Defense

Research Skills:

• Responsible Scholarship Training
• Computer Skills

Enrollment Hours:

• At least six per semester during two semester minimum residency.
• 18 after oral comprehensive (at least six in fall and spring and at least three in summer).
• Continuous enrollment after that.

Research Component:  Thesis

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Department of Mathematics

• Ph.D. in Mathematics
• General Information
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• M.S. in Mathematics
• Ph.D. in Mathematics with Thesis in Actuarial Science
• M.S. in Actuarial Science
• M.S. in Applied Financial Mathematics
• M.S. in Applied Financial Mathematics – Actuarial Science
• Course Schedule
• Preliminary Exams

Overview of Graduation Requirements

To graduate with a PhD in Mathematics, a student must satisfy all of the following requirements:

• 45 credits, including 15 doctoral dissertation research credits/GRAD 6950
• If you have a Master’s degree in mathematics at UConn, then 30 credits are required, including 15 doctoral dissertation research credits.
• Pass at least one prelim exam after each semester for the first three semesters of their graduate study.
• Finish all prelim exam requirements by the beginning of the spring semester in their second year.
• Finish all core course requirements by the end of their second year of graduate study,
• Choose a thesis advisor no later than the end of the fall semester of the third year.
• Form your advisory committee :  Major Advisor and at least two Associate Advisors or Co-Major Advisors and at least one Associate Advisor.
• Meant to further the student’s education, scholarship and professional development.
• Material should be in the broad area in which the student intends to write a dissertation, but should not focus on the actual thesis research.  Students present and answer questions about the material they have studied.
• The exam is normally taken at the end of the third year or beginning of the fourth year.
• Submit Dissertation Proposal
• Submit Plan of Study
• Apply for Graduation in Student Admin by the fourth week of the final semester.
• Dissertation Specifications under Dissertation Information
• Thesis templates (for LaTex) are available on our Thesis Formatting page
• Submit Approval Page – Step 7
• Complete Survey of Earned Doctorates – Step 7
• Add Diploma Address in Student Admin and Diploma Info.

Preliminary Examination and Core Course Requirements

Pure Mathematics

• Math 5111/Measure and Integration
• Math 5120/Complex Function Theory
• Math 5210/Abstract Algebra
• Math 5310/Introduction to Geometry and Topology I
• Math 5160/Probability Theory and Stochastic Processes I
• Math 5210/Abstract Algebra I
• Math 5211/Abstract Algebra II
• Math 5260/Mathematical Logic I
• Math 5360/Differential Geometry

Applied Mathematics

• Math 5410/Introduction to Applied Mathematics I
• Math 5510/Numerical Analysis and Approximation Theory I
• Math 5440/Partial Differential Equations
• Math 5520/Finite Element Solution Methods I

Past Prelim Exams

Ph.D. Program

Degree requirements.

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

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

Detailed Regulations

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

Course Requirements

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

Preliminary Examination

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

Qualifying Examination

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

PhD Program

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

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

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

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

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

Course Requirements for students starting in Autumn 2023 and later: 

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

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

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

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

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

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

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Guide to Graduate Studies

The PhD Program The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

• Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
• Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
• Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful.

University Requirements

The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the department. Most students complete the Ph.D. in four or five years. Please review the program requirements timeline .

There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Qualifying Exam

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here .

Students are expected to pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a Ph.D. dissertation advisor.

Minor Thesis

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

The topic is selected in consultation with a faculty member, other than the student’s Ph.D. dissertation advisor, chosen by the student. The topic should not be in the area of the student’s Ph.D. dissertation. For example, students working in number theory might do a minor thesis in analysis or geometry. At the end of three weeks time (four if teaching), students submit to the faculty member a written account of the subject and are prepared to answer questions on the topic.

The minor thesis must be completed before the start of the third year in residence.

Language Exam

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages. Accordingly, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevant to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here .

Non-native English speakers who have received a Bachelor’s degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper-level math courses, students can apply for a continuing Master’s Degree.

Teaching Requirement

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department’s teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship, students are paired with a member of the department’s teaching staff. Students attend some of the advisor’s classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor and from members of the class.

Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (an advanced undergraduate) to grade homework and to take a weekly problem session. Usually, there are several classes following the same syllabus and with common exams. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math concentrators.

Final Stages

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others and some variation in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member’s field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin research. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one’s nascent abilities and insights can blossom. Ph.D. dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. The ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren’t known; and (2) a somewhat fatalistic attitude concerning “creative ability” and recognition that hard work is, in the end, much more important.

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Doctor of Philosophy (PhD)

Program synopsis and training.

 The Doctor of Philosophy (PhD) in mathematics is the highest degree offered by our program. Graduates will have demonstrated their ability to conduct independent scientific research and contribute new mathematical knowledge and scholarship in their area of specialization. They will be well-supported and well prepared for research and faculty positions at academic institutions anywhere in the world. Owing to their independence, analytic abilities, and proven tenacity, our PhD graduates are also sought after by private and government employers.

Our PhD program offers two tracks, one for Theoretical Mathematics and one for Applied Mathematics . The tracks differ only in the course and  qualifying requirements during the first two years. Applicants are required to decide on one of the tracks and applications will be evaluated subject to respective criteria described below.

Once students have passed their Qualifying Requirements, the two tracks merge and there is no distinction in later examinations and research opportunities. In particular, the candidacy exam for both tracks consists of a research proposal, the graduate faculty available for advising is the same, and the final degree and thesis defense are independent of the initially chosen track.

Expected Preparations for Admission

Competitive applicants to the theoretical track are expected to have strong foundations in Real Analysis and Abstract Algebra, equivalent to our Math 5201 - 5202  and Math 5111 - 5112 sequences.

Expected preparations for the applied track include the equivalents of a rigorous Real Analysis course (such as Math 5201 ), a strong background in Linear Algebra, as well as an introductory course in Scientific Computing.

Besides these basic requirements, competitive applicants in either track submit evidence for a broad formation in mathematics at the upper-division or beginning graduate level. Relevant coursework in other mathematical or quantitative sciences may also be considered, especially for the applied track.

Prior research experiences are not required for either track, and we routinely admit students without significant research background. Nevertheless, applicants are encouraged to include accounts of research and independent project endeavors as well as letters of supervising mentors in order to be more competitive for fellowship considerations. The research component is likely to have greater weight in applications to the applied track.

These prepared documents serve to provide our admission committee with a narrative overview of the applicant's mathematical trajectory. Their primary focus should, therefore, be to enumerate and describe any evidence of mathematical ability and mathematical promise. The information included in the documents should be well-organized, comprehensive, informative, specific, and relevant. This will help our committee to properly and efficiently evaluate the high number of applications we receive each year.

Our Graduate Recruitment Committee will generally  not  consider GRE test scores for this Autumn 2024 admissions. If you have already taken the test, please do not self-report the scores to us. In exceptional circumstances students may have the option to report unofficially. 

International students whose native language is not English and are not exempt should score at least a 20 on the Speaking portion of the TOEFL or at least 6.5 on the IELTS Speaking portion.  We also recommend an overall score of at least 95 on TOEFL or at least 7.0 on IELTS.  For a list of exempt countries, please see  https://gpadmissions.osu.edu/intl/additional-requirements-to-apply.html

Qualifying Requirements by Track

The qualifying requirements for the theoretical track are fulfilled by passing our Abstract Algebra course sequence  ( Math 6111 , Math 6112 ) and  our Real Analysis course sequence ( Math 6211 , Math 6212 ), each with at least an A-, or  by passing a respective examination.

The qualifying requirements for the applied track combine a mandatory Scientific Computing course ( Math 6601 ), one of the algebra or analysis courses, and three additional courses chosen from  Math 6602 , Math 6411 ,  Math 6451 , and the courses comprising the algebra and analysis sequences.

The breadth requirements in the applied track are more flexible than in the theoretical track, but also include a mandatory graduate course in a non-math STEM department from an approved list. 

You can find more information about our PhD program requirement here .

Opportunities & Outcomes

The research opportunities and academic outcomes of our doctoral program are described in detail in the Graduate Program Prospectus  [pdf]. 

Our department has about 80 active graduate faculty on the Columbus and regional campuses. Virtually every area of mathematics is represented in our program, with a sampling displayed below.

• Commutative, Non-commutative, & Quantum Algebra
• Analytic, Algebraic, Computational Number Theory
• Algebraic Geometry, Tropical Geometry
• Applied Mathematics, Mathematical Physics
• Real and Complex Analysis
• Functional Analysis, Operator Algebras
• Combinatorics and Graph Theory
• Differential Geometry
• Dynamical Systems and Ergodic Theory
• Financial and Actuarial Mathematics
• Logic and Foundations
• Probability Theory, Statistical Mechanics
• Mathematical Biology
• Ordinary and Partial Differential Equations
• Representation theory
• Scientific Computing
• Topology, Topological Data Analysis

See also our  Applied Mathematics Topics List  [pdf].

Our program offers many support opportunities without teaching duties as well, to allow more time for scientific endeavors. These opportunities include university fellowships, external funding, and departmental fellowships and special assignments. See the  Financial Support  page for more details.

The median time to degree completion in our program is below six years but also varies significantly among our students, with as little as four years for students entering with substantial prior preparations. Funding is guaranteed for six years and can be extended to seven years with advisor support and the permission of the Graduate Studies Committee.  

Most of our graduates continue their careers in academia. Post-doctoral placements in the last two years include, for example, UCLA, Stanford, ETH-Zürich, Brown University, University of Michigan, Northwestern University, University of Vienna, EPF Lausanne, Free University at Berlin, Purdue University, and University of Utah. In recent years our graduates also went to Princeton University, IAS, University of Chicago, Yale University, University of Michigan, Cal-Tech, Northwestern University, University of Texas, Duke University, SUNY Stony Brook, Purdue University, University of North Carolina - Chapel Hill, and Indiana University. Recent non-academic placements include Google, Facebook, Amazon, NSA, and prestigious financial institutions.

Students also have access to training and networking opportunities that prepare them better for careers in private industry and teaching - for example, through the Erdős Institute  - and are regularly offered highly competitive positions in the industry. 

Nearly half of the graduate population consists of domestic students coming from both larger universities and smaller liberal arts colleges with a solid math curriculum. And as a program group member of the National Math Alliance , we are dedicated to enhancing diversity in our program and the scientific community. The International students in our program come from all parts of the world with a wide variety of educational backgrounds.

Prospective students:  [email protected]

Graduate Office Department of Mathematics The Ohio State University 231 W 18th Avenue ( MA 102 ) Columbus, Ohio 43210 United States of America                

Phone: (614) 292-6274 Fax: (614) 292-1479

[pdf] - Some links on this page are to .pdf files. If you need these files in a more accessible format, please email  [email protected] . PDF files require the use of Adobe Acrobat Reader software to open them. If you do not have Reader, you may use the following link to Adobe to download it for free at:  Adobe Acrobat Reader .

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Doctoral Degree Requirements

The PhD in Mathematics requires completing at least 36 credits of graduate coursework beyond the bachelor’s degree, preliminary written and oral examinations, 24 thesis credits, and a doctoral thesis with final defense. 

PhD students in Mathematics are expected to earn an MS degree upon completion of their preliminary oral examination.

Entering students

• Required to have completed an undergraduate degree prior to matriculation into the program. 
• Some students enter with previous graduate coursework which can include having completed a Master’s degree.

Course requirements for a doctoral degree

• 36 credits of coursework — Students normally complete the coursework for the PhD at the University of Minnesota. Students may transfer up to 12 credits of previously completed graduate coursework toward their PhD in consultation with their advisor and the Director of Graduate Studies upon submitting their degree plan. 
• MATH 8001: Preparation for College Teaching — All PhD students must complete MATH 8001 in the Fall semester of their first year.
• 18 credits of core coursework — Courses in Mathematics that are the foundation for advanced doctoral work in Mathematics. 
• 18 credits supporting coursework — Mathematics coursework in your research area or that supports this research. Students may take a maximum of 12 credits outside of Mathematics. Coursework used for a formal minor can be counted towards this requirement.
• 24 thesis credits — Taken upon the completion of the preliminary oral examination. Students are expected to complete their thesis credits no more than two semesters after successfully completing their preliminary oral examination.

+ Core courses

Core courses.

• General Algebra (MATH 8201/02)
• Manifolds and Topology (MATH 8301/02)
• Mathematical Modeling and Methods of Applied Mathematics (MATH 8401/02)
• Numerical Analysis and Scientific Computing (MATH 8441/42)
• Real Analysis (MATH 8601/02)
• Theory of Probability Including Measure Theory (MATH 8651/52)
• Complex Analysis (MATH 8701/02)

Preliminary written exam

Students must demonstrate proficiency in basic areas of mathematics by passing the preliminary written exams.

Students are expected to complete their preliminary written exams by the end of the second year of study. Extension to this timing may be approved through a meeting with the academic advisor and the Director of Graduate Studies.

What’s required

Students must pass two written examinations by either:

1. Taking the written examinations.

Given Fall and Spring semester in Real Analysis, Complex Analysis, Algebra and Manifolds/Topology.

2. Passing the final examination in the second semester of any of the core courses.

What to expect.

• Students with previous graduate coursework may transfer up to 18 credits to fulfill core course credit requirements upon passing preliminary written exams.
• Students without previous graduate coursework who complete core coursework requirements by passing exams will take up to 36 credits of other major coursework.  

Preliminary oral exam

Students complete a preliminary oral examination to demonstrate proficiency in their primary area of study and supporting program or minor program. The exam also serves as a final exam for a Plan B Master’s in Mathematics. 

• Typically completed after the third year of study.
• Must be completed prior to the last day of Spring semester in the fourth year. 
• Changes to this timing may be approved by meeting with the advisor and the Director of Graduate Studies.

The preliminary oral examination concentrates on the primary area of study and the supporting program or minor. 

• Examination committee — The examination committee of four faculty members consists of the major area advisor and three additional faculty members. Preliminary Oral Exam Committees require a faculty member from outside of Mathematics OR Outside of the University of Minnesota. Students who have declared a minor in another department are required to have a committee member from the department they are minoring in.
• Advisor — The advisor recommends coursework in the basic material, as well as expository and research papers for individual study.
• Expository paper — The study, including the recommended research papers, will be summarized in an expository paper (approximately 10 pages), with a substantial bibliography. This will demonstrate a knowledge of the definitions and results in the area, and indicate open problems which may form the basis for the PhD thesis.
• Registration and credits — Students taking their preliminary oral examinations during Fall or Spring semester are expected to register for six pre-thesis credits (MATH 8666). 
• The student should choose their primary area and decide on an advisor who will support them through these exams by the end of fall semester in the second year.
• In the semester prior to the oral exam, students should work with their advisor to determine a committee and expectations for their exam. 
• Students are expected to have begun work on their expository paper by the beginning of the semester in which they are planning to complete this exam. 

See the PhD degree (steps 1-7) and the Plan B Master’s degree (the preliminary oral examination serves as a Final Exam for a Plan B Master’s in Mathematics) completion steps:

Degree completion steps

Final exam/dissertation defense

Mathematics PhD students typically complete their degree in the sixth year of their program. The examination is a defense of the thesis that is the culmination of the student’s work in the program. 

• The doctoral final exam committee consists of four members, including the advisor(s).
• The chair of this committee must be a full mathematics faculty member who is not the student's advisor.  
• If there is a formal minor, this committee member must represent that program. In the case of multiple minors, there must be a separate thesis reviewer for each minor.

The doctoral final exam must include a public presentation of the candidate’s dissertation to the doctoral final oral examination committee and the invited scholarly community, followed by a closed session for questions by the examiners. 

See the PhD degree completion steps (steps 8-15):

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• Doing a PhD in Mathematics
• Doing a PhD

What Does a PhD in Maths Involve?

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

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

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

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

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

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

Browse PhD Opportunities in Mathematics

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

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

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

Additional Learning Modules

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

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

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

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

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

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

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

Tips to Consider when Making Your Application

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

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

How Much Does a Maths PhD Typically Cost?

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

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

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

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

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

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

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

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

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

What Jobs Can You Get with A Maths PhD?

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

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

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

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

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

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

How Much Can You Earn with A PhD in Maths?

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

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

Academic Lecturer

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

Actuary or Finance

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

Aerospace or Mechanical Engineering

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

Data Analyst

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

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

Facts and Statistics About Maths PhD Holders

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

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

Which Noteworthy People Hold a PhD in Maths?

Alan turing.

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

Ruth Lawrence

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

Emmy Noether

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

Other Useful Resources

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

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

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

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

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

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

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

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

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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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Doctor of Philosophy Program

Besides satisfying the general regulations of the Graduate School for the degree of Doctor of Philosophy, the student must comply with the requirements briefly outlined below. For complete details about these requirements see section IV of the Graduate Handbook .

Pass four Qualifying Examinations . The exams are based on material that is covered in the courses listed and on material from undergraduate prerequisites. Credit for passing a similar examination at another university cannot be transferred. See sections IV and VI of the Graduate Handbook for more information.

Advanced Topics Examinations. A student becomes eligible to take the Advanced Topics Examination after passing the Qualifying Examinations.

Plan of Study. The plan of study should be submitted electronically to the Graduate School through myPurdue by each student preparing to hold their Advanced Topics. 

Preliminary Examination. The preliminary examination for most students will only require the completion of a form for the Graduate School. An oral or written examination may be required by the student's advisory committee for admission to candidacy. Graduate School regulations require that at least two sessions (including summer sessions) must elapse between the preliminary examination and the thesis defense.

Admission to Candidacy. To be admitted to candidacy for the Ph.D. degree, the student must have fulfilled the requirements above which are detailed in section IV of the Graduate Handbook .

Dissertation. A thesis must be submitted in final form presenting new results of sufficient importance to merit publication.

Recommendation for the Ph.D. Degree. If the requirements are met within the time limits detailed in section IV of the Graduate Handbook , the candidate will be recommended to the faculty to receive the degree of Doctor of Philosophy.

For information about financial support and research in absentia see section IV of the Graduate Handbook .

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Ph.D. in Mathematics

We offer a ph.d. in mathematics which is for pure and applied mathematicians.  learn more about our applied math research online here., math ph.d. requirements, passing four core course sequences from algebra, applied mathematics, complex analysis, pde, topology, and/or real analysis. a sequence consists of all courses with the same course number except for an alphabetical suffix. any course without an alphabetical suffix is not part of a sequence., pass two written qualifying exams, a student is allowed to take the qualifying examination at most twice for each sequence., completion of four math graduate elective class. , completion of a 2 unit course in professional development., writing and defending a ph.d. dissertation which is a significant original contribution to the mathematical literature., normative time to degree: 15 quarters.

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Margarita Roman

Email:  [email protected]

Phone: (951)-827-3090

Graduate Student Services Advisor

Kathryn Aquino

Email: [email protected]

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Randy Morgan

Phone: (951)-827-3114

Graduate Advisor, Recruitment

Dr. Zhenghe Zhang

Email:  [email protected]

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Requirements

Requirements for admission to candidacy.

The following are required for admission to candidacy for the Ph.D., or to obtain an M.S. degree in Applied Mathematics:

• Prerequisite to the graduate program are familiarity with analysis and algebra at the advanced undergraduate level (e.g., MATH 4130-4140 and MATH 4330-4340). Students lacking either prerequisite (which may be determined by their special committee chair and/or the CAM Director of Graduate Studies), should take the appropriate courses within their first two years of study. No more than two of these courses can count towards meeting other CAM degree requirements.
• Students are required to take at least eight courses in mathematics and its applications that are approved by her/his special committee, at least 4 of which must be numbered 6000 or above. Suggested areas for these courses are given in the list of  Focal Areas  for Applied Mathematics.
• The courses taken to satisfy item (2) must include an advanced course in computational methods (focal area (a)). In order to achieve breadth in Applied Mathematics, courses from at least three other Focal Areas should normally be included. Should a course be listed under more than one focal area, then it will count towards only one such area as chosen by the student's Special Committee.
• Students are required to have minors in Mathematics and in another field relevant to their doctoral research. Note that the course requirements listed above may suffice to satisfy the requirements for a graduate minor in Mathematics.

Exceptions to these requirements can sometimes be made, if approved in advance by the student's Special Committee and the Director of Graduate Studies. All requirement courses must be taken for a letter grade.

Requirements for the Graduate Minor

A student majoring in another graduate field (excluding Mathematics) can obtain a Minor in Applied Mathematics by successfully completing four courses, drawn from at least two of the Focal Areas, each of which contains substantial mathematical content. Two of the courses must be at the graduate level, the others numbered 4000 or above. The courses must be approved by a member of the Field of Applied Mathematics who is a member of the student's Special Committee. Any course used to satisfy requirements of the student's major field of doctoral study may not be used to satisfy the Minor in Applied Mathematics. If field members or students feel that these requirements should be updated or altered, please inform the Director of Graduate Studies.

Graduate Study Guidelines

Selection of committees.

A committee chair (thesis advisor) must be selected by each Ph.D. student  before the end of the second semester  (i.e., before the end of Academic Year 1).

The committee chair will aid the graduate student in selecting a Special Committee, which shall consist of a minimum of three faculty members and which will directly supervise the graduate study and research of a student. The committee chair, who is also the thesis advisor, represents the major field of applied mathematics. The other faculty members represent two minor fields; one of these must be mathematics and the other is a field outside mathematics. This committee must be formed  before the end of the third semester .

Exceptions to the above requirements must be approved by the Director of Graduate Studies (DGS). Failure to comply with the above requirement may result in a hold being placed on your registration by the Graduate School.

A Special Committee Selection and Change form must be submitted to the Graduate School to establish a committee or change/add committee members. This can be done online through Student Center.

Selection of a Special Committee (including the thesis advisor) is a mutual agreement between a student, the chosen faculty members and the DGS. During the pursuit of the degree, changes in circumstances may cause the student and/or faculty members to desire the termination of this commitment.

The A-Exam must be scheduled  before the beginning of the seventh semester  unless a special petition is filed. The A-Exam should be scheduled well in advance of the exam and the appropriate form ( Schedule of Examination  form)  MUST  be submitted to the Graduate School at least 7 calendar days ahead. CAM graduate students are eligible for a Non-Thesis Masters Degree upon completion of the A-Exam. In order for this to be awarded, the Committee Chair must check the relevant box on the  Results of Examination Form . The form must be turned into the Graduate School within 3 days after the exam.

The B-Exam (thesis defense) is scheduled with the same form as the A-Exam and must be submitted to the Grad School at least 7 calendar days ahead.

View the timetable for fulfilling all degree requirements.

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PhD Requirements

Please direct inquiries about our graduate program to: [email protected]

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The PhD Degree in Mathematics

The basic requirements for the PhD degree in mathematics include demonstrating a mastery of a broad area of mathematics and writing a dissertation making an original and substantial contribution to mathematics. Specific requirements for the PhD degree in mathematics consist of:

• Qualifying Exams
• Oral Degree Audit (a.k.a. General Exam)
• Dissertation and Defense

Registration each semester

• Completion of at least 54 semester hours of graduate credit.

PhD Qualifying Exams

The Written PhD Qualifying Examination consists of three qualifier exams, each of three hours duration, chosen from the four qualifier exams covering the material in the Core courses (the exam in Math 7382 is available starting in January 2024). A PhD student may take the PhD Qualifying Exam as early as feasible, but must pass it no later than the beginning of the fourth regular semester of study, except by permission of the Graduate Committee. Meeting this requirement is part of the definition of satisfactory progress---a requirement for holding a graduate assistantship or fellowship. The PhD Qualifying Exams are scheduled twice per year: normally the week before the beginning of the fall semester, and the week before the beginning of the spring semester. An incoming student who passes a qualifier exam at the PhD level before the first semester of studies is accredited with completion of the corresponding Core course.

Qualifying Exam policy is that at least 50% of the credit on each exam will come from the test problem banks below. There will normally be approximately 6 to 8 problems offered on each exam, and students will typically need to turn in approximately 5 of these.

Syllabi and test bank problems for each of the Qualifying Examinations:

• Algebra Syllabus (Math 7210-7211) ; --> Algebra Syllabus and Test Bank
• Analysis Syllabus and Test Bank
• Topology Syllabus and Test Bank
• Applied Math Syllabus and Test Bank

Copies of past Qualifying Exams can be found in the Exam Archive .

Use this Registration Form to sign up to take a Qualifying Exam.

Oral Degree Audit (General Exam)

Description. The Degree Audit, a.k.a. General Exam, is an oral exam for Ph.D. students that tests if a student is prepared to begin dissertation research. This exam must be passed prior to the start of the fourth year of study, unless a postponement is allowed by the Graduate Committee. The exam is conducted by the student's Advisory Committee, which will consist of at least three Graduate Faculty Members plus a Dean's Representative appointed by the Dean of the Graduate School. The committee is to be chosen in consultation with the intended dissertation advisor, who will serve as Chair of the Advisory Committee, and must be approved by the Graduate Director. The Advisory committee shall determine whether the student passes the General Exam. If the student fails, the Advisory Committee will determine whether the exam can be retaken. The exam can be taken at most one additional time.

Content and Format. The specific content of the general examination is established by the individual student's Advisory Committee in consultation with the student, and it is formalized in the Exam Syllabus Form. The format of the exam will consist of an optional presentation by the student (typically no longer than 30 minutes), and an oral exam on the content of the exam syllabus. The exam should last at most two hours.

Note. The student must be enrolled in classes (at least one credit hour) in the term in which the oral exam is scheduled; this includes the Fall and Spring semesters and the Summer. The Fall and Spring semesters begin the first day of classes, not the week before.

Instructions on scheduling your Degree Audit can be found at the Instructions link.

Original Dissertation and Defense

Writing an original dissertation and passing the Final Examination which is the doctoral dissertation defense.

Instructions on the steps to graduation and the forms to be filled out the semester of your graduation can be found in the Instructions link.

The Mathematics Department requires each doctoral student to consult with their Faculty Mentor or Dissertation Advisor each semester concerning courses for the following semester and progress toward the PhD.

• Consultation. Student and advisor consult concerning courses for the following semester and progress in the PhD program.
• Registering. The student registers for the courses which have been agreed upon during consultation. See below for independent study courses 7999 (reading course) and 9000 (dissertation research) and for how to audit a course. The student then sends the advisor an electronic copy of the official schedule of classes for the following semester.
• Approval. The advisor submits the Advising Form to the Director of Graduate Studies. This includes the advisor's acknowledgement of approval of the student's course schedule and the advisor's comments on the student's progress.

Here is how to enroll in independent study courses and how to audit a course.

MATH 7999 and MATH 9000: Reading courses, Math 7999, carry maximally 3 credit hours. Each must have a different topic, and the topic may not coincide with that of a regularly scheduled course. To enroll in an independent study course, be it a MATH 7999 reading course or MATH 9000 dissertation research, the professor of that course must request its creation by filling this IS Form . Then the graduate secretary will create the course; sometimes this takes a day or two to be completed.

AUDIT: To audit a course, the student must fill out this Audit Form . Then the graduate secretary will add the course to the student's schedule for Audit.

• Printable View

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. Requirements

In addition to the information on this page, the Mathematics Graduate Program Handbook provides comprehensive policies regarding satisfactory progress towards satisfying the Ph.D. requirements.  Students should refer to the graduate catalog, available on the Graduate College website , for more details on graduate college requirements for PhD candidates.

Graduate Students are expected to follow the policies and procedures for both the UA Graduate College and for the Department of Mathematics. Policies are updated frequently and it is the student’s responsibility to comply with current policies. Graduate College policies can be viewed on-line at https://grad.arizona.edu/new-and-current-students ; university policies can be found at https://catalog.arizona.edu/ .

Course Requirements

Students are required to complete 36 units of graduate credit in the major and 12 units in a supporting minor, which may be declared in Mathematics, although outside minors are encouraged.  Units may not be counted towards both the major and minor. In addition, 18 units of dissertation (Math 920) must be completed.  Students cannot register for 920 until they have passed their oral comprehensive exam.  This rule can be waived by the Associate Head for the Graduate Program in exceptional circumstances.

Traditional Core Courses

For each of the traditional core courses, Algebra, Real Analysis, and Geometry–Topology, students must either take the course and receive a grade of B or better in both semesters or earn a high pass on the corresponding written qualifying exam.  The material in these courses is essential knowledge for all mathematicians, and it is assumed in all further advanced course work in the department.

Further Mathematics Coursework

Two year-long Mathematics course sequences that are not dual-numbered and are not part of the required core of algebra, real analysis, and geometry-topology are required.  Students should seek advice on appropriate courses from their advisor (if they have one already), faculty in the area in which they plan to do research or the director of the graduate program.  For many students Complex Analysis is a good choice for one of these sequences.

Outside of Department Courses or Internship

Students who entered prior to Fall 2022 , unless they elect to adopt the new Community and Professional Development requirement (see below), must take two courses (6 units) outside the mathematics department. These may be applied toward the minor, if appropriate. The spirit of this requirement is that students should learn to communicate with and appreciate the perspectives of users and producers of mathematics in other disciplines.  The requirement may either be satisfied by either taking two courses (6 units) outside the Mathematics Department or by doing an internship.

Courses which fulfill this requirement should (a) have significant content in mathematics or mathematics education; and (b) not be substantially equivalent to courses in the mathematics department. We maintain a list of a priori acceptable courses . For courses not on this list students should ask the Associate Head for the Graduate Program if they would fulfill the requirement. A priori unacceptable courses include those cross listed in mathematics or taught by a mathematics faculty member. An exception is that courses offered by the math department in mathematics education may be used to satisfy the outside course requirement by students whose dissertation is not in mathematics education.

An internship with a company or government lab may satisfy this requirement if it involves mathematics in a significant way. Students should consult with the Associate Head for the Graduate Program before such an internship to see if it would satisfy the requirement. The student's internship supervisor may be asked to provide documentation of the amount and mathematical nature of the work involved in the internship.

Students who entered the program Fall 2022 or later are encouraged but not required to take out-of-department courses or internships. If taken, these may be applied towards their course units and/or Professional Development requirements (see G2. Professional Development Requirements).

The University requires that PhD students declare a minor. PhD students in mathematics may declare their minor in mathematics or in a supporting discipline. Requirements for the minor are determined by the minor department. Up to 12 units of course work may be in the minor. Students contemplating a minor other than Mathematics should consult with the Director of Graduate Studies and their thesis advisor regarding the suitability of their plans. Typical minors include Statistics, Computer Science, Education, Physics, and Applied Mathematics, though others may be considered.

Program of study

The Plan of Study should be completed after the student has passed their qualifying exams. Each student must present a coherent collection of courses in which the work outside of Mathematics is related to part of the studies in Mathematics. There are many such possibilities, including: algebra, and computer science or discrete methods in operations research; probability, and statistics or reliability/quality control; numerical mathematics, and computer science or computational science; mathematical foundations and history, and education; analysis, and physics or optics; etc.

Graduate Faculty Advisor

All students in the Mathematics Ph.D. program are required to have a Graduate Faculty Advisor (also called Major Professor) in order to maintain Satisfactory Progress. First year students should select a faculty advisor and have it approved by the Director of Graduate Studies (DGS). If the student cannot find an advisor, the DGS will work with the student to appoint one. Students may change their advisor in consultation with the DGS at any time. Once an advisor is chosen, the student will inform the Graduate Coordinator. By the time a student starts to prepare for their Comprehensive Exam, it is expected that the chair of Comprehensive Committee will take over the role of Graduate Faculty Advisor will  It is typical for the Dissertation Advisor to serve as Graduate Faculty Advisor once the research area is selected. The primary responsibilities of a Graduate Faculty Advisor include:

1. Be a source of academic information for their graduate students

2. Provide assistance with details in determining the Plan of Study

3. Be proficient in inputting, managing, and approving forms in GradPath as needed to assure smooth progression to final degree

4. Meet periodically with their students and provide regular, timely input to determine academic progress. This may include review of Plan of Study and Prospectus as prepared by the student.

 When selecting the Graduate Faculty Advisor, the student should contact the faculty member to discuss expectations for both the faculty member and the student. The two shall meet throughout the academic year and at the end of each semester the student will complete the end of year conversation form to discuss with the advisor. A survey will be filled out at least twice yearly to report progress to the Graduate Office.

Research Tutorial Groups

Students must enroll in MATH 596G and complete a research tutorial group (RTG) project starting in the spring semester of their first or second year of enrollment. In the spring, MATH 596G is a one-unit course in which faculty members present short lecture series on research topics of current interest. In the following fall, students choose one of the proposed topics and work with the corresponding faculty member on a research project. This project and a presentation of it at the end of the fall semester is the basis for three more units of credit in MATH 596G. The RTG project is meant to be an early introduction to research in mathematics and forms part of the evaluation of the qualifying exam.

Qualifying Examination

The qualifying examination is based on the following assessment options:

• A written exam in algebra
• A written exam in analysis
• A written exam in geometry and topology
• An MS thesis

Students must attempt at least three assessment options. Two of the assessments must be chosen from the traditional core exams (the first three options). Each written exam is offered in August and January. There is no limit to the number of attempts for the written exams. Students may attempt more than three assessment options. Students with prior preparation may attempt the examinations upon entrance to the program, or after one semester.

Each of the options has three possible grades: fail, pass, and high pass. In general, a grade of high pass indicates the student is ready to go on to advanced course work and to prepare the comprehensive exam. For the MS thesis option the meaning of pass is that the thesis is acceptable for the MS degree. The meaning of high pass is that the quality of the thesis indicates the student is capable of PhD level work. The thesis need not contain original work, but the quality should indicate that the student has the potential for such work. The grade for the MS thesis is determined by the thesis committee. Prior to scheduling your thesis defense, you will need to get your MS committee approved by submitting the Committee Approval form to the Graduate Office. Once your committee is approved, you will need to print out the Results of the MS thesis form and take this to your thesis defense for a final grade.  Both forms can be found on the forms page on our website .

The written exams in algebra, analysis and geometry/topology  cover material from the traditional core courses, Algebra (MATH 511A-B), Real Analysis (MATH 523A-B), and Topology–Geometry (MATH 534A-B). They also include a small amount of undergraduate level material. For the algebra exam this undergraduate material is from linear algebra. For the analysis exam it is from rigorous advanced calculus. For the geometry/topology exam it is from undergraduate complex analysis. Short lists of topics on the exams and copies of recent examinations are available on the web .

To successfully complete the Ph.D. qualifying examination, a student is expected to obtain a result of high pass in two of the assessment options and a result of pass or high pass in a third. The Graduate Committee will be responsible for making the final determination as to whether the student has successfully completed the Ph.D. qualifying examination and may take into account all factors relating to the student's work.  

Students must successfully complete the qualifying exams before the end of their sixth semester to continue in the PhD program.

Comprehensive Examination

The purpose of the comprehensive examination is to determine whether the student has mastered the necessary general and specialized knowledge required to carry out dissertation research. The comprehensive exam has written and oral parts. To complete the written part, students write a short paper which may give an account of a research problem of interest, a significant example, or significant computations. The written part must be approved by the examining committee, which consists of a minimum of 4 tenured or tenure-track faculty, at least two weeks before the oral examination. The oral examination consists of a representation by the student, typically lasting one hour, followed by questions from the examining committee.

As part of the comprehensive examination, students are encouraged to prepare a detailed plan for the last years of their program. This plan should include a discussion of courses to take, seminars to participate in, faculty beyond the dissertation advisor to interact with, and possibly conferences to attend and professional development activities to undertake.

The Oral Comprehensive Examination is primarily, but not exclusively, on material in the area of concentration. The examination covers background material for the general area together with advanced references in a more specific sub-specialty.

After completing the comprehensive exam, students are expected to prepare a prospectus in consultation with their advisor. The prospectus is a detailed plan for the last years of their program. This plan should include a discussion of the research being undertaken as well as courses to take, seminars to participate in, faculty beyond the dissertation advisor to interact with, and possibly conferences to attend and professional development activities to undertake. Students must complete a departmental approval form.

G1. Professional Development Requirements (for students entering prior to Fall 2022)

PhD students must complete two professional development requirements chosen from this list:

• a foreign language requirement,
• a computing requirement, and
• a communication skills requirement.

Details of each requirement are given below. The requirements have been designed so that to a great extent they should be satisfiable by activities that would normally be undertaken by any good PhD student. The need for foreign language and computing skills varies considerably among fields of mathematics and so students should consult with their advisors on which requirements would be the best choice. Advisors may also suggest that students complete more than the minimum of two of these requirements. Students are urged to complete the professional development requirements as early in their programs as possible. In all cases, they must be completed before advancement to candidacy.

Foreign Language Requirement

A substantial portion of the mathematical literature is written in languages other than English, principally French, German, and Russian. Knowledge of Spanish is important for some fields of Mathematics Education. Being able to read and accurately translate these texts is a valuable skill in Mathematics and Mathematics Education research.

In order to fulfill the foreign language requirement, students will demonstrate their abilities to read and accurately translate mathematical texts in French, German, or Russian, (or, for students in Mathematics Eduction, texts relevant to that field in Spanish) by passing an examination given by a faculty member approved by the graduate committee. The student will prepare a careful, written translation of a text chosen by the examining faculty member (typically 5–10 pages) in a limited amount of time (typically 48–72 hours), with the aid of a dictionary and language reference works, but without the assistance of computers or other people. As a minimum standard, the translation must be mathematically accurate. We maintain a list of approved examiners .

Grading of language examinations is a significant burden on faculty and students should not make frivolous attempts to pass the examination without sufficient preparation. Faculty members may administer an oral “pre-test” to gauge whether the student appears to be prepared for the examination.

Results of foreign language examinations should be communicated to the graduate office by the examining faculty member using the language examination form .

Computing Requirement

Machine computation is an increasingly important component of mathematical research. Students for whom such computation will be relevant should master the needed programming skills and software packages during their graduate careers.

To fulfill the computing requirement, students should demonstrate their mastery of the relevant skills by carrying out a significant computing project under the supervision of a mathematics faculty member. This project might be tied to course work, the student's MS thesis, or his or her dissertation research. The precise nature of the project will be determined by the student and the sponsoring faculty member, but projects must have substantial mathematical content and should typically involve the following aspects of computing:

• formatted input and output
• appropriate use of data structures and algorithms
• use of structured programming techniques, possibly including calls to externally provided subroutines or functions.

Projects may be written in a standard programming language such as C or Fortran, or may use software packages such as Matlab, Maple, GAP or Pari. 

At the conclusion of the project, working code and documentation must be made available on the department's web site. The completion of the requirement should be communicated to the graduate office by the sponsoring faculty member using the computing examination form .

Communication Skills Requirement

The ability to communicate effectively, both verbally and in writing and to audiences of varying levels of sophistication, is essential to a successful career in industry, research, or teaching. The communication skills requirement gives students an opportunity to develop their capabilities in a variety of directions. To complete the requirement students must:

• prepare a basic web page containing information on the student's research, teaching, and other professional activities and make this page available on the department's web site
• prepare a professional CV and post it on the web site
• write articles or proposals and give lectures or presentations for audiences of various levels of sophistication so that at least one activity occurs in each row of the following table of examples. At least one of these activities must be verbal, and at least one must be written.

The entries in the table are meant to be illustrative and do not exhaust the possibilities. Written components should use TeX or other scientific text processing software. Verbal components may involve the use of such technologies as overheard transparencies or presentation software. Each component must be sponsored by a faculty member who will review the text or presentation and provide constructive feedback. When the sponsoring faculty member is satisfied with a student's performance on a component of the requirement, this fact should be communicated to the graduate office using the communication skills progress form .

G2. Professional Development Requirements (for students entering Fall 2022 or later or who opt in)

Students entering Fall 2022 or later will be subject to the following professional development requirements. Students who entered prior to Fall 2022 have the option of following these requirements or the requirements in the handbook listed under “Professional Development Requirements prior to Fall 2022.” Students who wish to use the new requirements must formally “opt in.” Note that those who opt in to the new requirements will be exempt from both the prior Professional Development requirement (G1) and the out-of-department course requirement.   These requirements are to be reviewed again during the 2024-2025 academic year.

Context: Students graduating from the Mathematics PhD programs move on to many different careers, including those in the academic, industrial, and government fields. In each of these fields, it is essential that applicants have more skills than those developed singularly through PhD research and teaching. Successful applicants to academic positions must be able to contribute to multiple axes of the academic endeavor, including mentoring, service, and leadership. Successful applicants to industrial positions must be able to work in teams with both their supervisors and colleagues and be able to communicate effectively with stakeholders. It is notable that most recently there has been a significant trend among graduating PhD students away from academic positions and toward industry regardless of the field of mathematical study.

Purpose:  The overarching goal of these requirements is to help students develop into well-rounded individuals who will gain important skills and competencies that will help them be competitive toward multiple career paths.  

The goal of professional development and community development requirements are to develop important “soft skills” necessary for future employment and job success. These include, but are not limited to communication, teamwork, critical thinking, organization, networking, problem solving, and leadership. Many of these are interpersonal skills which require interaction with people outside the mathematics community. Professional and community development are done throughout one’s career. The goal of the professional development requirement is to set a precedent for students to become lifelong learners toward bettering themselves as professionals. The goal of the community development requirement is to set a precedent for students to engage in their communities and work to improve them.

Requirements: In order to help balance and guide students through this process and create an individualized experience that allows students to handle changing circumstances throughout their time in the program, students will work with advisors each year to plan and assess the amount of work to be devoted to professional development and community development each year. For instance, if a student serves as an officer in an organization that takes significant time one year, it may be prudent to do community development with a lower time commitment the following year. Some years require more time to devote to studies (such as during qualifying exams or when finishing a dissertation), and this should be taken into consideration when planning, and also revisited during the year if changes need to be made. The student-advisor relationship will be relied upon to ensure that the student is able to complete these activities taking into account any mental or physical limitations such as anxiety, and to ensure students find a program that promotes their development without getting in the way of other necessary pieces of the graduate program such as exams and defenses.

Students will discuss a plan for community development and professional development at the beginning of each year and touch base with their advisor through the year with updates. It is expected that students will explore a few areas of work with the community (each with a low time commitment) in their initial years and then take a leadership role in later years in one or more areas of their choice. This will provide a wealth of different experiences and help the students develop a narrative for use in later job applications and interviews. Professional and community development activities may be paid; being paid does not preclude them from being considered as professional development or community development requirements for the year. The advisor and the student will need to agree on use of such requirements. It is generally expected that the time commitment for all of these requirements averages no more than 1-2 hours per week, and any commitments that will be longer (for instance, taking an out-of-department course) should be specifically discussed in this context as to whether it is a good plan for the student.

The Graduate Committee will review all plans each Fall semester in order to ensure that the plans are not too onerous on the student and satisfy the spirit of the requirements as stated above.

The following constitute the requirements:

Professional development requirement

Professional development is required yearly after year 1. In jobs, this is sometimes called training and sometimes people get certification. Possible examples are:

Community development requirement:

It is expected that the student will start to take leadership roles in the later years of their program. Here are several communities that may be considered:

Broader Impact requirement :

All National Science Foundation grants require the applicants to describe BOTH the proposal’s “Intellectual Merit” and “Broader Impact.” Intellectual Merit is the scientific content, and the research work would satisfy this piece. Broader Impacts are areas that benefit society. All students are expected to participate in at least one Broader Impact activity prior to completion of the PhD, which will likely come from the community and/or professional development activities. Some possible areas of broader impacts could be:

Evaluation:

At the end of each year, students will be evaluated in terms of the amount and quality of their community development and professional development and create recommendations for the following year. Part of this procedure will require keeping a current CV and writing a short narrative (not a list) on their community development and professional development activities as well as broader impact efforts. The evaluation will take place in the yearly Career Conversations between the Student and their Advisor, to be submitted to the Graduate Office. Professional and community development for the year will be evaluated at Satisfactory, Needs Improvement, or Unsatisfactory. Students with fewer than four years of Satisfactory community and professional development will need to petition for completion of the requirements and may be asked to remediate. The broader impact requirement can be satisfied at any time and completion will be submitted via form to the Director of Graduate Studies.

Advancement to Candidacy

A student may advance to candidacy once they have completed all program requirements other than the dissertation.  Students must advance to candidacy at least one semester prior to their dissertation defense. Failure to do so will delay the date of the dissertation defense.

Dissertation

The dissertation is a polished written account of a substantial new contribution to the mathematical sciences, publishable in a reputable journal. It is evaluated by an internal committee of at least 4 members who must be tenured or tenure-track faculty members or approved as equivalent by the Graduate College. One member may come from the minor department. Otherwise the members must be from the Mathematics Department. (Exceptions to this last rule may be granted by the Graduate Committee.) The dissertation committee approves the dissertation after a final oral defense. Students must give a copy of the dissertation to each member of the committee at least four weeks prior to the oral defense. Students have the option of also including an external reviewer who is not on the faculty of the University of Arizona. The inclusion of such an outside reviewer can provide the student with valuable feedback as well as help make the student's research known outside the local community. Students should register for Math 920 while working on their dissertation. The Graduate College requires 18 units of Math 920.

Students are encouraged to form their dissertation committee as soon as possible after the comprehensive exam. Requirements for how often this committee must meet may be found in the handbook .

The dissertation is by far the most important component of the PhD program and its quality and originality will have a major impact on the beginning of the student's professional career. Writing a quality dissertation should be the student's top priority.

Final Oral Examination

The final oral examination is a presentation and defense of the student's dissertation; the first part of the exam is open to the public.

Sample Plans

The following gives an example of how students might schedule their coursework, community & professional development, teaching, and other milestones. It is meant to be a guideline and not prescriptive.

Note that underlined items are possible Broader Impact activities.

Typical electives by research area (important core courses are in parentheses):

Ph.D. Degree Requirements: MATHEMATICS EDUCATION OPTION

The course requirements are 36 units of graduate credit in the major and 12 graduate units in a minor in Education (or related field) and 18 units of dissertation (Math 920).

Courses in Mathematics

Students will normally either take the first year graduate core courses in Algebra ( MATH 511A-B ), Real Analysis ( MATH 523A-B ), and Topology–Geometry ( MATH 534A-B ), or otherwise learn this material by the end of their first year of Ph.D. studies for the Qualifying Examinations. The remaining 18 units will be chosen in consultation with an advisor. These remaining units will include one year-long Mathematics course sequence that is not dual-numbered and is not part of the required core of algebra, real analysis, and topology-geometry. Some of the units will include relevant courses in Mathematics Education research (to be discussed with an advisor).

Courses in Education (or related field)

The 12 units in Education (or related) will be chosen in consultation with an advisor to ensure a coherent program of study. The courses will primarily be in Education. Courses in psychology, anthropology, sociology, women's studies, etc., may also be appropriate, depending on the student's research focus. Some suggested Education courses are listed below. EDUC 500, 501, 600, 601, 602; TTE 521, 524, 532, 545, 621, 640. Two courses in research design and methods (e.g., EDUC 600, 601, 602, or appropriate research methods courses in other fields such as sociology, anthropology, agriculture, ...) are required.

Teaching Experience or Practicum

Two or more years of pre-college teaching experience are required. Students can fulfill this requirement through 9 units of practicum in local schools. Such students will take 3 units per semester to complete one practicum at the elementary school level, one at the middle school level, and one at the high school level. (Note: these 9 units do not apply toward the required 36 units of mathematics nor the 12 minor units.)

Program Of Study

The same stipulations as given for the Ph.D. program in Mathematics .

Same as for the Ph.D. program in Mathematics .

Similar guidelines to those for the Ph.D. program in Mathematics , but the area of concentration will be in Mathematics Education.

Professional Development Requirements

Same as for the Ph.D. program in Mathematics except that the foreign language requirement may be satisfied in Spanish or American Sign Language as well as French, German, or Russian.

Same guidelines as for the Ph.D. program in Mathematics . The dissertation will be in Mathematics Education.

Doctor of Philosophy in Mathematics

• Fall January 10
• Spring September 10

International students may need to surpass the Graduate School’s minimum English language proficiency exam scores for this program. If the graduate program has unique score requirements, they will be detailed below. Otherwise, please refer to the Graduate School’s minimum score guidelines.

• 540 TOEFL Minimum score for admission
• 75 TOEFLI Minimum score for admission
• 6.5 IELTS Minimum score for admission
• 105 Duolingo Minimum score for admission

Degree Description:

PhD in Mathematics This degree is awarded in recognition of distinctive scholarship and original contributions to knowledge in Mathematics. The PhD program is especially designed to prepare the student for teaching at the graduate level and doing mathematical research in academic, industrial and business settings. Students studying various fields within the realm of pure mathematics would be included in this PhD program.

PhD in Mathematics (Applied Mathematics Option) The specialization of modern academic disciplines provides both a challenge to those who wish to do research at the interface of mathematics and its areas of application and many opportunities to make valuable contributions. The Applied Mathematics Option allows students from a range of backgrounds to pursue a traditional applied mathematics program, while retaining the option to thoroughly learn an area of application. Entering students may not necessarily have a bachelor’s degree in Mathematics. However, they will be required to demonstrate a grasp of the core areas of advanced calculus and linear algebra at the level of a bachelor’s degree in Mathematics. They will then be given great latitude to take specialized courses in Mathematics and their area of application.

PhD in Mathematics with Education Emphasis The degree of PhD in Mathematics with Education Emphasis is awarded in recognition of scholarship and original contributions to the teaching and learning of mathematics. The main difference from the other PhD choices is in the research focus. The requirements for this PhD include competence in core mathematics, as well as study in the research methodologies applicable to research in mathematics education.

Admission Requirements:

Students should have taken upper-level analysis and linear algebra courses before applying, and have the equivalent of an undergraduate degree in mathematics, statistics, or a related field. Additional upper-level courses like abstract algebra, functional or complex analysis, optimization, applied mathematics, topology, etc. will be taken into account.

For international students, either a TOEFL, IETLS or Duolingo score is required. The minimum score for admission is listed above. The minimum score required for assistantship consideration is TOEFL score of 100, IELTS of 7, or Duolingo of 130. Exceptions can be found here:  https://gradschool.wsu.edu/international-requirements/

Students applying to the PhD Mathematics program will be automatically considered for an assistantship position.

The application will require:

• Unofficial transcripts from all previous institutions.
• GPA above a 3.0 on a 4.0 scale.
• Email contact information for 3 references.
• Personal statement. The personal statement is your chance to highlight relevant experiences and discuss your future research interests.
• GRE scores are not required but encouraged.

Career Opportunities:

Academia (tenure-track positions at universities and colleges worldwide), Bio-statistics (health and pharmaceutical companies), Risk analysis (financial and insurance companies, investment management), Research (state or federal government, software development), Mathematics education (publishing, consulting and developing of educational software).

Career Placements:

Graduates from the Mathematics PhD program begin careers in both academia and government or industry. A few examples of career placements in academia from the past few years include:

• Postdoc, USDA Northwest Climate Hub and WSU Center for Sustaining Agriculture
• Assistant Professor, Central Washington University
• Assistant Professor, Marian University (WI)

A few examples of career placements in government and industry from the past few years include:

• Regulatory Data Scientist, Office of Regulatory Innovation and Assistance, Washington State Government
• Quantitative Model Validation Analyst, First Tech Federal Credit Union (OR)
• Data Scientist, Emsi Labor Market Analytics (ID)

Contact Information:

• Graduate Coordinator [email protected]

Colorado State University

College of natural sciences, department of mathematics, phd in mathematics.

Entering students will have a diagnostic and advising interview with the Graduate Director to plan an appropriate program of study.

General Requirements

• A minimum of 72 credit hours beyond the bachelor’s degree – all at the 300 level or above
• Course work in mathematics must be at the 400 level or above in order to be used towards the 72 credit hour requirement
• 3.0 GPA must be maintained in all course work
• 3.0 GPA must be maintained in all mathematics courses at the 400 level or above, excluding work in seminars and doctoral research
• No more than 24 credit hours of research/thesis may be counted towards the 72 credit hour requirement
• The Seminar Requirement must be satisfied – at most two additional credits beyond the two allowed for the MS may be counted towards the 72 hour requirement
• At most 1 hour of MATH 584 (Supervised College Teaching) may be counted towards the 72 credit hour requirement
• MATH 530 may not be counted toward the 72 hour requirement

Students entering with a master’s degree in mathematics may receive up to 30 credit hours towards the 72 credit hour requirements. For these students, a minimum of 42 credits must be earned at CSU after admission to a doctoral program, and at least 21 credits beyond the master’s degree must be earned in courses numbered 500 or above. For students enrolled in a continuous master’s/Ph.D. program at Colorado State University, all courses taken during the master’s program may be applied to the doctoral degree, even if the total master’s degree credits exceed 30.

It must be stated on the Ph.D. program of study that all credits earned on the MS will be counted toward the PhD; furthermore, this must be approved by the student’s advisory committee, the Department of Mathematics, and the Graduate School. For more detailed information on university-wide Ph.D. requirements, please consult the Graduate School website:  http://graduateschool.colostate.edu/

The faculty has passed a new breadth requirement for the PhD that is effective for incoming students Fall 2021.  Students who entered the program before Fall 2021 can also opt into this breadth requirement.  The policy is that students opting into this requirement must fulfill the whole requirement.  There will be no hybrid options .

New Breadth Requirement: 

Math Course Requirement: every graduate student must pass, with a grade of B- or higher, at least 10 courses in the math department of 3+ credits each, at the 500+ level.  

Breadth seminar requirement: every graduate student must pass the following two seminars within their first two years at CSU.   

• Seminar on applications of math . 1-credit. Fall.   
• Seminar on diversity of mathematics and mathematicians in the 20th century.  1-credit. Spring

Old Breadth Requirement:  To fulfill the breadth requirement for the Ph.D., all students will be required to take and pass, with a grade of B- or higher, at least 14 mathematics courses (each three or more credits) at the 500 level or above excluding:

Students entering with a master’s degree in mathematics from another institution may use courses approved to count towards the 72 credit hour requirement to also count towards this requirement.

At least two courses must be chosen from each of the lists below. This breadth requirement may be completed over the course of the student’s graduate program. Credit earned for comparable courses taken at other institutions may be evaluated by the Graduate Director, at the student’s request, to determine suitability for substitution for particular course(s) in the lists below.

List I:  MATH 501, 502, 566, 567, 570, 571, 601, 602, 666, 667, 672, 673

List II:  MATH 517, 519, 617, 618, 619, 717, 718

List III:  MATH 510, 520, 535, 540, 545, 546, 560, 561, 620, 621, 640, 645, 646

The faculty has passed a new Ph.D. Qualifying System effective for incoming students Fall 2022. 

For Ph.D. students admitted before fall 2022:

The departmental Ph.D. qualifying exam consists of two examinations, parts I and II.

PART I OF THE QUALIFYING EXAMINATION:

Purpose of Part I of the Qualifying Examination

The student will complete Part I of the Qualifying Examination in order to present evidence of performance at an appropriate level to the faculty. This evidence will be used as a basis for judging the candidate’s qualifications for continuing in the Ph.D. program in mathematics.

The Suite of Qualifying Courses: Overview

Each student seeking to pursue a Ph.D. in Mathematics will propose a Suite of Qualifying Courses. The Graduate Committee will review each proposal. There are Standard Suites of courses that are automatically approved. A student may alternatively propose a Special Suite of courses

What is a Proposal for a Suite of Qualifying Courses?

A Suite proposal always consists of:

A list (Suite) of specified courses that the student has completed, or will complete, during his or her graduate studies,

For each course on the list, a specification of the documentation that will be provided by the student for assessment purposes. If the course is a QE course*, this documentation MUST be the student’s completed final examination in the course.

*A QE course (Qualifying Examination course) is any course in a Standard Suite.

There are two types of Suites of Qualifying Courses (described below): Standard Suites and Special Suites.

How is a Suite proposal made?

• The student, together with a faculty advisor, will complete the Suite Proposal form and return it to the Graduate Coordinator. Note that the Graduate Director will act as the default proposal mentor for all Standard Suite proposals.
• The signature of the proposal mentor, indicating support for the proposal, is necessary to complete each Suite Proposal.
• The Graduate Committee will review all Special Suite proposal for approval (or not). The Graduate Director will automatically approve all Standard Suite proposals , if properly made.
• Each approved Special Suite Proposal will be announced to the faculty

More details on policies and procedures for the review of Suite proposals are included in the department policies and procedures manual.

What is a Standard Suite of Qualifying Courses?

The following four groups of Suites of courses are those recognized as Standard Suites through which a graduate student may complete Part I of the Qualifying Examination for the PhD program.

Analysis/Algebra/Geometry Suites

MATH 517, MATH 566 plus two of MATH 519, MATH 567, MATH 570, MATH 617

Applied, Computational, Interdisciplinary Suites in Algebra/Geometry

MATH 517, MATH 566 plus one of MATH 519, MATH 560 plus one of MATH 519, MATH 540, MATH 560, MATH 561, MATH 567

Applied, Computational, Interdisciplinary Suites in Analysis, Differential Equations and Dynamical Systems

MATH 517, MATH 560 plus one of MATH 540, MATH 545, MATH 617 plus one of MATH 510, MATH 519, MATH 540, MATH 545, MATH 546, MATH 561, MATH 617,MATH 618, MATH 640

Combinatorics/Algebra Suites

MATH 501, MATH 517, MATH 566 plus one of MATH 502, MATH 560, MATH 567

Using one of the Standard Suites as Part I of the Qualifying Examination is the appropriate choice for most students in the program.  The Graduate Director will act as the default proposal mentor for all Standard Suite proposals.  The Graduate Director will automatically approve all Standard Suite proposals, if properly made.

What is a Special Suite of Qualifying Courses?

Any other sort of Suite is a Special Suite. Special Suites may include QE courses.

For example:

• A student who has successfully completed a graduate-level course, comparable to one of the QE courses at another university, may request that this course be included in a Special Suite, OR that a more advanced CSU course in the same area, or a closely related one, be included in a Special Suite.
• A student’s CSU faculty advisor may recommend the inclusion of a particular graduate- level course (not necessarily a CSU course) in a Special Suite to the student. Such a course does not have to be a mathematics course. It must be a graduate-level course.

A proposal for a Special Suite must specify the documentation that the student will submit to allow assessment of performance in each non-QE course in the Suite.

How does a student complete Part I of the Qualifying Examination?

Once a student’s Suite of Qualifying Courses is approved by the Graduate Committee, the student must provide the assessment documents specified in the student’s Suite Proposal to a Qualifying Examination Committee (The role of the QEC is defined in the department policies and procedures manual) after each course in the Suite is completed. A student must receive a Pass- rating or higher from a QEC for each course in his or her Suite in order to complete that component of his or her Part I Examination. For more details on policies and procedures for assessment and evaluation of courses in Suites, consult the department policies and procedures manual.

PART II OF THE QUALIFYING EXAMINATION:

After the selection of a permanent advisor, the student together with the Ph.D. advisor, and sometimes in consultation with additional faculty, will devise a second evaluation to be completed by the student before the preliminary examination, and after the completion of Part I of the qualifying examination. This evaluation must be completed within two academic years following the passing of Part I of the qualifying examination. The student should demonstrate proficiency on material approximately at the 600-level. An MS completed at CSU will fulfill Part II, but students coming in with an MS need to fulfill this requirement locally.

Departmental Qualifying System for Ph.D. students admitted fall 2022 or after:

The qualifying exam consists of Parts (A) and (B).  It is followed by the Research Prelude, which is described below after Parts (A) and (B).  The Departmental Preliminary Exam and the PhD Dissertation and Final Examination are as described above.

Policies for Part (A) and (B) are listed in detail following this brief summary.

Time-Line: recommended timeline; required timeline.

  Part (A):

Each student must pass the final exam in 3 of the qualifying system (QS) courses, subject to requirement R1.

The QS course are:

Requirement R1:  each student must pass the final exam in at least one of MATH 517 or MATH 566.

Part (B):  

Every student must demonstrate conceptual understanding and technical mastery of a topic which builds on but is not covered in their courses. To do this, a student must provide written and verbal explanation of material contained within 1-2 chapters of a graduate level textbook or 1-2 research papers, or an alternative source.

A graduate student will qualify for the doctoral program by completing parts (A) and (B). The qualifying exam committee will meet every semester to determine which graduate students have completed parts (A) and (B) successfully.

Part (B) Agreement form

Part (B) Evaluation form

Exceptions:

Part (A): if a graduate student has already succeeded in a course that can be considered equivalent to one of these 8 courses at another institution, they can request to use the final exam from that course to fulfill a portion of the part (A) requirement. This request can be made for up to 3 courses and may include either or both of 517 and 566. The graduate committee will evaluate whether or not to approve each such request, on a course-by-course basis.

Part (B): no exceptions.  

Policies for Part (A):

For each of the QS courses:

There will be a syllabus of recommended topics and a list of recommended textbooks, which will be considered current if it was approved by the math faculty in the last 10 years. If the syllabus is current, the faculty member is required to cover the recommended topics for at least 10 weeks of the course.

The faculty member will choose another faculty member as a course consultant, preferably someone who previously taught the course. The course consultant will be responsible for approving the following material developed by the faculty member: the syllabus, the choice of textbook, and the structure, sources, and content for the final exam.

Students will be assessed on the material from the syllabus of recommended topics at the end of the course. For each QS course except MATH 574, the qualifying exam will be an in-class 2-hour exam at the time scheduled for the course final by the university and the final exam will only cover topics from the syllabus of recommended topics. Students may be expected to complete problems about material discussed in class, material from homework, or new material, as long as it is closely related to the material on the syllabus of recommended topics.

The faculty member will assign one of the following grades to each final assessment: strong pass, pass, weak pass, or fail . The course consultant will be responsible for approving the cut-off between a passing and failing grade on the qualifying exam.

To determine grades for the course, the faculty member may choose to evaluate students’ understanding of material not on the list of recommended topics and outside of the course final exam. The faculty member has complete decision-making authority for the content and method of this evaluation, as long as it is clearly communicated in the syllabus.

The department will encourage faculty to invest in these courses and will enable students to meet potential PhD advisors by rotating a broad range of faculty through the QS courses.

Policies for Part (B):

A student will ask 2 faculty members from the math department to be on their part (B) evaluation committee.  The two faculty members are responsible for evaluating the student’s part (B) written work and presentation, but are not committed to being the student’s future advisor.

The student and faculty members will fill out the part (B) agreement form.  In particular, they will decide on the content, references, and time-line for the part (B) evaluation. The expectation is that the material in the part (B) project should be approximately equivalent to five weeks of a graduate course.

They will decide on the length, content, and format of the written assessment. For the alternative sources, valuable ways to provide written explanation might include an analysis of technical reports and real data, instructional materials for new techniques, a survey about a series of interviews with an expert, or documentation for new code.

They will decide on the audience, length, content, and focus of the verbal assessment. The part (B) presentations will not be announced by the department. The student and faculty members may choose to invite several people, or to have a private meeting.

The faculty members will evaluate the student’s part (B) written and verbal material and decide whether the student passed or failed. In either case, the faculty will provide the student with written feedback, for example, by filling out the questions on the part (B) evaluation form. A student who attempts but does not pass part (B) may try again, with either the same or with different faculty members, but must pass part (B) by the second attempt.

The committee recommends that Part (2) of the current qualifying system be renamed and re-interpreted as a requirement for the doctoral degree process.

Research Prelude:

Every doctoral student must propose a research area and general topic to their doctoral committee. With this proposal, the student should demonstrate the ability to synthesize research concepts with a written paper and presentation, including examples, data, or proofs. This may involve either an analysis of an important reference, an analysis of a recent publication, or new material generated by the student.

A doctoral advisor may decide that a masters defense completed at CSU can fulfill a student’s research prelude requirement. If a student switches advisers, it is up to the discretion of the new advisor whether the student should complete another research prelude.

Required for all PhD students:

Departmental preliminary examination:.

At some time after a Ph.D. advisor and committee have been assigned, the student will be required to pass an oral preliminary examination prepared and administered by the student’s committee.

This examination must be passed at least one academic year (two semesters; the summer session may be included) before defense of the dissertation. The purpose of this oral examination is to evaluate the student’s proposed research project.

Two weeks prior to the examination, the student must provide to the Ph.D. committee a written copy of the dissertation proposal, including a survey of the relevant literature, a description of the problem to be investigated, and an outline of methodology to be considered. The student either passes, passes provisionally (certain additional conditions must be met), or fails. In case the student fails the examination, it may be repeated once subject to Graduate School regulations. After this examination has been passed, the student becomes a Ph.D. candidate.

Doctoral students in the mathematics department at CSU are considered to achieve “candidacy” for the degree upon passage of preliminary examinations.

A written report of the examination will become part of the student’s file.

PhD Dissertation and Final Examination

Each Ph.D. candidate must prepare a dissertation containing original mathematical research that is acceptable to the students committee. The student must successfully defend this dissertation in an open oral examination before the committee as well as other interested parties. This examination can be given no sooner than two semesters (including summer) after the student becomes a Ph.D. candidate.

The Dissertation:

• Dissertations will be submitted electronically through ProQuest/UMI though the Colorado State University libraries ETD submission website. Instruction for submitting a thesis are detailed on the Graduate School’s website and can be found here .
• The format of the thesis must strictly adhere to the requirements of the Graduate School. A requirements checklist, formatting guide and sample pages are available on the Graduate School website and can be found here .
• The thesis must be submitted to the Graduate School by published deadlines .
• PhD students must also submit the Survey of Earned Doctorates confirmation certificate with the Thesis/Dissertation Submission form.

The Graduate Coordinator will assist with room scheduling once the date and time for the defense have been established with your committee.

The student’s committee must have adequate time to review the thesis before the final examination. Therefore, the committee must have the thesis at least two weeks (including two weekends) before the final examination which meets the following standards:

• The Thesis must be in final form
• The Print is letter quality, and a uniform style has been used throughout.
• Figures, printouts, etc. are clear and easy to read.
• Text is double-spaced and printed on only one side of the paper.

Results of the final oral examination will be reported on the GS24. Voting is limited to the members of the student’s committee, and a majority vote is necessary to pass the examination. A tie vote is interpreted as failure to pass the examination. All committee members must vote to either Pass or Fail the student; there is no option to withhold a signature. Committee members who are not academic faculty do not have a vote on the final examination.

If significant revisions to the thesis are required by the committee after the final oral examination is complete, a reasonable amount of time must be given for committee members to re-review the thesis once the revisions have been made.

A candidate who fails the final examination may be reexamined once and, for the reexamination, may be required to complete further work. The reexamination must be held not later than 12 months after the first examination. The examination must not be held earlier than two months after the first examination unless the student agrees to a shorter time period. Failure to pass the second exam results in dismissal from the Graduate School.

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Ph.D. Program

• Ph.D. Requirements
• Ph.D. Coursework
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The Mathematics Department offers a selective and focused doctoral program for talented students specializing in three broad research areas: Geometry/Topology, Number Theory/ Representation Theory, and Algebraic Geometry.

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Ph.D. Theses

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Applying to the Ph.D. Program

Application requirements.

Applications received by January 20 will receive full consideration for admission and Teaching Assistantships. 

 A complete application includes:

• Application forms
• Official transcripts
• An abstract of mathematics courses taken at other institutions 
• GRE scores (including the Mathematics subject test) -  GRE scores are optional for the upcoming admission cycle due to COVID-19
• Three letters of recommendation, preferably from undergraduate teachers of Mathematics
• A detailed description of coursework .
• Official TOEFL scores, for students whose native language is not English
• The application fee.  

Some Advice

At a minimum, students should have had at least one year of Abstract Algebra and one year of Analysis (real and complex) in their undergraduate preparation. It is helpful to list the textbooks used for these courses. Since our research is centered around Number Theory/Representation Theory, Geometry/Topology/Dynamics, and Algebraic Geometry, our aim is to accept only those students with an interest in at least one of these areas. The Statement of Purpose is a good place to describe this.

Financial Aid

We offer financial aid stipends and tuition remission for qualified students. It is anticipated that support will be provided for five years of study, given reasonable progress toward the degree and acceptable performance of TA/TF duties.

Teaching Fellowships

Fellowship award basics.

Fellowship awardees are designated as a TA (Teaching Assistant) in the first year. Under normal circumstances, assuming reasonable progress towards the degree and acceptable performance as a TA, awardees are typically supported for subsequent years as TFs (Teaching Fellows). It is anticipated that support will continue through the fifth year of study. Each TA/TF award includes a complete tuition scholarship, plus stipend paid over the nine-month academic year, plus office space equipped with networked computer (shared with one other TA/TF).

Fellowship Availability

All Ph.D. applicants are considered for fellowships.

TA/TF Responsibilities

TAs usually work together with full-time faculty in large lecture classes and conduct recitation sections, problem sessions, and perhaps some tutoring. TFs usually are responsible for teaching one section of a core-level Mathematics course in each semester. These activities require about 15 hours per week, above and beyond your attention to your own coursework. 

TA/TF Scheduling

Generally, TA assignments in the fall semester of the first year are based on incoming students' course schedules. In subsequent semesters, scheduling preferences for the upcoming term are solicited from everyone involved. We try to accommodate everyone as best we can although some compromises are necessary each semester.

Teaching Excellence Awards

Teaching Fellows who achieve the greatest success as a TF will be considered for a Donald White Teaching Excellence Award, a cash prize given to outstanding graduate student teachers at Boston College.

Graduate Teaching Seminars

The department requires Teaching Assistants and Teaching Fellows to participate in a weekly seminar designed to help students develop and refine their classroom skills. A major component of this seminar is the use of case studies.

Appreciation

As a Teaching Assistant or Teaching Fellow in the Mathematics Department, you will receive extensive support from the department in your teaching endeavors. We recognize the tremendous job done by our TA/TFs; year after year they play an essential role in our undergraduate Core curriculum and bring an exciting energy to the fifth floor of Maloney Hall.

You will work in an environment where your contributions are valued by everyone around you, including other students, faculty, and staff. Regardless of your future career, your experience as a TA/TF will contribute to your mathematical, management, and communication skills, and you will enjoy the experience.

Mathematics in Boston

Boston is a world-renowned center for Mathematics and provides a vibrant intellectual climate in which graduate students thrive, with many outdoor and cultural opportunities in the beautiful New England area.

Boston College is a member of the Graduate Consortium, and BC mathematics graduate students may cross-register for one course each semester at Boston University, Brandeis, or Tufts.

Faculty Research Areas

Our internationally recognized faculty publish research in top journals. The department also sponsors the BC-MIT Number Theory Seminar, the Algebraic Geometry Northeastern Series, and the Hamilton Geometry and Topology Workshop at Trinity College. 

Geometry/Topology

Geometry/topology .

The research areas of our faculty  in Geometry/Topology include:

• Heegaard-Floer and Khovanov homology;
• Hyperbolic geometry;
• Kleinian groups;
• Knot theory;
• Spectral geometry; and
• Three-dimensional manifolds and their geometry.

Faculty interested in this research area 

Number theory/representation theory, number theory/representation theory .

The research areas of our faculty in Number Theory/Representation Theory include:

• Analytic number theory;
• Automorphic forms and representations;
• The cohomology of arithmetic groups;
• Galois representations;
• Geometric representation theory;
• The Langlands Conjectures;
• Multiple Dirichlet series;
• Representations of p-adic groups; and
• Shimura varieties.

Algebraic Geometry

Algebraic geometry .

The research areas of our faculty in Algebraic Geometry include::

• Algebraic geometry and dynamics;
• Birational geometry;
• Derived categories;
• The geometry of moduli spaces; and
• Logarithmic geometry.

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PhD in Systems Science & Mathematics

Learn how you can earn your PhD in Systems Science & Mathematics at the McKelvey School of Engineering.

Earn your PhD in Systems Science & Mathematics at the McKelvey School of Engineering for the opportunity to develop and optimize complex systems including information systems, quantum systems and telecommunication networks.

Start your PhD application

Full Support & Funding

Our PhD students are fully funded , including full tuition support and health insurance.

As a doctoral candidate, you will also receive a generous stipend to cover living expenses. This support is guaranteed as you continue to make satisfactory progress towards your degree.

Additionally, doctoral students may also qualify for one of the fellowships:

• The Ann W. and Spencer T. Olin – Chancellor’s Fellowship
• McDonnell International Scholars Academy
• Dean's International Award

St. Louis offers big-city amenities at a small-town price allowing you to pursue your professional goals without financial strain.

See how the cost of living in STL compares (Nerd Wallet)   

Requirements

The PhD in Systems Science and Mathematics is an academic doctoral degree designed for full-time students interested in an academic, laboratory or industrial research and development career in a specialized area of systems, control or applied mathematics.

Students pursuing the degree must complete a minimum of 72 credit hours of post-baccalaureate study consistent with the residency and other applicable requirements of Washington University in St. Louis and the Graduate School. These 72 units must consist of at least 36 units of course work and at least 24 units of research, and may include work done to satisfy the requirements of a master's degree in a related discipline. Up to 24 units for the PhD may be transferred to Washington University in St. Louis from another institution.

Degree Progression

The following milestones provide an example of how PhD students are expected to progress toward their degree. Students who enter the program with a master's degree may be able to shorten this timeline by one year or more.

• Complete at least 36 hours of post-baccalaureate course work
• Pass an oral qualifying examination, to be taken before the second academic year of the program
• Pass an oral preliminary research examination, to be completed within two years of passing the written qualifying examination, and at least one year prior to completion of the dissertation
• Satisfy the general residency requirement for the Graduate School 
• Satisfy the general teaching requirement for PhD degrees offered by the Graduate School
• Write a doctoral dissertation that describes the results of original and creative research in a specialization within electrical engineering or systems science and mathematics
• Pass a final oral examination in defense of the dissertation research
• Take ESE 590: Electrical and Systems Engineering Graduate Seminar each semester

Admission Information

The deadline to apply for the PhD in Systems Science & Mathematics is Dec. 15 . 

As COVID-19 is making it difficult for many students to be able to safely take the GRE General test, the test is optional for this program in Fall 2023.  

Visit the Application Checklist for complete details on what's required in your application packet. If you have any questions, check out our admissions FAQs or contact  Stacia Burd, Graduate Program Advisor, at  [email protected] .

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Our faculty strive to advance basic science and solve cutting-edge engineering problems relevant to society in the areas of:

• Applied physics
• Devices and circuits
• Signals and imaging
• Systems science

Discover our research areas

Madi Hester Graduate Program Coordinator [email protected] Green Hall, Room 1101

Stacia Burd Graduate Program Advisor [email protected] Green Hall, Room 1101

General Inquiries [email protected]

*For questions regarding your application please contact  Graduate Student Services

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Higher test scores needed to graduate will impact 37% of county seniors. See the new rules

This year's seniors will be held to a higher standard, particularly in reading. here's what to know..

Thousands of Palm Beach County students in line to take an alternative path to graduation by using their college entry exam scores are at risk of falling short as the state embraces repeatedly delayed, more challenging standards.

For years, Florida has allowed students who fail the state's high school exams in math and reading to earn a diploma using their results on other tests like the ACT or SAT . But in an effort to increase academic rigor, the state Department of Education raised the graduation requirements before the class of 2024 even entered high school.

For the last several years, legislators delayed the implementation of those new requirements in an effort not to kick students who were already down due to the pandemic. The state even waived some testing requirements for graduating classes in 2020 and 2021.

However, this year's seniors, including roughly 13,000 in Palm Beach County, will be held to the higher standard — particularly in reading.

"(The state) has been trying to do this for a number of years," said Adam Miller, a senior chancellor at the state Department of Education, speaking at a Feb. 21 meeting of the school district's academic advisory committee. "For last school year, at the end of the school year, they did relent and leave things mostly as they were, which was great, and we see that reflected in our graduation rates."

From 2023: Last-minute change in Florida graduation requirements helps 700 Palm Beach County seniors

Last year, district records show that 37% of seniors across Palm Beach County used the alternative test scores, often referred to as "concordant scores," to get their diplomas. But for some groups of students, that number was far higher:

• 71% of students learning English used concordant scores to graduate.
• 48% of Black students used concordant scores to graduate.
• 45% of students who qualify for free or reduced-priced lunches used the concordant score requirements to get a diploma.

The higher bar to reach a diploma doesn't necessarily mean all of those students won't get to graduate, but Palm Beach County's data shows the new requirements will disproportionately affect the ability to graduate for students of color.

What are new reading requirements in Florida for 2024 seniors?

A majority of graduates will get their diplomas because they've met Florida's requirements: They've kept a 2.0 unweighted cumulative grade-point average, completed course credit requirements and achieved passing scores on the 10th grade Florida Standards Assessment for English Language Arts and the Algebra 1 or Geometry End-of-Course Assessment.

If a student does not pass the state reading or math tests, they can earn a concordant, or comparative, score on the common college entrance exams and still earn their diploma.

On the reading side, students who don't pass the language arts exam will now need to score a 480 on the SAT's reading and writing section. Last year, the required score was 430. The standard remains the same as last year for the ACT English and reading sections: Students must score an 18.

This year's requirements remove the SAT's reading-only section and the ACT's reading-only section as potential concordant scores. But they also added a new test, the Classic Learning Test, as an option. The state requires a student to score a 36 or higher on that test.

The Classic Learning Test is an exam that was first launched in 2016 and is popular with home-schooled and privately schooled students as an alternative to the SAT or ACT. The test markets itself as a step toward "reviving great education" in classical studies and texts. Its board of academic advisers includes conservative activists and people affiliated with religious schools like Hillsdale College in Michigan, NPR reported.

Last fall, Florida's state university system approved using the Classic Learning Test score for undergraduate admissions, making Florida the first state in the country to do so.

What are Florida's new math requirements for 2024 seniors?

On the math side, graduation requirements remain the same as last year for the following exams:

• PSAT Math section: Students must score 430 or above.
• SAT Math section: Students must score 420 or above.
• ACT Math section: Students must score 16 or above.

This year's requirements remove the option to use the Postsecondary Education Readiness Test, given to all high school sophomores, as a graduation alternative. Students caught in a lurch in 2023 took the test repeatedly in the final weeks of school to score the 114 required to graduate.

Finally, the quantitative reasoning section of the Classic Learning Test was added as a means to earn a concordant score this year. Students need to score 11 or higher to graduate.

Will there be any changes to graduation requirements in 2024?

School leaders confirmed this month that they don't expect any last-minute changes to the graduation requirements this year, but said they're confident in their test-prep programs for students who will use concordant scores to graduate.

"We're going to be under the new rules and expectations," Ed Tierney, the district's chief of schools, said at a March 6 school board meeting. "We've put a lot of time into graduation rates. ... High schools do a great job of preparing students for that. We push hard on the SAT and the ACT and refine our instruction on that."

Tierney also mentioned the school district has "found a path" to helping students prepare for the Classic Learning Test, specifically in the math section. He didn't provide any more details at the March 6 meeting.

Katherine Kokal is a journalist covering education at The Palm Beach Post. You can reach her at [email protected]. Help support our work, subscribe today!