The Mathematics PhD program prepares students with a broad base in pure, applied and industrial mathematics.

The Doctor of Philosophy degree in Mathematics is intended to provide a broad base in applied and industrial mathematics. The goal of the program is to produce students who will attain distinction in their fields of research. In order to achieve this, the program has required core courses as well as a set of electives providing cross-disciplinary subjects.

Students in the program can specialize in one of many aspects of mathematics, including Approximation Theory, Applied and Computational Harmonic Analysis, Big Data and Mathematical Statistics, Combinatorics and Graph Theory, Commutative Algebra and Algebraic Geometry, Control and Optimization, Differential and Symplectic Geometry, Fluid and Plasma Dynamics, Functional Analysis, Inverse and Ill-posed Problems, Machine Learning, Mathematical Biology, Mathematical Finance, Nonlinear Waves and Nonlinear Dynamics, Numerical Analysis, Orthogonal Polynomials, Partial Differential Equations, Probability and Stochastic Analysis, and Tomography and Medical Imaging. Responding to this wide variety of interests, the program offers flexibility in the composition of the core courses as well as the candidacy examination. The program is comprehensive with opportunities for students to pursue research in a variety of disciplines.

The Mathematics PhD program consists of 75 credit hours of course work beyond the bachelor's degree. The program includes 12 credit hours of required courses, 12 credit hours of restricted electives in two 2-semester sequences, and 36 hours of unrestricted electives. Lastly, 15 credit hours of dissertation research (7980) are required. Courses taken as required courses and electives must include a minimum of 45 hours of formal course work exclusive of independent study, directed research, and internship.

Master en route is permitted and encouraged.

**Total Credit Hours Required: 75 Credit Hours Minimum beyond the Bachelor's Degree**

## Program Prerequisites

Bachelor's degree in related field.

Students entering the graduate program with regular status are assumed to have a working knowledge of undergraduate calculus, differential equations, linear algebra (or matrix theory), and maturity in the language of advanced calculus (at the level of MAA 4226).

## Degree Requirements

### Required Courses

12 Total Credits

- Complete all of the following
- All students are required to complete the following courses with grade of "B" or better.
- Complete the following:
- MAA5237 - Mathematical Analysis (3)
- MAT5712 - Scientific Computing (3)
- MAS5145 - Advanced Linear Algebra and Matrix Theory (3)
- MAP6385 - Applied Numerical Mathematics (3)

### Restricted Electives

12 Total Credits

- Complete 2 of the following
- All students are required to complete two 2-semester sequences. Sequences are pairs of related courses that give advanced knowledge in an area of mathematics. Each sequence must be approved by the graduate program director. The following shows examples of acceptable sequences using current courses. We expect that other sequences will be developed as our program grows. An examination on two such sequences will be required as part of the candidacy examination (see more details in Candidacy Examination section)
- Complete at least 2 of the following:
- MAA6238 - Measure and Probability I (3)
- MAA6245 - Measure and Probability II (3)
- MAP6218 - Stochastic Calculus (3)

- Complete at least 2 of the following:
- MAA6306 - Real Analysis (3)
- MAA6506 - Functional Analysis (3)
- MAA6508 - Hilbert Spaces with Applications (3)

- Complete the following:
- MAA6405 - Complex Variables (3)
- MAA6404 - Complex Analysis (3)

- Complete at least 2 of the following:
- MAA6416 - Topology (3)
- MTG5256 - Differential Geometry (3)
- MTG6345 - Algebraic Topology (3)

- Complete the following:
- MAD5205 - Graph Theory I (3)
- MAD6309 - Graph Theory II (3)

- Complete at least 2 of the following:
- MAP5336 - Ordinary Differential Equations and Applications (3)
- MAP6356 - Partial Differential Equations (3)
- MAP7359 - Advanced Topics in Partial Differential Equations (3)

- Complete the following:
- MAP5641 - Financial Mathematics I (3)
- MAP6642 - Financial Mathematics II (3)

- Complete at least 2 of the following:
- MAP6111 - Mathematical Statistics (3)
- MAP6469 - Bayesian Analysis and Approximation Theory (3)
- MAA7239 - Asymptotic Methods in Mathematical Statistics (3)

- Complete the following:
- MAP6195 - Mathematical Foundations for Massive Data Modeling and Analysis (3)
- MAP6197 - Mathematical Introduction to Deep Learning (3)

- Complete the following:
- MAP6207 - Optimization Theory (3)
- MAP6387 - Numerical Linear Algebra (3)

- Complete the following:
- MAP6416 - Applied and Computational Harmonic Analysis (3)
- MAP6445 - Approximation Techniques (3)

- Complete the following:
- MAS5311 - Algebra I (3)
- MAS6312 - Algebra II (3)

### Unrestricted Electives

36 Total Credits

- Earn at least 36 credits from the following types of courses: Electives are chosen in consultation with the student's advisory committee and may be chosen from the suggested options: Discrete Mathematics, General Applied Mathematics, Mathematical Computer Tomography, Mathematical Finance, Mathematical Physics, Pure Mathematics, and Mathematical Statistics. A list of elective course options can be obtained from the graduate program director. Students are strongly encouraged to take more than two course sequences listed in the restricted electives and to use some of them as unrestricted electives. At most 15 credit hours of Independent study or directed research can be used as Unrestricted Electives. Courses taken outside the Mathematics department must be approved by the graduate program director. At most 9 credit hours of graduate courses outside the math department can be used as Unrestricted Electives.

### Dissertation

15 Total Credits

- Earn at least 15 credits from the following types of courses: XXXX 7980 Dissertation Research 15 Credit Hours (minimum)

### Examinations

0 Total Credits No Rules

### Qualifying Examination

0 Total Credits

- The qualifying/comprehensive examination is based on the core course work (MAA 5237 -Mathematical Analysis and MAS 5145 - Advanced Linear Algebra and Matrix Theory). To continue in the PhD program students must pass the examination on Mathematical Analysis and on Matrix and Linear Algebra, with both at the PhD level. Two attempts are permitted. The examination will be administered twice a year: one in the Fall semester and the other in the Spring semester. To take the examination, students must have earned a "B" or better in each core course, must have a minimum grade point average of 3.0 (out of 4.0) in the program, or must obtain permission from the graduate program director. Students will normally take the examination after taking the core courses MAA 5237 and MAS 5145, and are expected to have passed it by the end of the second year of study unless a written request for a postponement has been approved by the Graduate Committee at least two months before the examination date. The student must pass the Qualifying Examination in at most two attempts. It is strongly recommended that the student select a dissertation adviser by the completion of 18 credit hours of course work, and it is strongly recommended that the student works with the dissertation adviser to form a dissertation committee within two semesters of passing the Qualifying Examination.

### Candidacy Examination

0 Total Credits

- The Candidacy Examination consists of two examinations based on the materials from two of the selected two-semester sequence courses taken by the students beyond the core courses on Mathematical Analysis and Advanced Linear Algebra (MAA 5237, MAS 5145), with possible exam waiver on one two-semester course sequence. A committee formed or selected by the Graduate Committee or the graduate program director is responsible for preparing and grading the written examinations. After passing the candidacy examination and meeting other requirements, the student can register for Doctoral Dissertation (MAP 7980 or MAA 7980). A minimum of 15 Doctoral Dissertation credit hours are required. The Candidacy Examination can be attempted after passing the qualifying examination. The Candidacy Examination must be completed within three years after passing the qualifying examination. A student must successfully pass the Candidacy Examination within at most two attempts.

### Admission to Candidacy

0 Total Credits

- The following are required to be admitted to candidacy and enroll in dissertation hours: Completion of all course work, except for dissertation hours. Successful completion of the candidacy examination. The dissertation advisory committee is formed, consisting of approved graduate faculty and graduate faculty scholars. Submittal of an approved program of study.

### Dissertation

0 Total Credits No Rules

### Dissertation Proposal Examination

0 Total Credits

- After passing the candidacy examination, the student will prepare a dissertation proposal and orally present it to the dissertation advisory committee for approval. The proposal will include a description of the research performed to date and an agenda for the research planned to be completed for the dissertation. In addition to standards of correctness, indicating a suitable level of mastery of the material of the area of the dissertation, and suitability of the proposed dissertation topic, the presentation must meet current standards for professional presentations within the discipline of mathematics. For the successful completion of the Dissertation Proposal Examination the presentation must be judged as passing the requirements for the examination by the majority of the dissertation committee. This exam must be passed within 18 months of passing the candidacy examination and not later than the end of the sixth year of graduate study. A candidate must pass this examination within at most two attempts.

### Dissertation Defense

0 Total Credits

- Upon completion of a student's research, the student's committee schedules an oral defense of the dissertation. Most students complete the program within five years after obtaining their bachelor's degree. Students are expected to complete the dissertation in no more than seven years from the date of admission to the program.

### Independent Learning

0 Total Credits

- The required 15 credit hours of dissertation will provide ample opportunities for students to gain the independent learning experience through studying published research papers and deriving, on their own, new and meaningful research results.

#### Grand Total Credits: **75**

## Application Requirements

## Application Deadlines

## Financial Information

Graduate students may receive financial assistance through fellowships, assistantships, tuition support, or loans. For more information, see the College of Graduate Studies Funding website, which describes the types of financial assistance available at UCF and provides general guidance in planning your graduate finances. The Financial Information section of the Graduate Catalog is another key resource.

## Fellowship Information

Fellowships are awarded based on academic merit to highly qualified students. They are paid to students through the Office of Student Financial Assistance, based on instructions provided by the College of Graduate Studies. Fellowships are given to support a student's graduate study and do not have a work obligation. For more information, see UCF Graduate Fellowships, which includes descriptions of university fellowships and what you should do to be considered for a fellowship.

The department offers over 20 Graduate Teaching Assistantships every year on a competitive basis. A few Graduate Research Assistantships are also available for qualified students.