## 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 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.)

## 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.

## UCI Mathematics

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

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

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

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

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

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

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

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

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

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

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

- Applied & Computational Mathematics
- Geometry & Topology
- Probability

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

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

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

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

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

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

Areas of Specialization and Their Corresponding Advancement to Candidacy Courses

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

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

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

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

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

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

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

*PhD Requirements Summarized*

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

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

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

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

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

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

Graduate Program in Mathematical and Computation Biology (MCSB)

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

http://mcsb.uci.edu/

- MyU : For Students, Faculty, and Staff

## 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. Students who have declared a formal minor are required to have a committee member from the minor program.
- 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.
- At least two major field thesis reviewers are required.
- One committee member must represent a field outside of mathematics. 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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## 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.

© Stanford University . Stanford , California 94305 .

## Quick Links +

Math phd requirements starting fall 2021.

Course Requirements – Core and Electives (48 units) (see attached Progress Checklist)

Students must acquire proficiency in mathematics by completing 48 units or passing twelve graduate courses (MAT 280 and below). Three of these may be taken outside of mathematics if they are related to the student’s area of specialization.

- MAT 201AB (Analysis) (4 units each)
- MAT 250AB (Algebra) (4 units each)
- MAT 215A (Topology) (4 units)
- MAT 239 Differential Topology (4 units)
- Similar courses completed at another institution may substitute for all or part of the Math 201AB, Math 215A, Math 239, and Math 250AB, if the GPC gives its approval.
- Elective Courses (24 units) : All courses at the 200 level (MAT 280 and below) count towards elective units. No more than 6 units of MAT 280 (Topics in Mathematics) may be counted toward the M.A.
- Research – Students may take research units (299) when doing research with a faculty member. There is no specific requirement for a minimum number.
- English Language Requirement - Students who have not obtained a previous degree at an approved English-medium institution or demonstrated English- language proficiency through an appropriate exam (e.g. TOEFL) are required to complete appropriate English-language courses, as described in the policy Graduate Student Course Requirements – English as Second Language (GC2018- 02). Courses taken in satisfaction of this requirement do not count towards the units required for graduation
- Summary - 24 units of core coursework, and 24 units of electives are required for a total of 48 units. Full-time students must enroll for 12 units per quarter including research, academic and seminar units. Courses that fulfill any of the program course requirements may not be taken S/U unless the course is normally graded S/U. Once course unit requirements are completed, students can take additional classes as needed while completing their dissertation, although the 12 units per quarter are generally fulfilled with a research class (299) and perhaps seminars. Per UC regulations students should not ordinarily enroll in more than 12 units of graduate level courses (200) or more than 16 units of combined undergraduate and graduate level (100, 200, 300) courses per quarter.

Teaching Skills

The department has a commitment to develop outstanding teaching skills in its Ph.D. students. All Ph.D. students are required to be teaching assistants for at least one quarter. Exceptions require approval of the Graduate Program Committee. Students beyond their first year are encouraged to apply for positions as Associates In mathematics to develop and improve their lecturing skills. The department makes every effort to give all students exhibiting solid teaching skills the opportunity to serve at least one quarter as an Associate In mathematics.

Advancement to Candidacy

Students entering with a BA or BS should advance to candidacy by the end of the ninth quarter. Students entering with a MA or MS or equivalent should advance to candidacy by the beginning of their seventh quarter. Before advancing to candidacy for a doctoral degree, a student must have satisfied all requirements set by the graduate program, must have maintained a minimum GPA of 3.0 in all course work undertaken (except those courses graded S or U), and must have passed a Qualifying Examination before a committee appointed to administer that examination. Normally, students advance by the end of the ninth quarter. The student must file the appropriate paperwork with the Office of Graduate Studies and pay the Candidacy Fee in order to be officially promoted to Ph.D. Candidacy. Refer to the Graduate Council policy on Doctoral Qualifying Examinations (GC2005-02) for further details.

Preliminary Examination, Qualifying Examination and Dissertation requirements

Preliminary Examination is a written exam, which comprises of graduate material in Analysis, Algebra, and Topology as covered in the following six graduate courses: 201AB, 250AB, 215A and 239. The exam consists of three parts: Analysis (201AB), Algebra (250AB), and Topology (215A, 239). The exam is written and administered by the GPC. The exam is offered twice yearly, normally at the beginning of the Fall and Spring quarters. Students in the Ph.D. program may take any or all three parts, during a given offering of the exam. Students in the Ph.D. program must pass two of the three parts of the Preliminary Exam by the beginning of the student’s 7th quarter. Passing two of the three parts is considered fulfilling the Preliminary Examination requirement.

Qualifying Examination (QE ) must be passed by the end of the student’s 9 quarter. This is the final requirement for the advancement to candidacy, all other requirements must be completed, and a Dissertation Advisor selected, before a student can petition for this examination.

General Information - The purpose of this examination is to determine if the student is capable of independent research. In consultation with the Dissertation Advisor, the student proposes to the GPC a date for this exam, the material to be included, and a committee of four examiners. Normally three of the members are members of the Department of Mathematics. Per Graduate Council guidelines, at least one member must be external to the Department. The Dissertation Advisor can be a member of the committee but cannot be chair.

Written Portion of the Exam –Dissertation Prospectus-Thedissertation proposal should be between one and three pages in length and should contain an outline of the general context of the thesis research, a description of the specific problem(s) to be addressed, and an indication of the methods and techniques to be used. The proposal must be submitted to GPC for their approval at least 6 weeks before the proposed date of the exam. After approving the proposal, the GPC will recommend the appointment of the QE Committee to Graduate Studies. Example QE Proposal here

Oral Portion of the Exam - Normally the QE is given in the form of a seminar talk on the candidate’s proposed research topic, in which the committee members have the opportunity to ask in-depth background questions and evaluate the breadth of the candidate’s knowledge. The talk should last no longer than 45 minutes, but, the entire QE could last up to three hours.

- “Pass” (no conditions may be appended to this decision)
- “Not Pass” (the Chair’s report must be submitted to Graduate Studies within 72 hours and specify whether the student is required to retake all or part of the examination, list any additional requirements, and state the exact timeline for completion of requirements to achieve a “Pass”), or
- "Fail"
- If the committee is unable to reach a unanimous decision, the QE Report must be submitted to Graduate Studies within 72 hours and specify the opinions of the majority and minority of the committee.
- If a unanimous decision takes the form of “Not Pass” or “Fail,” the Chair of the QE committee must include in its report a specific statement, agreed to by all members of the committee, explaining its decision and must inform the student of its decision. If a student does not pass the QE on the first attempt, then, at the recommendation of the QE Committee, and subject to the approval of Graduate Studies, the student may repeat the exam once, within six months of the first attempt. Failure to pass on the second attempt disqualifies the student from continuing in the Ph.D. program.

Dissertation Requirements : Ph.D. candidates will complete a written dissertation that represents an original and significant contribution to the scientific body of knowledge. The dissertation committee will evaluate when this dissertation requirement has been met. We do not have any program-specific requirements, such as length or presentation format. The Thesis Advisor provides information about appropriate length and formatting guidelines as expectations vary within subfields of mathematics.

Ph.D. Students are required to give a 50- minute seminar presentation, open to the public, on their dissertation subject. After the seminar, the student’s dissertation committee may meet privately with the student to discuss the contents of the dissertation. Satisfaction of this requirement must be verified by the Dissertation Committee Chair.

Normative Time to Degree

Students entering with a BA or BS should accomplish their PhD by the end of their fifth year. Students entering with a MA or MS should accomplish their PhD by the end of their fourth year.

## PhD Requirements

The purpose of the Doctor of Philosophy program is to prepare the student for research and teaching.

It is expected that each graduate student who is a U.S. citizen or permanent resident should begin the process of establishing North Carolina residency during the first semester of graduate study. Detailed information can be found here.

## Time and residency requirements

First year courses.

Courses taken during the first year are usually chosen from the following list, which is designed to prepare students for the PhD qualifying examinations. Master’s students will typically take these courses as well in the first year before focusing in the second year on the requirements for a Master’s degree.

## Qualifying Exam Requirements

There are five qualifying exams:

- Algebra: Math 676 and 677
- Analysis: Math 653 and 656
- Geometry-Topology: Math 680 and 681
- Methods of Applied Mathematics: Math 668 and 669
- Scientific Computation: Math 661 and 662

PhD qualifying exams are given twice each year, near the beginning of classes in August and January. PhD students are encouraged to take the PhD qualifying exams early and in principle would be ready to take them in August after the first year. The qualifying exams, sometimes called comps, are subject to the following rules:

- A PhD student can pass either the Pure Math option for the for the qualifying examination or the Applied Math option.
- To pass the Pure Math Option, a student is required to pass three of the five qualifying exams by the beginning of the sixth semester. Any three of the five can be passed.
- To pass the Applied Math option, a student is required to pass Methods of Applied Math and Scientific Computation by the beginning of the sixth semester.
- In each examination period a student can take any number of qualifying examinations, from one to five.
- A graduate student who passes a PhD qualifying examination, as determined by the Graduate Committee, will receive credit for that examination toward the PhD requirements.
- If a student does not pass at least one PhD exam by the beginning of the fourth semester, the department will not guarantee financial support past the end of the fourth semester. However, if such a student passes one exam by the beginning of the fifth semester, then reinstatement of financial support will be considered.
- To remain in the PhD program, a graduate student must pass the written qualifying exam, with either the Pure Math or the Applied Math option, by the beginning of the sixth semester. A student who does not do this will be out of the PhD program.
- Any exceptional cases will be decided by the graduate committee.

## Course Requirements

All Ph.D. students must take and receive a grade of P or higher in at least six courses from the following two lists:

- Comprehensive courses that are not basic courses for any of the three comprehensive exams passed by the student
- Algebra: Math 641, 643, 771, 774, 775
- Analysis: Math 657, 751, 753, 754, 857
- Applied Mathematics: Math 635, 761, 762, 768, 769, 892
- Geometry-Topology: Math 781, 782, 773, 775, 776

Furthermore, of these six courses every PhD student must take and receive a grade of P or higher in three courses numbered over 700 from the second tier list

A student taking a pure math option, respectively an applied math option, may replace one, respectively one or more, course/s in the second tier requirement with other graduate level courses inside or outside of the department. For each replacement course, the student must obtain permission, prior to registration, from both the adviser and the graduate director.

Any incomplete grades must be resolved within one year (ideally within one semester).

## Oral Examination Requirement

When the student is prepared and has selected a direction for the PhD dissertation, the PhD candidate must pass an oral examination on material basic to the proposed dissertation area. The examination will be conducted by the PhD committee of the student, which consists of five faculty members and is chaired by the student’s adviser. The committee will recommend action to correct any deficiencies noted during the oral examination.

After passing the oral examination the candidate continues working in the area of concentration to complete the PhD dissertation.

## Dissertation Defense

Computer language requirement.

A Master’s or PhD student must pass a computer language requirement by demonstrating a certain level of programming ability. Please note the following guidelines:

- Computer language requirements are the same for PhD and Master’s students.
- Passing the following courses at UNC will be sufficient to satisfy the computer language requirement: MATH 565, MATH 566, MATH 661, MATH 662, MATH 761, MATH 762, COMP 110, COMP 116, COMP 121, COMP 401
- Any computer science course which lists one of these courses as a prerequisite is also acceptable.
- A Master’s or PhD student may also satisfy the computer language requirement by passing an approved one semester undergraduate course on computer programming at any university.
- If the course title title on the transcript is not self-explanatory, then a syllabus, text or other information may be required. In all cases where there is some question about whether a course fulfills the requirement the Graduate Director will decide in consultation with the Graduate Committee.

## Teaching Requirement

Students are required to take and successfully pass the TA Teaching Seminar, a special section of Math 920, during their first fall semester of their program. Students are also required to perform a minimum of two semesters of instructional service.

A semester of instructional service can be satisfied by any of the following:

- Teach one course: 12 hours
- Lead 4 recitations (includes 110L and 231L): 3 hours each
- Lead 2 labs (383L, 528L, 529L): 6 hours each

## For Additional Information Contact

Ann Van Elsue Graduate Student Services Manager Phillips Hall 331A 919-962-4178 [email protected]

## Program Requirements

The Department of Mathematics offers graduate courses on various levels, all of which are oriented toward research. There are numerous seminars that encourage research even more directly. The content of courses varies considerably from year to year, and the course descriptions below should be read only as a rough guide. Students usually acquire the standard beginning graduate material primarily through independent study and consultations with the faculty.

## Ph.D. Requirements

To earn the Doctor of Philosophy (Ph.D.), the student must pass a language requirement and both portions of the general examinations, submit an acceptable dissertation and sustain a final public oral examination.

## "Incidental" M.A. Requirements

To qualify for the "Incidental" Master of Arts (M.A.), the student must pass the language requirement and the first part of the general examination, and be recommended by the faculty. (Note: It is expected that ALL students will pursue their Ph.D..)

## One-Year in Residence

The student must be in residence for at least one academic year before standing for the general examination. If, however, a student wishes to take his/her general examination during either the October or January generals period during their first year, it must be approved by the DGSs and the Graduate School. Otherwise, the more acceptable time is during the April/May exam period.

## Language Requirement

The student must satisfy the language requirement by demonstrating to a member of the mathematics faculty a reasonable ability to read ordinary mathematical texts in at least one of the following three languages: French, German, and Russian. The language test must be passed before the end of the first year and before standing for the general examination.

## General Examination

In the first two years, students acquire a background in mathematics. Depending upon individual preparation, a student may take the general examination in the first or the second year of study.

The student must stand for an oral exam administered by a committee of three professors, including the advisor who serves as chair of the committee. A typical exam can last 2 to 3 hours. Areas covered are algebra, and real and complex variables.

The student must also choose two (2) special or advanced topics. These two additional topics are expected to come from distinct major areas of mathematics, and the student's choice is subject to the approval of the Department. Usually, in the second year, and sometimes even in the first, the student begins investigations of his/her own that lead to the doctoral dissertation.

The department collaborates with the Department of Physics in offering work in mathematical physics and leading to an advanced degree. For a student interested in mathematical physics, the general examination is adjusted to include mathematical physics as one of the two special topics.

A plan of study also may be coordinated with the Program in Applied and Computational Mathematics (PACM). See their program description for more information.

There are three general examination periods each academic year--October, January, and April/May. It has been a tradition of the students to post their exams as a resource and study guide for other students, see Graduate Students' Guide to Generals .

## Dissertation and Final Public Oral (FPO) Exam

The student must prepare an acceptable doctoral dissertation (thesis), which must demonstrate that the student has achieved a high level of understanding of his/her topic/field and is capable of doing independent research, and which must expand upon what was previously known or present a significant new interpretation of known materials. The Final Public Oral Exam is the successful presentation of the oral defense of the dissertation.

## Preparation and Procedures for FPO

Please follow the Graduate School Advanced Degree guidelines . The Graduate Program Administrator will provide additional department specific information to all PhD candidates at the start of the spring semester.

Thesis LaTex template Graduate students wishing to use LaTeX to write their doctoral thesis can use a premade LaTeX style file puthesis. Puthesis style has appropriate preset margins, title page and other settings that should help format the thesis.

Puthesis style files consist of

- puthesis.cls - the actual style file
- puthesis.sample.tex - first sample LaTeX file that shows how to use puthesis.cls
- thesis.tex - second sample LaTeX file

You should begin by copying at least the puthesis.cls file into the directory where you will be writing your thesis. You can do that by either clicking on puthesis.cls in the above list and downloading the file to appropriate directory, or else copy the file on the Linux/Unix systems from /usr/finehall/tex/puthesis/latest, for example:

cp /usr/finehall/tex/puthesis/latest/puthesis.cls my_thesis_directory/

Please take a look at the two sample files to get an idea on how to use puthesis style, or even better, begin by adapting one of sample files to your needs.

## Jill LeClair

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

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

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

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

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

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

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

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

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We offer Ph.D. programs in both mathematics and applied mathematics. Concentrations are available in computational and interdisciplinary mathematics.

A candidate for the Ph.D. degree must (i) complete course requirements, (ii) pass written qualifying exams, (iii) pass a preliminary oral examination and (iv) conduct thesis research, write a thesis and pass a final oral examination. These steps are described below.

All students take a minimum of 30 credit hours of graduate courses in mathematics , numbered 500 and up. Among these should be at least one course each in continuous, discrete and computational mathematics.

## more on course requirements

Qualifying exams.

The Ph.D. written qualifying examinations are written exams in three subjects selected by the student from a list of twelve possibilities. The purpose of the exams is to ensure that each Ph.D. student studies three subjects to a depth that gives adequate preliminary background to begin a Ph.D. project. Each subject is represented by a two-semester sequence of courses.

## more on qualifying exams

Select three one-year sequences from among the following.

- For students without a prior masters degree in Math or Applied Math : Such students must attempt as many exams as necessary to complete the requirement by August before the start of the third year. If they do not pass all exams at that time (and have allowed retakes left, see below), they have until January of the third year to pass all remaining exams.
- For Students with a Prior Masters Degree in Math or Applied Math: Such students must attempt as many exams as necessary to complete the requirement by August before the start of the second year. If they do not pass all exams at that time, they have until January of the second year to pass all remaining exams. A student with a prior masters degree who does not pass any exams when entering the program should enroll in 3 qualifying exam sequences in the first year so that they can attempt 3 exams in the August before the start of the second year.
- Each exam is written and graded by two faculty members.
- At the end of the Spring Semester, the two exam writers for a qualifying exam will come up with a study guide for the qualifying exams that year. That study guide will be given to the students in the course, and will be given to the DGP so that any students who wants to take the exam and who were not in the course will be able to prepare for the exam.
- Each exam is a three hour long written exam.
- One retake is allowed for each of the three examinations (with exceptions described below). Students are advised to schedule retakes as soon as possible; retakes must be done within 12 months of the date the examination is first taken. If a student fails an examination twice, he/she is considered to have failed the written qualifying examinations.
- The retake does not have to be the same exam as the one initially failed. However, if an examination for a specific sequence is failed and retaken later, the second examination must be considered a retake of the first.
- Incoming students are allowed to take qualifying exams in the August that they arrive. A fail at that time does not count against the student.
- The number of examinations taken at any given exam period cannot exceed the remaining number of passes need to reach a total of three.
- January exams are reserved for retakes, or in rare instances, special circumstances (please consult DGP). If a student takes a January retake in the first year or second year (first year only for students with a previous master’s degree), fails that happen at that time do not count against the student.
- It is recommended that students attempt at least one to two exams by the end of their first year. By the end of their second year (first year for students entering with a MS in Mathematics), students have to attempt a number of exams equal to the number of exams they have yet to pass.
- It is possible to “drop” an exam one has registered for any time up to two weeks before the exam (no questions asked). After that, we recommend students talk to the director or the administrator of the programs. Dropping an exam only applies to students who do have the option to “wait”, i.e., are not required to take the exam at that exam session.
- For part-time students, years will be counted using credit hours, with one year equal to 18 credit hours. For full-time students, calendar years are used. Students who start the graduate program in the spring semester should have a statement from the DGP put into their file specifying the date by which their exams must be taken. Depending on prior coursework, whether they are a transfer student, etc., it will be after three, four, or five semesters.

## suggested paths

Faculty have provided these “suggested paths” of courses that students can take, depending on their area of interest. Note, however, that the particular courses most relevant for an individual student will vary depending on which faculty the student works with, and what project the student works on. Hence, it’s important to talk to faculty early to figure out the best courses that are right for each student.

## Thesis Research and Defense

Our Ph.D. programs typically take five years. More information about timetable and milestones, including the preliminary and final oral examinations, can be found in our road to graduation .

## Summer Internships

We encourage Ph.D. students at all stages of their studies to seek summer employment at government or industry facilities. Through such experiences you will expand your understanding of the mathematical sciences, discover possible areas for thesis research, and enhance your career options.

In the past few years our students have worked as summer interns at national laboratories including Argonne, Lawrence Livermore, Los Alamos, Oak Ridge, PNNL and Sandia; at federal agencies such as the Environmental Protection Agency, the National Security Agency, the Naval Research Laboratory, the Naval Sea System Command and the Army Corps of Engineers; at international research institutions such as the Chinese Academy of Science, INRIA-Rocquencourt (France), Osaka University (Japan), RIKEN (Japan), the University of Canterbury (New Zealand) and Xiamen University (China); and at companies such as Chimerix, Merck, M&T National Bank and SAS.

Advisors, faculty members and the director of the graduate programs can assist in finding summer internships.

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## PhD Degree Requirements

Updated on August 15, 2023

A PhD candidate must satisfy the requirements for a doctoral degree as laid out by the UW Graduate School. Some of the key Graduate School requirements and additional requirements from the Mathematics Department are listed below. If you have questions about the requirements, please contact your preliminary advisor, Student Services or the Graduate Program Chair.

Three years of full time study, two of which must be at the University of Washington.

All PhD students must take twelve 500-level numerically graded courses in Mathematics, Applied Mathematics, or Statistics in their first two years of study. Students are expected to take seven quarters of core courses; at least five of them in their first year. The current list of core courses consists of MATH 504, MATH 505, MATH 506; MATH 524, MATH 525, MATH 534; MATH 544, MATH 545, MATH 546. Courses from other departments may be included in the twelve with approval of the Graduate Program Coordinator. Students who have already met their core course requirement may take additional core courses for a grade of S/NS. It is the student’s responsibility to select S/NS credit if they w ish to use this option .

The seven quarters of core courses requirement only applies to students who are entering our PhD Program in September 2023 or later.

## Writing Milestone :

By the end of the second year, each student must produce a carefully written mathematical document. This could be in any area, not just a presumed area of specialization, with any faculty member, not just a potential advisor. Such a document can be an expository paper coming from deep study of material in a reading course with a faculty member, a write-up of a project done under the supervision of a faculty member, an existing research paper that the student has done could also count, if a faculty member agrees. In general, this is flexible but the resulting document must be substantial .

The writing milestone requirement only applies to students who are entering our PhD Program in September 2023 or later.

## General Examination:

The General Examination is an oral examination on a special area of intended research, given by a committee after the student has passed the preliminary exams. This exam can be given only after two years of graduate study, to students who have formally declared an advisor, and have formed their thesis committee with the Graduate School. Normally, it should be taken by the middle of the student's fourth year. In extenuating circumstances, the exam can be postponed if approved by the Graduate Committee. In addition, the students must prepare a written General Paper and distribute it to the committee at least two weeks before the date of the General Exam. The content of this paper is decided upon in consultation with the committee. For example, this might be a 10-20 page expository account of the student's research area, culminating in a problem or list of problems to be studied, together with a discussion of some of the relevant literature.

## A PhD thesis must be a well-written original contribution to the scientific literature. This document will be archived by the University of Washington Library. During the preparation of the thesis, a student must take 27 credits of Math 800 required over a period of at least three quarters.

Final examination:.

The PhD thesis defense is the Final Examination. The defense is an oral exam given by the thesis committee consisting of the advisor as the chair of the committee, two additional members of the department who have been appointed to the Graduate School, plus one member of the committee from outside the Math Department called the Graduate School Representative (GSR). Successful completion of the exam is required for the PhD.

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## Georgia Institute of Technology College of Sciences

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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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## PhD Requirements

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

## Graduate Program

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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 (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.

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## Ph.D. Program in Mathematics

Degree requirements.

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

## NYU Shanghai Ph.D. Track

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

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

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

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

Probability

PDE/Analysis

## The Written Comprehensive Examination

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

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

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

## The Oral Preliminary Examination

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

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

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

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

## The Dissertation Defense

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

## Summer Internships and Employment

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

## Mentoring and Grievance Policy

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

## Visiting Doctoral Students

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

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## Ph.D. 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

## 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. Program Requirements

The Doctor of Philosophy program at the College of Education prepares students for careers of research or scholarly inquiry and teaching at the college/university level. The program consists of: (1) continuous research and faculty discussion inquiry, (2) courses in education and related fields designed to develop a comprehensive academic basis for future work in research and teaching, and (3) teaching and other related experiences tailored to individual needs and career goals.

## Table of Contents

- Enrolling in First & Second Year Courses
- Research & Teacher Preparation
- Advancing to Prospective Candidacy
- Forming a Supervisory Committee
- Research and Inquiry Conference
- Eligibility
- General Exams
- Completing the Oral General Exam
- Dissertation Credits
- Preparing the Dissertation Proposal
- Forming the Reading Committee
- Conforming to Stylistic Standards
- Completing the Final Exam (Dissertation Defense)
- Submitting Your Dissertation to the Graduate School
- Maximum Allowable Time

## 1) Enrolling in First & Second Year Courses

Upon admission to the Ph.D. program, you are designated "Post-Master's," meaning that you have been assigned to an adviser, but do not yet have a doctoral Supervisory Committee. The goal of the post-master's phase is to arrange research/inquiry experiences and coursework that will qualify you for Prospective Candidacy. You are assigned a first-year adviser whose research and scholarly activities are in your field of intended specialization. During the first year of study, your adviser will be a central figure, helping you plan academic life.

Working with your adviser, you will: (1) identify a research topic and secure ways and means for participating in the selected project, (2) select first-year courses, and (3) prepare documentation for advancement to Prospective Candidacy. Although the role of faculty advisers is designed to assist you in completing the Ph.D. degree, it is your responsibility to follow all procedures of the Graduate School and College of Education.

In the College of Education's LSHD program, post-bachelor's students may be admitted to work toward a Ph.D. without formally completing a master’s degree program. Post-bachelor's applicants to the Ph.D. track are expected to have research experience and/or research potential, as well as research interests that align with faculty expertise. Post-bachelor's students in the LSHD Ph.D. program may choose to complete an M.Ed. along the way. Those who would like to complete their M.Ed. along the way must meet the minimum 45 credit Graduate School requirements for the LSHD M.Ed. program. The 45 credits include a minimum of 21 credits in EDPSY coursework, 18 minimum numerically graded credits at the 400 or 500 level, and 18 minimum credits at the 500 level or above.

If you are a post-bachelor's student working within the prospective Ph.D. track and plan to obtain your M.Ed. along the way in LSHD, you will complete a qualifying paper no later than the quarter in which you complete 45 credits. The qualifying paper is designed to be the equivalent of a master’s final exam or thesis in quality, and must be evaluated by two members of the graduate faculty. This paper must be separate from your R&I paper.

## 2) Research & Teacher Preparation

A number of useful methods exist for inquiry into educational problems and issues. You will need to develop an appreciation for the diversity of options available. Initial preparation consists of studying the fundamental differences and similarities among various approaches to inquiry in education through the required Educational Inquiry Seminar Series (EDLPS 525 and 526; see the General Catalog for course details). Please note that these courses are sequential; EDLPS 525 is the prerequisite for EDLPS 526. You should complete this sequence as early in the program as possible, preferably in your first year.

Additionally, you will be required to complete a minimum of four additional 500-level courses (combined total of no less than 12 credits) relating to methods of educational inquiry; in these four courses, you must earn a grade of at least 3.0 (or written verification that you would have received a 3.0 in courses that are offered C/NC). You are strongly encouraged to select coursework representing at least two broad approaches to inquiry (quantitative, qualitative, philosophical, historical, etc.) offered both inside and outside the College of Education. The final selection of appropriate courses will be made with the advice and consent of your adviser. The required Inquiry series must be completed prior to your advancement to Prospective Candidacy; two of the four additional research courses must be completed prior to your Research and Inquiry Presentation.

Each Supervisory Committee will design experiences to promote excellence for students who will seek teaching positions. The nature of these experiences will vary according to your prior experience. Some students come to programs in education with substantial experience as teachers, and for them, fewer graduate school experiences may be required.

For some students, the annual Research and Inquiry Presentation will be enough to polish their instructional skills and to demonstrate mastery of instructional approaches. Other students may need to serve as teaching assistants, either formally or informally. Your Supervisory Committee will see that you have appropriate, supervised experience as needed to promote effective teaching skills.

The advancement to Prospective Candidacy process--including the materials and discussions involved in it--is an opportunity for students, advisers, and the broader faculty to evaluate the student’s progress up to that point and to plan for future course taking, committee member selection, and dissertation interests.

You may be considered for advancement to Prospective Candidacy after completing 24 credits of study, including the Inquiry Seminar Series if required (EDLPS 525 and 526) and a minimum of nine credits within your chosen field(s) of study. Individual programs may require additional coursework, and your adviser will inform you of any additional requirements early in your first quarter of study.

Once you meet the minimum requirements, your adviser will help you prepare documents for presentation to the faculty. Those documents include (1) a course of study form (including grades received in each course), and (2) a revised goal statement. You will revisit and revise the goal statement you wrote when you applied for your program to reflect your current thinking and goals. Your adviser may require other materials, such as a curriculum vita or a paper from a course. Check with your adviser to see if additional materials are necessary. Together, the student and the adviser are required to meet to discuss the materials and to make any appropriate changes before the adviser presents the student’s case to the larger faculty for consideration. Advancement to Prospective Candidacy needs to be completed before you can do your R&I.

The faculty in your program will review your work, judge the adequacy of your progress, offer suggestions about future course taking, and make a recommendation on Advancement to Prospective Candidacy to the Graduate Program Coordinator (the Associate Dean for Graduate Programs). While we encourage as much faculty input as possible, a minimum of one faculty member besides your advisor will take part in this review. Advisers are then required to meet with the student to provide a summary of the collective input gathered from the larger program faculty meeting.

Once you have advanced, you should initiate the Prospective Candidacy Form to notify the Office of Student Services about completing this milestone.

A summary of the process is below: 1. Meet minimum requirements for advancing to prospective candidacy. 2. Prepare course of study, revised goal statement, and whatever materials your advisor or program requires. 3. Meet with advisor to go over documents and revise as needed. 4. Advisor meets with program faculty and presents the student’s case for consideration. 5. Faculty in program review work, judge adequacy of progress, offer feedback, and make recommendation on advancement. 6. Advisor meets with student to give feedback and decision of the faculty. 7. Student initiates the Prospective Candidacy Form online. Once signed by the faculty advior, the completed form is then automatically submitted to the Office of Student Services.

Probationary language: If, after reviewing the student’s case, the program faculty decides that the student will not be Advanced to Prospective Candidacy, the student will be warned or placed on probationary status per the Graduate School's policy on Unsatisfactory Performance and Progress. At that time, the advisor must call a meeting with the student, one other faculty member, and the Associate Dean for Graduate Programs. This group may require additional materials (i.e. course papers), and the student may offer additional materials as well. The meeting should take place no later than the second week of the following academic quarter. At this meeting, the faculty members and student will discuss what is necessary to lift probationary status. Examples might include: improving grades, revising the goal statement further, and requiring certain courses.

## 4) Forming a Supervisory Committee

Once you have been advanced to Prospective Candidacy, you should direct your attention to forming a Supervisory Committee. In concert with your adviser, you should explore which members of the graduate faculty would be willing to serve on your Supervisory Committee. Each member of a Supervisory Committee will devote substantial time to working with you and should formally indicate willingness to serve. The chairperson of the Supervisory Committee, who must be a graduate faculty member from the College of Education, should express the willingness and availability to supervise a dissertation, since this is normally the most time-consuming responsibility.

Supervisory Committees will be formed in accordance with Graduate School policy

- A minimum of four voting faculty (at least three with graduate faculty appointments) must represent, respectively, your (a) specialization within their broad areas of study, (b) first cognate, (c) second cognate, and (d) specialization outside of the College of Education (definitions of broad area, specializations, and cognates can be found ( here ).
- No more than two voting faculty from your broad area may be on the committee.
- An additional graduate faculty member, the Graduate School Representative (GSR), must also serve on the committee. GSRs must be members of the graduate faculty with an endorsement to chair doctoral committees, and must have no conflict of interest (such as budgetary relationships or adjunct appointments) with the College of Education. Members of Supervisory Committees representing students’ specializations outside of the College of Education may also serve as GSRs, provided they are qualified to serve in both roles.

Once you have identified appropriate graduate faculty who are willing to serve, their names should be submitted to the Office of Student Services using the Committee Formation Request Form . Your faculty adviser must approve the form to indicate their approval.

NOTE: The Graduate School requires each doctoral student who is forming a committee for the first time to submit a Use of Animal and Human Subjects Form to the Office of Student Services.

You should form a Supervisory Committee no later than the quarter prior to your General Exam. It is not imperative that the Supervisory Committee be formed before your Research and Inquiry Presentation. It is necessary, however, for you to have arranged for a group of faculty to evaluate your Research and Inquiry work.

The next task is to meet with your Supervisory Committee to develop a research program for the Research and Inquiry Presentation and to plan a course of study in preparation for the General Exam. Between Supervisory Committee meetings, your chairperson is responsible for serving as your adviser.

The Supervisory Committee may recommend against continuation in the program if your progress toward the degree is unsatisfactory. This may include, but is not limited to, an excessive number of course withdrawals or incompletes, a grade point average of less than 3.0, unsatisfactory performance in field placements, or unsatisfactory performance on the General Exam.

## 5) Completing the Research & Inquiry Presentation

Research preparation is the foundation of the Ph.D. program, as research will play a paramount role in students’ professional careers. Training to be an effective researcher requires (a) concentrated focus to learn the various methods of inquiry and practice, and (b) employment of these methods in various research projects while pursuing your degree. You will begin research activities during the first year of the program, and will continue to develop skills by conducting various research projects, culminating with a dissertation. The Research and Inquiry milestone consists of two major components: A major product of your research preparation effort is the R&I paper and presenting at the Research and Inquiry Conference annually during autumn quarter.

The purposes of R&I are to:

- Immerse you in issues of content and method directly pertinent to your chosen specialization.
- Provide you with practical experience in the use of methods and the application of content learned in coursework.
- Convey aspects of substance and method that characterize the topic studied, but are not taught in general method or content courses.
- Afford an opportunity for you to present research to a professional audience and for the audience to learn about the research.

The design, implementation, and presentation of the R&I research shall be under the supervision of your chair and at least two additional faculty members or your Supervisory Committee. At least three faculty members must approve a thoroughly developed research papers prior to taking the General Exam.

In general, your R&I paper should hold substantial promise of contributing to preparation for a dissertation, and at its inception should have a good chance of being publishable in a juried journal. At each meeting, members of the Supervisory Committee will reassess the extent to which your R&I activities are contributing to stated goals, and will provide advice in accordance with their assessment. Between committee meetings, the chairperson will assume primary responsibility for advising and assisting you with preparation of your R&I plan.

After successful completion of the written portion, Students will be required to present at the annual CoE R&I Conference held in autumn quarter.

## 5.1) Research and Inquiry Conference

The R&I Conference is a half-day event where students will present their research in two types of session formats. All formats provide a means for grouping related papers into sessions, with different opportunities for moderators and audience participation. Students, with the approval of their advisor, determine which format is optimal for future preparation. Successful participation of in the Research ad Inquiry Conference is required prior to defending a dissertation.

The purposes for R&I conference:

- To mentor student research experience.
- To support professional practices toward becoming part of a community of scholars.
- To build community in the College

Session format options:

Panel presentations typically group together 2-5 student presenters with similar topics for a shared presentation and discussion opportunity. Each student will present an abbreviated version of her/his R&I paper, followed by summarizing comments from the moderator and then facilitated audience discussion and questions. A typical structure for a session allows approximately 5 minutes for the moderator’s introduction to the session, 10 minutes per presenter, another 5 minutes for moderator comments and summary, and finally 15 minutes for audience discussion. Individual presenters must be attentive to the time allocation for presenting their work in paper sessions.

Structured poster sessions combine the graphic display of materials with the opportunity for individualized, formal discussion of the research. Depending on how many individuals plan to participate and how many intellectual areas will be presented, there could be anywhere from 1- 4 individuals in a 60 minute session. These sessions begin with attendees viewing poster presentations, then move into brief oral presentations to the audience gathered as a group, followed by direct discussion with poster presenters. Posters are linked conceptually in terms of education research issues, problems, settings, methods, analytic questions, or themes.

## 5.2) Eligibility

To be eligible to participate in the R&I Presentations, you must meet the following requirements:

1. You must be registered as a graduate student at the University of Washington during the quarter of the R&I Presentation. 2. You must have completed the following research course requirements: six credits of the Inquiry series (EDLPS 525 and 526), plus two additional research methodology courses at the 500-level. 3. You must have been advanced to Prospective Candidate status through your academic area.

4. You must have identified a group of faculty who have agreed to evaluate your R&I work. In some cases, this group will be your Supervisory Committee; it is not imperative, however, that you formally establish your Supervisory Committee before R&I. As an alternative, a group of three faculty members can agree to evaluate your R&I work.

5. Some papers might require might need Human Subjects Form approval. If you and your advisor have determined you need this, you must have a Human Subjects Form approved prior to starting the research if the investigation is conducted with human subjects. See Louise Clauss in 115J Miller hall if you have questions regarding Human Subjects applications.

6. The final copy should be submitted to the faculty evaluators and the Office of Student Services with the approval of three faculty members (or instructors). Instructions on completing the R&I submission process can be found on the Graduate Student Forms page . Please keep in mind that the faculty members have other time constraints. It is to your benefit to submit your research paper for evaluation as early as possible.

## 6) General Exams

When both you and your Supervisory Committee concur that you are prepared and have completed all course requirements (except the dissertation) — including the completion at least 60 credit hours of coursework, per Graduate School requirements (or 30 hours if you already completed a master’s degree that will be less than 10 years old at the time of graduation from the UW) — your Course of Study and research activities will be evaluated through Written and Oral Exams conducted by the Supervisory Committee.

The General Exam is given in two parts. The first part is written and examines content area in your broad area, specialty areas, and cognates. Upon satisfactory completion of the written portion of the General Exam, the oral portion may be scheduled. During the Oral Exam, members of the graduate faculty may ask any questions they choose. By majority vote, the Supervisory Committee will rule on whether you pass.

## 7) Completing the Oral General Exam

You are responsible for scheduling the oral portion of the General Exam (locating an adequate room, determining a date and time that is acceptable to all members of the Supervisory Committee, etc.), as well as submitting a Request for General Exam to the Graduate School. You should submit the request after forming your Supervisory Committee (see above) and at least three weeks prior to the date of the General Exam by using the Graduate School’s online process. During the Oral Exam, members of the graduate faculty may ask any questions they choose. By majority vote, the Supervisory Committee will rule on whether you pass. Once you have passed, the Office of Student Services will convey the exam results to the Graduate School. This will result in Candidacy being awarded at the end of the quarter in which you pass your Oral Exam.

## 8) Candidacy

After successfully completing the General Exams, you enter the Candidacy stage of your program. The main tasks of this phase include preparing a dissertation proposal, completing dissertation research, writing the dissertation, and conducting your final defense.

## 9) Dissertation Credits

When you and your adviser determine that you are completing dissertation-related work, you may register for dissertation credits (EDUC 800). The Graduate School requires a minimum of 27 dissertation credits for degree completion, and these credits must be taken over a minimum of three quarters.

## 10) Preparing the Dissertation Proposal

Upon successful completion of the oral portion of the General Exam, you and your Supervisory Committee will shift attention to the dissertation proposal. The purpose of the dissertation proposal is to provide you with constructive criticism from the entire Supervisory Committee prior to the execution of your dissertation research. The written dissertation proposal should be approved unanimously by the Supervisory Committee members; approval will be indicated by completing the Dissertation Proposal Form . Approval does not guarantee that the Supervisory Committee will approve the dissertation at the Final Oral Exam, but it does guarantee that the committee may not later disapprove the dissertation on the grounds that the research was poorly conceived. The approved proposal becomes the working paper for conducting your dissertation research.

Once the proposal receives Supervisory Committee approval, you will likely need to submit an application for review and approval by the Human Subjects Division. On its website, the College of Education has summarized some of the most important aspects of the Human Subjects Review Process . You should also consult the website of the UW’s Human Subjects Division .

For additional information about the process, the type of review suitable for a given project, application forms, and general assistance, contact Louise Clauss at [email protected] or 206-616-8291.

## 11) Forming the Reading Committee

The Reading Committee will be composed of a minimum of 3 members of your Supervisory Committee members, including the chairperson. It is also advisable to include a member who is knowledgeable in the chosen research methodology. The Reading Committee will read and review your dissertation in detail and make a recommendation to the larger Supervisory Committee about readiness to schedule the Final Exam. Once you identify appropriate graduate faculty who are willing to serve on the Reading Committee, their names should be submitted to the Office of Student Services using the Committee Formation Request Form on the Graduate Student Forms page .

## 12) Conforming to Stylistic Standards

It is your responsibility to ensure that your dissertation meets current Graduate School formatting requirements. You may find information about these requirements on the Graduate School Dissertation page .

## 13) Completing the Final Exam (Dissertation Defense)

You are expected to pass the Final Exam. The final defense of the dissertation is intended as an opportunity for all involved to celebrate the good results of their work during your career in the College of Education.

You should schedule the Final Exam after submitting your dissertation to the Supervisory Committee. You are responsible for scheduling the Final Exam (locating an adequate room, determining a date and time that is acceptable to all members of the Supervisory Committee, etc.), as well as submitting a Request for Final Exam to the Graduate School. You should submit the request after forming the Reading Committee and at least three weeks prior to the date of the Final Exam by using the Graduate School’s online process. You should also note that you must be enrolled for credit hours during the quarter of the Final Exam. If your Final Exam occurs during a period between academic quarters, then the Final Exam will be considered to have taken place the following quarter, and you must register for that quarter.

The Final Exam will cover your dissertation and related topics, and it may also cover other areas deemed appropriate by the Supervisory Committee. While the committee alone votes on acceptance of the dissertation, any member of the graduate faculty may participate in the Final Exam.

## 14) Submitting Your Dissertation to the Graduate School

Once you pass the Final Exam and complete any revisions requested by the Supervisory Committee, the remaining step is to submit your dissertation to the Graduate School.

In preparation for submitting your dissertation, you should keep the following Graduate School policies in mind:

- If you wish to submit your dissertation in the same quarter as your Final Exam, make note of the submission deadlines established by the Graduate School.
- You may submit your dissertation up to two weeks after the end of a quarter without having to register for the following quarter by using the Registration Waiver Fee . The Registration Waiver Fee option is available to a student who has completed all other degree requirements except submission of the dissertation. You will then be permitted to graduate the following quarter by paying a $250 fee in lieu of registering for credit hours.
- Submission of the dissertation is done electronically and involves several steps. You should carefully review the degree completion information available from the Graduate School. All Reading Committee members must approve the dissertation online and you must also complete the Survey of Earned Doctorates .

Specific questions about the electronic submission of dissertations should be directed to Graduate Enrollment Management Services (GEMS) at 206-685-2630.

## 15) Maximum Allowable Time

In planning your program of study and timeline, keep in mind that all requirements for the Ph.D. must be completed within a 10-year time limit.

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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 ...

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...

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;

Doctor of Philosophy (PhD) in Mathematics. To earn a PhD in Mathematics one must satisfy the following requirements: ... Math 270ABC, Math 271ABC, Math 272ABC, Math 274 *PhD Requirements Summarized* By the beginning of the 2nd year: Pass at the MS level two exams in real analysis, complex analysis, algebra or applied math.

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

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.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements. During the first year of the Ph.D. program: 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 ...

Overview. The Department of Mathematics graduate program has minimal requirements and maximal research and educational opportunities. It differentiates itself from other top mathematics institutions in the U.S. in that the curriculum emphasizes, from the start, independent research. Our students are extremely motivated and come from a wide ...

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 PHD Requirements Starting Fall 2021. Course Requirements - Core and Electives (48 units) (see attached Progress Checklist) Students must acquire proficiency in mathematics by completing 48 units or passing twelve graduate courses (MAT 280 and below). Three of these may be taken outside of mathematics if they are related to the student ...

The PhD program in mathematics is intended for students with exceptional mathematical ability. The program emphasizes a solid mathematical foundation and promotes innovative scholarship in mathematics and its many related disciplines. ... Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and ...

PhD Requirements MS and MA Requirements Residency Requirements Courses Prospective Students Current Students Events Calendar Analysis & PDE Applied Mathematics

The Department of Mathematics offers graduate courses on various levels, all of which are oriented toward research. ... Ph.D. Requirements. To earn the Doctor of Philosophy (Ph.D.), the student must pass a language requirement and both portions of the general examinations, submit an acceptable dissertation and sustain a final public oral ...

Welcome to the Math PhD program at Harvard University and the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences. Learn more about Harvard's Math community and our statement on diversity and inclusion.

A candidate for the Ph.D. degree must (i) complete course requirements, (ii) pass written qualifying exams, (iii) pass a preliminary oral examination and (iv) conduct thesis research, write a thesis and pass a final oral examination. These steps are described below. Coursework

A PhD candidate must satisfy the requirements for a doctoral degree as laid out by the UW Graduate School. Some of the key Graduate School requirements and additional requirements from the Mathematics Department are listed below. If you have questions about the requirements, please contact your preliminary advisor, Student Services or the ...

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 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: Coursework. Qualifying Exams.

PhD students are required to take 4 of 5 core courses, one of which must be MATH 675. The core courses are MATH 675 (Linear Analysis) MATH 621 (Algebra) MATH 631 (Topology) MATH 677 (Ordinary Differential Equations) MATH 685 (Numerical Analysis) Students should also take the Graduate Seminar (1 credit) for at least six semesters (maximum nine).

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.

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 Qual Courses Required: 727 or Qual in Probability/Statistics 765 or Qual in Analysis

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.

PhD Requirements. Below is a summary of the requirements for the Ph.D. in Mathematics. General requirements for graduate degrees at Rutgers are governed by the rules of the Graduate School - New Brunswick and are listed in the current catalog. These include how and when credit can be transferred and how many credits can be taken each semester.

The Doctor of Philosophy program at the College of Education prepares students for careers of research or scholarly inquiry and teaching at the college/university level. The program consists of: (1) continuous research and faculty discussion inquiry, (2) courses in education and related fields designed to develop a comprehensive academic basis for future work in research and teaching, and (3 ...