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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.

### Overview

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. These courses 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. Upon completing the minor requirement, a student should immediately complete the Doctoral Minor form.

**Breadth requirements.** The students must demonstrate:

*Basic understanding*of 3 subjects- Basic understanding of 2 subjects must be demonstrated through passing the subjects' written comprehensive exams.
- 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.

*Exposure*to 2 other subjects- 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:*

1. Algebra 6112 Advanced Linear Algebra 6121 Algebra I 6122 Algebra II 6421 Algebraic Geometry I 6422 Algebraic Geometry II 2. Analysis 6321 Complex Analysis 6337 Real Analysis I 6338 Real Analysis II 6580 Introduction to Hilbert Spaces 7334 Introduction to Operator Theory 7337 Harmonic Analysis 7338 Functional Analysis *Area II subjects:*

3. Differential Equations 6307 Ordinary Differential Equations I 6308 Ordinary Differential Equations II 6341 Partial Differential Equations I 6342 Partial Differential Equations II 4. Discrete Mathematics 6014 Graph Theory 7012 Enumerative Combinatorics 7016 Combinatorics 7014 Advanced Graph Theory 7018 Probabilistic Methods in Combinatorics 5. Geometry and Topology 6441 Algebraic Topology 6452 Differential Topology 6455 Differential Geometry 6458 Intro. to Geometry and Topology II 6457 Intro. to Geometry and Topology I 6. Numerical Analysis 6640 Intro. to Numerical Methods for PDE 6643 Numerical Linear Algebra 6644 Iterative Methods for Systems of Eqns 6645 Numerical Approximation Theory 6646 Numerical Methods for ODE 7. Probability and Statistics 6241 Probability I 6242 Probability II 7244 Stoc. Processes and Stoc. Calculus I 7245 Stoc. Processes and Stoc. Calculus II 6262 Statistical Estimation 6263 Testing Statistical Hypotheses 6266 Linear Statistical Models 6267 Multivariate Statistical Analysis

**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

Coursework completed at other universities may be counted towards breadth and hours requirements. 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 Graduate Director.

### 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.

### Transfer to the PhD Program

A graduate student enrolled in another School at Georgia Tech who wishes to be admitted to the PhD program in Mathematics should direct his or her request to the School's Director of Graduate Studies. The minimum requirement for such a transfer is the passing of one of the written comprehensive exams. In addition, a transfer request should be accompanied by: (1) three letters of recommendation including one from a prospective advisor; (2) a statement of purpose; (3) most recent copies of transcripts. Transfer requests will be considered by the School's Graduate Committee.