# Graduate Program Details

### Masters Degree

##### The master of science is designed to prepare students for positions in business and industry. It also provides preparation for further graduate study leading to a doctoral degree.

#### REQUIREMENTS FOR DEGREE — THESIS PROGRAM

**Credit hours**(30): Minimum 24 coursework hours in approved graduate proposed graduate mathematics*with a grade of C+ or better in each*, and six thesis hours (Math 699R). Three of the course hours must be from one of three areas of Applied Mathematics, Analysis, or Algebra, and another three course hours must be from one of the remaining two areas. With the approval of the graduate coordinator, graduate-level courses in other departments can be used toward the coursework hours.**Examination:**The examination requirement in the thesis program may be met by way of coursework, or by way of the written examination as explained below, in two of three broadly defined areas of Applied Mathematics, Analysis, and Algebra, by the end of year two. Failure by a master’s student to meet the examination requirement by the end of year two will result in the discontinuation of financial support for that master’s student at the end of year two.Continuing master’s students have the option of meeting the examination requirement under the current or new system.

**Coursework,**To meet the examination requirement by way of coursework, a master’s student will need to earn an A- or better in two courses, or their equivalents at another institution, with one course in one of the three areas and another course in one of the remaining two areas. Courses in the area of Applied Mathematics include, but are not limited to, Math 510, Math 511 , Math 521 , Math 522 , Math 611, and Math 636 . Courses in the area of Analysis included but are not limited to Math 532 , Math 540 , Math 541 , Math 565 , Math 641 , and Math 655 . Courses in the area of Algebra include but are not limited to Math 561 , Math 571 , Math 572 , Math 586 , Math 656 , and Math 676 . A graduate course, or its equivalent at another institution, can only be used to count towards one of the three areas. For example, Math 636 (Probability 1) is both Applied Mathematics and Analysis, but it can only be counted for one of these two areas. The Graduate Committee will maintain a list of associations of graduate courses with the three areas. - A master’s student may meet the examination requirement by way of graduate coursework completed as an undergraduate.
**Written Examination.**To meet the examination requirement by way of written examination, a master’s student will need to pass written examinations in two of the three areas. The written examinations will be based on published syllabi associated with the basic courses listed below. If a master’s student has earned an A- or better in one course associated with one of the three areas, but not in another course in another area, the master’s student need only pass one written examination in an area not met by coursework. Written examinations will be offered in August/September, January, and May.1. Applied Mathematics: Math 521

and Math 522

2. Analysis: Math 540and Math 541

3. Algebra: Math 571and Math 572 - The Graduate Committee may change the basic courses associated with any of the three areas, but there will always be two basic courses associated with each of the three areas. No written examinations are available for other courses.

- A master’s student may meet the examination requirement by way of graduate coursework completed as an undergraduate.
**Graduate Program of Study:**Students are required to submit a program of study before the end of the third week of their second semester. The university now uses the GRADPROG system for recording programs of study and other information having to do with a student’s graduate program. Instructions for using GRADPROG may be found at https://math.byu.edu/wp-content/uploads/2020/07/GradProgInstructions.pdf**Thesis****Oral Defense of thesis**

#### REQUIREMENTS FOR DEGREE — NON-THESIS PROGRAM

**Credit hours**(32): Minimum 30 coursework hours in approved graduate proposed graduate mathematics*with a grade of C+ or better in each*, and two hours for the project (Math 698R). Three of the course hours must be from one of three areas of Applied Mathematics, Analysis, or Algebra, and another three course hours must be from one of the remaining two areas. With the approval of the graduate coordinator, graduate-level courses in other departments can be used toward credit hours.**Examination:**The examination requirement in the thesis program may be met by way of coursework, or by way of the written examination as explained below, in two of three broadly defined areas of Applied Mathematics, Analysis, and Algebra, by the end of year two. Failure by a master’s student to meet the examination requirement by the end of year two will result in the discontinuation of financial support for that master’s student at the end of year two.Continuing master’s students have the option of meeting the examination requirement under the current or new system.

**Coursework,**To meet the examination requirement by way of coursework, a master’s student will need to earn an A- or better in two courses, or their equivalents at another institution, with one course in one of the three areas and another course in one of the remaining two areas. Courses in the area of Applied Mathematics include, but are not limited to, Math 510, Math 511 , Math 521 , Math 522 , Math 611, and Math 636 . Courses in the area of Analysis included but are not limited to Math 532 , Math 540 , Math 541 , Math 565 , Math 641 , and Math 655 . Courses in the area of Algebra include but are not limited to Math 561 , Math 571 , Math 572 , Math 586 , Math 656 , and Math 676 . A graduate course, or its equivalent at another institution, can only be used to count towards one of the three areas. For example, Math 636 (Probability 1) is both Applied Mathematics and Analysis, but it can only be counted for one of these two areas. The Graduate Committee will maintain a list of associations of graduate courses with the three areas. A master’s student may meet the examination requirement by way of graduate coursework completed as an undergraduate.

**Written Examination.**To meet the examination requirement by way of written examination, a master’s student will need to pass written examinations in two of the three areas. The written examinations will be based on published syllabi associated with the basic courses listed below. If a Master’s student has earned an A- or better in one course associated with one of the three areas, but not in another course in another area, the master’s student need only pass one written examination in an area not met by coursework. Written examinations will be offered in August/September, January, and May.- Applied Mathematics: Math 521
and Math 522 - Analysis: Math 540
and Math 541 - Algebra: Math 571
and Math 572

The Graduate Committee may change the basic courses associated with any of the three areas, but there will always be two basic courses associated with each of the three areas. No written examinations are available for other courses.

- Applied Mathematics: Math 521
**Graduate Program of Study**

Students are required to submit a program of study before the end of the third week of their second semester. The university now uses the GRADPROG system for recording programs of study and other information having to do with a student’s graduate program. Instructions for using GRADPROG may be found at https://math.byu.edu/wp-content/uploads/2020/07/GradProgInstructions.pdf**Project and presentation:**Write a paper about the project (Math 698R) which should focus on an area of advanced mathematics and present a 45-minute talk based on the paper.

## PhD Degree

### The doctoral program prepares students for a career in research and teaching at the university level or in basic research in a non-academic setting.

#### REQUIREMENTS FOR DEGREE

**Credit Hours**(54): Minimum 36 coursework hours in mathematics courses numbered 600 or above with a grade of B or better in each plus 18 dissertation hours (Math 799R).**Required Courses**: Complete at least 3 hours each in algebra, analysis, applied mathematics, and geometry/topology.**Written Examinations:**This requirement has two parts, foundational and advanced, each of which can be met by coursework or written examination, as explained below. The time frame for a Ph.D. student to complete the Requirements is two years from the beginning of the semester of admission to the Ph.D. program.**Foundational Requirement:**A Ph.D. student meets the foundational requirement by earning an A or A- in four courses, or their equivalents at another institution, associated to two of the three Foundational Options List below, or by passing written examinations at BYU that similarly cover the same material as in these courses:**Foundational Options List:**

1. Applied Mathematics: Math 521 and Math 522

2. Analysis: Math 540 and Math 541

3. Algebra: Math 571 and Math 572**Foundational Written Examinations:**A Ph.D. student, after having taken at BYU four courses associated to two of the three foundational options listed, or their equivalents at another institution, and not meeting the Foundational Requirement by earning A or A- in the four courses, may petition the graduate coordinator to meet the Foundational Requirement by way of written examination. For each of the two options in which the Ph.D. student did not earn an A or A- in both of the associated courses, a written examination will be administered. Written examinations for the options of the Foundational Requirement will be offered as needed three times a year in August/September, January, and May.**Advanced Requirement:**A Ph.D. student meets the advanced requirement by earning an A or A- in three of the courses in the Advanced Courses List below or by way of written examination at BYU. For one of the three courses, one substitution is possible as explained below. A Ph.D. student may in part meet the Advanced requirement by way of coursework completed as part of a Masters degree for up to two of the required three courses. A Ph.D. student must take at BYU at least one of the courses on the Advanced Courses List (the other two courses may come from transfer credits from a previous degree at another institution).**Advanced Courses List:**

1. 611 (Finite Element Methods for Numerical Partial Differential

Equations—to be created, see New Courses to be Created for proposed description and learning outcomes),

2. 621 Matrix Theory 1 (to be possibly modified)

3. 622 Matrix Theory 2 (to be modified into an applied math course)

4. 634 Ordinary Differential Equations

5. 635 Dynamical Systems

6. 636 Probability 1

7. 641 Measure and Integration Theory (to be renamed this)

8. 647 Partial Differential Equations

9. 655 Differential Topology

10. 656 Algebraic Topology

11. 663 Algebra Geometry 1

12. 676 Commutative Algebra (currently a prerequisite for 663)

13. 682 (Modular Forms—to be created, see New Courses to be Created for proposed description and learning outcomes)

14. 686R, 687R (Topics in Number Theory)The Graduate Committee may add, remove, or replace courses on the Advanced Courses list with other current or newly created 600 level courses.

With approval of the graduate coordinator, one of the three advanced courses may be substituted with another 600-level mathematics course at BYU or a 600-level course in another department at BYU, provided the Ph.D. student earns an A or A- in it, and it appears on the Ph.D. student’s Program of Study. No written examination is available for a substituted advanced course.

**Prospectus:**The Prospectus for a Ph.D. student is a 30-50 minute oral presentation held before the Ph.D. student’s committee (virtual or in-person) as well as an extended abstract or brief description (2-3 pages will suffice). There should be time during and/or after the presentation for the committee to ask the Ph.D. student questions about the oral presentation. The Advisor should work closely with the Ph.D. student before-hand to ensure a well-prepared oral presentation.The oral presentation is to be held after the Ph.D. student has fulfilled the Ph.D. Qualifying Exam requirement but no later than the end of the first semester of the third year in the Ph.D. program. The purpose of the oral presentation is to outline the objective(s) of the Dissertation, the background and history associated with the objective(s), and the proposed approaches and/or techniques that may be used in obtaining the main results. This will also ensure that the student has adequate support from their advisor and committee.

With approval from the graduate coordinator and in accordance with the desires of the dissertation committee, the student may be given a second chance to complete the prospectus requirement within 2 months of the initial attempt. Approval of the Prospectus by the committee is required for the Ph.D. student to continue in good standing in the graduate program.

**Graduate Program of Study:**Doctoral students should submit their programs of study during the first year, and in no case later than the third week of the beginning of the second year of study. A doctoral committee must consist of a chair and at least four members The university now uses the GRADPROG system for recording programs of study and other information having to do with a student’s graduate program. Instructions for using GRADPROG may be found at https://math.byu.edu/wp-content/uploads/2020/07/GradProgInstructions.pdf**Defense of Dissertation**: A final oral defense of the dissertation is conducted by a faculty committee consisting of the student’s research advisor, two other readers of the dissertation (one of whom may be an outside examiner), and two other members of the faculty.**Dissertation**