# Graduate Programs

Welcome to the webpage for the graduate program of the Department of Mathematics at the University of Utah. We offer a Ph.D. and Master's degrees in many different areas of mathematics. The faculty at the University of Utah are active and involved with the students and have many diverse research projects. Our former graduate students have gone on to mathematical careers both in academia and in industry. We invite you to explore our graduate program information, our math department, university, and living in Utah.

See the links below for more information about our programs.

# DEGREE OPTIONS

# Ph.D. Program in Mathematics

The Ph.D. Program in the Department of Mathematics enjoys an excellent reputation in a diversified set of subjects, such as algebraic geometry, commutative algebra, differential geometry, geometric topology, group and representation theory, and number theory in pure areas, and materials and fluids, mathematical biology, mathematical finance, numerical analysis, partial differential equations, probability, and statistics in applied areas.

### Degree requirements/COURSEWORK

The course requirements for the Ph.D. degree consist of at least seven sequences (each sequence contains two consecutive courses on a subject) numbered 6000 or above, or their equivalent, approved by the student’s supervisory committee. Exceptions must be approved on an individual basis by the Graduate Committee upon the recommendation by the student’s supervisory committee.

The following is a list of departmental requirements, in addition to the requirements of the Graduate School, for candidates seeking a Ph.D. degree in mathematics.

- At least 42 credit hours numbered 6000 or above
- At least 14 credit hours of the 42 credit hours should be courses numbered 7800-7970 (topics courses, seminars, and thesis research).

- Three written qualifying examinations
- An oral qualifying examination
- Two semesters of teaching
- The final oral examination

More details can be found in the Graduate Bulletin.

#### WRITTEN QUALIFYING EXAMINATIONS

The written qualifying examinations are given twice a year, in January and August, just before the start of the Spring and Fall semesters. A Ph.D. candidate is required to pass three written qualifying examinations chosen from the following set of subjects (corresponding course numbers are included in parenthesis):

- algebra (6310-6320)
- applied mathematics (6710-6720)
- differential equations (6410-6420)
- numerical analysis (6610-6620)
- probability (6040)
- real and complex analysis (6210-6220)
- statistics (6070)
- topology and geometry (6510-6520)

Each exam lasts three hours and is written at a first-year graduate level. Entering students are recommended to take the 6000-level course corresponding to a particular exam and use the course materials to prepare for the exams. Students with particularly strong backgrounds can choose to take the exams without taking the courses. Copies of past exams from the last ten years can be obtained by contacting the Graduate Coordinator.

#### ORAL QUALIFYING EXAMINATION

This exam is conducted by the student’s supervisory committee and it can take one of the following two formats: a general exam to measure the student’s overall mathematical maturity and breadth, or a presentation of the proposed dissertation project. The exam also evaluates the student’s skill at chalkboard exposition and verbal exchange.

#### TEACHING REQUIREMENT

Each Ph.D. candidate is required to teach a minimum of two courses or equivalent tutorials, or supervise laboratory sessions.

#### FINAL ORAL EXAMINATION

This examination consists of a public thesis defense that summarizes the candidate's thesis work.

ADMISSION REQUIREMENTS

Admission to graduate status in the Ph.D. program requires that students hold a Bachelor's
degree in Mathematics, or its equivalent, with a **grade point average of at least 3.0** and that they show promise of success in graduate work. The prerequisites for the
various programs vary. Qualified applicants typically possess a Bachelor's or Master's
degree in mathematics. Students should be prepared to take the courses that directly
lead to qualifying exams.

### GRE Exams

For the 2021 academic year (Fall, Spring, or Summer) we will not require nor evaluate GRE exam scores as part of the application. We understand that taking the general test may be difficult, and that mathematics subject exams will not be offered until April 2021. Although not required, it is helpful if in your Personal Statement you add a short sentence indicating that you were unable to take the exams due to the COVID pandemic.

### English Tests

All applicants must meet the University of Utah's minimum requirements for English
proficiency, click here.

Additionally, successful applicants for Teaching Assistantships must be proficient
in their spoken and listening English skills. Those skills are usually reflected in
the submitted TOEFL scores (or equivalent). In most cases, for those subsections of
the TOEFL, we look for scores of at least about 20, and optimally, of at least about
23.

## FILLING OUT THE APPLICATION

For instructions on filling in the Graduate Mathematics Application Form:

**Questions?**

**Elena Cherkaev, Director of Graduate Studies- elena@math.utah.edu**

**Paula Tooman, Graduate Program Coordinator- tooman@math.utah.edu**

Master of Arts (M.A.) and Master of Science (M.S.) in Mathematics

DEGREE REQUIREMENTS/COURSEWORK

The M.A. and M.S. degree requirements are identical except that the M.A. degree requires an additional language proficiency. The following is a list of departmental requirements for candidates seeking a M.S. or M.A. degree in mathematics.

**Minimum Degree Hours: 30**

**FOR THE MASTER'S DEGREE IN PURE MATHEMATICS:**

- MATH 5210 - Introduction to Real Analysis 4 Credit(s)
- MATH 5310 - Introduction to Modern Algebra I 3 Credit(s)
- MATH 5320 - Introduction to Modern Algebra II 3 Credit(s)
- One 6000-level sequence consisting of two one-semester courses
- Four additional one-semester courses at the 5000- or 6000-level

**FOR THE MASTER'S DEGREE IN APPLIED MATHEMATICS:**

- MATH 5210 - Introduction to Real Analysis 4 Credit(s) and
- one 6000-level course
**OR** - two 6000-level courses

#### WRITTEN QUALIFYING EXAMINATIONS

The written qualifying examinations are given twice a year, in January and August, just before the spring or the fall semester starts. A candidate can fulfill the graduation requirements by passing two written qualifying examinations, chosen from the following set of subjects (corresponding course numbers included in parentheses):

- algebra (6310–6320)
- applied mathematics (6710–6720)
- differential equations (6410–6420)
- geometry and topology (6510–6520)
- numerical analysis (6610–6620)
- probability (6040)
- real and complex analysis (6210–6220)
- statistics (6070)

Each exam lasts three hours and is written at a first year graduate level. Entering students are required to take the 6000-level course corresponding to a particular exam and use the course materials to prepare for the exams. Copies of past exams are posted (as PDF files) on the Graduate Program homepage (see Qualifying Examinations).

#### ORAL COMPREHENSIVE EXAMINATION AND ADDITIONAL REQUIREMENTS

An oral comprehensive exam can be chosen as an option to fulfill the Master’s degree requirements. This exam is conducted by the student’s supervisory committee. In case a candidate chooses the option of oral comprehensive exam, the student must complete nine (9) additional credits of a Masters Project, which can be one of the following:

ADMISSION REQUIREMENTS

Admission to graduate status in the Master's program requires that students hold a
Bachelor's degree in Mathematics, or its equivalent, with a **grade point average of at least 3.0** and that they show promise of success in graduate work. The prerequisites for the
various programs vary. Master of Arts / Master of Science degrees, qualified applicants
typically possess a Bachelor's or Master's degree in mathematics. Students should
be prepared to take the courses that directly lead to qualifying exams.

### GRE Exams

For the 2021 academic year (Fall, Spring, or Summer) we will not require nor evaluate GRE exam scores as part of the application. We understand that taking the general test may be difficult, and that mathematics subject exams will not be offered until April 2021. Although not required, it is helpful if in your Personal Statement you add a short sentence indicating that you were unable to take the exams due to the COVID pandemic.

### English Tests

All applicants must meet the University of Utah's minimum requirements for English
proficiency, click here.

Additionally, successful applicants for Teaching Assistantships must be proficient
in their spoken and listening English skills. Those skills are usually reflected in
the submitted TOEFL scores (or equivalent). In most cases, for those subsections of
the TOEFL, we look for scores of at least about 20, and optimally, of at least about
23.

## FILLING OUT THE APPLICATION

For instructions on filling in the Graduate Mathematics Application Form:

**Questions?**

**Elena Cherkaev, Director of Graduate Studies- elena@math.utah.edu**

**Paula Tooman, Graduate Program Coordinator- tooman@math.utah.edu**

Master of Statistics (Mathematics)

The Master of Statistics (M-Stat) program is administered by the University Statistics Committee and applications should be made through the Graduate Admissions Office. Upon admission by the University Statistics Committee, the student is also admitted to the Mathematics Department Master's program. The degree of Master of Statistics (Mathematics) is awarded by the Mathematics Department.

Degree requirements/coursework

- MATH 5010 - Introduction to Probability 3 Credit(s)
- MATH 5080 - Statistical Inference I 3 Credit(s)
- MATH 5090 - Statistical Inference II 3 Credit(s)
- MATH 6010 - Linear Models 3 Credit(s)
- MATH 6020 - Multilinear Models 3 Credit(s)
- MATH 6070 - Mathematical Statistics 3 Credit(s)
- MATH 6960 - Special Projects 1 - 6 Credit(s) (Masters Project)
- Electives approved by the supervisory committee, 12 credits
- Final project approved by supervisory committee

## ADMISSION REQUIRMENTS

- Bachelor’s Degree in Mathematics, or equivalent
- MATH 3070, 3080, or equivalent

## FILLING OUT THE APPLICATION

For instructions on filling in the Graduate Mathematics Application Form:

**Questions?**

**Thomas Albert, Associate Professor - alberts@math.utah.edu**

**Paula Tooman, Graduate Program Coordinator- tooman@math.utah.edu **

MASTER OF SCIENCE (M.S.) IN MATHEMATICS TEACHING

## APPLICATIONS FOR 2019-2021 ARE NOW CLOSED

The Department of Mathematics at the University of Utah offers a general Master of
Science degree as well as some specialized master’s degrees. One such degree is the **Master of Science (M.S.) in Mathematics Teaching **(also known as **Mathematics Teaching M.S.**), a two-year program primarily addressed to in-service secondary teachers of mathematics.

The main goal of the Master of Science in Mathematics Teaching program is to extend and enrich secondary teachers' understanding of mathematics and to help teachers acquire a deeper and broader mathematics background.

The program is structured to best meet the needs of Utah’s secondary teachers of mathematics. Coursework is designed to extend teachers’ mathematical knowledge for teaching at the secondary level. Courses are offered at times that fit working educators’ schedules, with a majority of courses offered over the two summer semesters of the program, and other courses offered once a week during evening hours in the fall and spring semesters.

The student is admitted to the Master of Science in Mathematics Teaching program upon admission by the Mathematics Education Committee. The degree is awarded by the Mathematics Department.

Tuitions are significantly reduced to make the program more accessible, and we are pleased to be able to offer this program to practicing Utah teachers for $150 per credit hour.

ADMISSION REQUIREMENTS

1. Licensed teachers with a level 4 endorsement (level 3 teachers will be considered, but may need additional course work).

2. Completion of the Praxis 5161 Mathematics (post-2012) or Praxis 0061 (pre-2012) Math Content Knowledge Exam. A score of at least 165 (5161) or 143 (0061) is required.

3. At least two years of teaching experience as of the application deadline. Applicants
must be recommended by** two professional educators** who can judge their performance.

4. Satisfy University requirements for Graduate School admissions.

**The Mathematics Education Committee may advise candidates to take additional courses
to ensure readiness for the required course work in the program.**

### PLANNED SCHEDULE FOR 2019 COHORT

The Master of Science in Mathematics Teaching program requires a total of 39 credit hours: 30 are core courses, and the remaining 9 credits are to be electives with at least 6 from the discipline-specific mathematics courses at the 5000 level or above. This meets the requirements for the Master of Science degree set by the Graduate School at the University of Utah. In addition, candidates must take and pass two mid-program exams and submit and defend a final program project. The program for the current cohort is structured as follows:

#### FIRST ACADEMIC YEAR

**Summer Semester (weekday mornings from the beginning of June to the beginning of August)**

**Mid-Program Exam (end of Summer/beginning of Fall)**

**Fall Semester (1 night a week on Wednesdays)**

**Spring Semester (1 night a week**

**on Wednesdays**)

#### SECOND ACADEMIC YEAR

**Summer Semester (weekday mornings from the beginning of June to the beginning of August)**

**Mid-Program Exam (end of Summer/beginning of Fall)**

**Fall Semester (1 night a week**

**on Wednesdays**)**Spring Semester (1 night a week**

**on Wednesdays**)**Final Defense (preferably in Spring, possibly in Summer)**

### Questions?

**Tommaso de Fernex, Professor - defernex@math.utah.edu**

### Elena Cherkaev

**Dir. of Graduate Studies**

**Can Answer Questions About:**

• Ph.D. in Mathematics

• M.A. and M.S. Mathematics Degrees

• Master of Statistics

### Paula Tooman

**Graduate Program Coord.**

**Can Answer Questions About:**

• Ph.D. in Mathematics

• M.A. and M.S. Mathematics Degrees

• Master of Statistics

### Thomas Alberts

**Associate Professor**

**Can Answer Questions About:**

• Master of Statistics

### Tommaso de Fernex

**Professor**

**Can Answer Questions About:**

• M.S. in Mathematics Teaching