The Mathematics PhD program provides a comprehensive and rigorous foundation in both pure and applied mathematics, with a strong focus on research excellence. It aims to cultivate the next generation of mathematical thinkers, capable of discovering, integrating, and applying advanced knowledge to solve real-world problems.
This program goes far beyond technical training, fostering the development of advanced mathematical expertise and deep analytical thinking. It equips students to tackle complex challenges with a critical and innovative mindset, empowering them to make meaningful contributions across a broad range of projects and disciplines. With a curriculum designed to address the technological challenges of the 21st century, graduates are exceptionally prepared for academic careers and leadership roles in industry, government, and beyond.
The program offers high-quality instruction in several core mathematical areas, including Analysis, Geometry & Topology, Algebra, Discrete Mathematics, Computational Mathematics, Probability and Financial Mathematics, and Applied Mathematics. Students can tailor their education by specializing in one or more cutting-edge research topics, such as:
- Big Data and Mathematical Statistics
- Cryptography
- Combinatorics and Graph Theory
- Commutative Algebra and Algebraic Geometry
- Control and Optimization
- Differential and Symplectic Geometry
- Financial Mathematics
- Fluid and Plasma Dynamics
- Free Boundary Problems
- Functional Analysis
- Harmonic Analysis
- Inverse Problems
- Machine Learning
- Mathematical Biology
- Nonlinear Waves and Dynamics
- Numerical Analysis
- Partial Differential Equations
- Probability and Stochastic Analysis
- Tomography and Medical Imaging
To accommodate this wide range of interests, the program offers flexibility in core course selection and candidacy examinations, ensuring students receive a tailored educational experience. With opportunities to engage in interdisciplinary research, the program equips students to thrive in both traditional and emerging fields.
The Mathematics PhD program requires 72 credit hours of coursework beyond the bachelor’s degree. In addition, the program offers a Financial Mathematics Track, providing specialized training for students interested in the intersection of mathematics and finance. Details about this track can be found at the bottom of this page.
The program is structured into three key phases: (i) Foundational Training in Mathematics, (ii) Pathway to Specialized Expertise, and (iii) Dissertation. Successful completion of each phase marks significant milestones in the student’s academic journey: qualification, candidacy status, and graduation, respectively.
In the first phase, students engage with a broad range of mathematical topics designed to establish a strong and comprehensive foundation in key concepts. These courses emphasize mathematical rigor and play a critical role in shaping students' analytical skills, ensuring they develop the intellectual framework needed for advanced study.
The second phase focuses on advanced and specialized coursework tailored to deepen students' expertise in their chosen field. This stage is essential for facilitating research readiness and equipping students with the knowledge required for meaningful inquiry. Completion of this phase signals the transition to Ph.D. candidacy, after which students begin accumulating dissertation hours.
The third and final phase is the dissertation, during which students collaborate closely with their Ph.D. advisor(s) and committee to develop a substantial and original research project, that makes a meaningful contribution to their field of study.
Master en route is permitted and encouraged.
Total Credit Hours Required: 72 Credit Hours Minimum beyond the Bachelor's Degree
Program Prerequisites
Bachelor's degree in mathematics or closely related field.
Students admitted to the graduate program with regular status are expected to possess a solid foundation in undergraduate mathematics, including the calculus sequence, differential equations, and linear algebra. Additionally, they should demonstrate proficiency in constructing and understanding mathematical proofs, exhibiting a level of mathematical maturity comparable to that attained in an Advanced Calculus course, such as UCF’s MAA 4226.
For students needing additional preparation or further development of mathematical skills, the program offers remedial courses in the first semester, structured as split-level classes.
Degree Requirements
Required Courses
27 Total Credits
- Complete at least 9 of the following:
- MAA5237 - Mathematical Analysis (3)
- MAA6405 - Complex Analysis I (3)
- MAS5311 - Algebra I (3)
- MAD5205 - Graph Theory I (3)
- MAA6416 - Topology (3)
- MTG5256 - Differential Geometry (3)
- MAS5145 - Advanced Linear Algebra and Matrix Theory (3)
- MAT5712 - Scientific Computing (3)
- MAP5336 - Ordinary Differential Equations and Applications (3)
- MAP6356 - Partial Differential Equations (3)
- MAA5238 - Measure and Probability I (3)
- MAP6111 - Mathematical Statistics (3)
Electives
30 Total Credits
- Earn at least 30 credits from the following types of courses: advanced mathematical courses at the 6000 or 7000 level. This may include up to 9 credit hours from Independent Study (XXX 6908), Directed Research (XXX 6918), or Special Topics/Seminars (XXX 6939). Of the total required credits, at least 24 credit hours must be earned from courses offered by the Department of Mathematics. Any coursework taken outside the department requires prior approval from the graduate program director. 6000-level core courses that were not used to meet the core course requirement may count toward the 30-credit-hour minimum of Electives. Students' coursework must be planned in coordination with their Ph.D. advisor or the graduate program director. The following is a list of regularly offered Elective courses: MAA6245 - Measure and Probability II (3) MAA6306 - Real Analysis (3) MAA6404 - Complex Analysis (3) MAA6506 - Functional Analysis (3) MAA7239 - Asymptotic Methods in Mathematical Statistics (3) MAD6309 - Graph Theory II (3) MAP 7386 - Numerical Solutions of PDE (3) MAP6118 - Introduction to Nonlinear Dynamics (3) MAP6195 - Mathematical Foundations for Massive Data Modeling and Analysis (3) MAP6197 - Mathematical Introduction to Deep Learning (3) MAP6207 - Optimization Theory (3) MAP6218 - Stochastic Calculus (3) MAP6385 - Applied Numerical Mathematics (3) MAP6387 - Numerical Linear Algebra (3) MAP6416 - Applied and Computational Harmonic Analysis (3) MAP6445 - Approximation Techniques (3) MAP6469 - Bayesian Analysis and Approximation Theory (3) MAP7119 - Advanced Nonlinear Dynamics (3) MAP7359 - Advanced Topics in Partial Differential Equations (3) MAS6312 - Algebra II (3) MTG6345 - Algebraic Topology (3)
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
- To complete the first training phase and advance in the Ph.D. program, students are required to pass two written Qualifying Examinations, administered by the Graduate Program Committee with support from Mathematics Graduate Faculty. These examinations are intended to assess the student’s mathematical rigor and analytical maturity by testing foundational concepts in Analysis and Linear Algebra, corresponding to contents covered in MAA 5237 – Mathematical Analysis and MAS 5145 – Advanced Linear Algebra and Matrix Theory. The Qualifying Examinations are held twice yearly: just before the start of the Fall semester and at the beginning of the Spring semester. Eligibility to sit for the examination requires that students maintain a minimum program GPA of 3.0 on a 4.0 scale and submit a formal written request to take the examination at least thirty (30) days prior to the scheduled date. Students are expected to pass both qualifying examinations by the end of their first year. Passing both exams is required to continue in the program beyond the second year. Although there is no limit on the number of attempts allowed for students into take each exam, students who do not pass both exams by the end of their second year will be dismissed from the program but may be eligible for an MS degree en route.
Candidacy Examination
0 Total Credits
- The Candidacy Examination is a pivotal step taken prior to a student’s formal admission to Ph.D. Candidacy. Unlike exams based purely on textbook material, this examination focuses on advanced topics closely aligned with the student’s prospective dissertation project, often encompassing recent developments and ongoing research in the field. Its purpose is to ensure the student is well-prepared to undertake independent research and contribute meaningfully to the discipline. This exam is administered by the Candidacy Examination Committee, composed of the student’s Ph.D. advisor and two other members of the Graduate Faculty. It comprises of two parts, a written report and an oral examination. The written component of the exam must be submitted as a research report to the Candidacy Examination Committee, which will evaluate its breadth, depth, and relevance. If the report receives approval, the oral examination will be scheduled. The oral examination is conducted privately and is not open to the public. The Candidacy Examination Committee determines the structure and duration of the exam. During this examination, students are expected to demonstrate mastery of the topics presented in their research report by responding to questions related to the subject matter. Students must complete the Candidacy Examination before the end of their fourth year. Students are granted a maximum of two attempts to successfully pass the examination. Students who do not pass the Candidacy Examination within two attempts and before the end of the fourth year will be dismissed from the program but may be eligible for an MS degree en route.
Seminar Participation
0 Total Credits
- Ph.D. students in regular status are required to attend a significant portion of the seminars and colloquia hosted by the Mathematics Department. Students in their third year and beyond are also expected to present their research or independent study annually.
Admission to Candidacy
0 Total Credits
- Admission to Candidacy marks a significant milestone in a Ph.D. student’s academic journey, signifying the successful completion of required coursework and examinations needed for independent research. This status represents a transition from coursework to original research, granting students greater independence and responsibility in their academic pursuits. It reflects faculty confidence in the student's readiness to contribute meaningfully to their field, paving the way for intensive research opportunities, leading to the successful completion of their dissertation, the pinnacle of their academic achievement. Requirements for Admission to Candidacy 1. Completion of Coursework: Students must complete all required coursework, maintaining a minimum program GPA of 3.0 on a 4.0 scale. This includes both foundational and advanced classes, totaling a minimum of 57 credit hours. Of these, at least 21 credit hours must be earned from courses at the 6000 and 7000 levels. 2. Candidacy Examination: Students must pass the Candidacy Examination, which, as described above, includes a written research report and an oral component. 3. Advisory Commitment: The dissertation advisory committee must be submitted and approved by the graduate program director. It consists of the PhD advisor, assisted by at least three other graduate faculties or scholars, with at least one of them external to the Department of Mathematics. 4. Program of Study: An approved program of study must be submitted. Once admitted to candidacy, students are expected to focus on their dissertation research.
Dissertation
0 Total Credits No Rules
Dissertation Proposal Examination
0 Total Credits
- After successfully passing the Candidacy Examination, the student is required to prepare a Dissertation Proposal and present it orally to the Dissertation Advisory Committee for approval. The proposal should comprehensively outline the research conducted to date, as well as provide a detailed agenda for the research intended to be completed for the dissertation. In addition to demonstrating a high level of mastery in the relevant area of study, the proposal must adhere to rigorous academic standards regarding correctness and the suitability of the proposed dissertation topic. The presentation must also meet the current standards for professional presentations within the discipline of mathematics. To successfully complete the Dissertation Proposal Examination, the presentation must be evaluated and approved by a majority of the Dissertation Committee as meeting the examination requirements. This examination must be completed within 12 months of passing the Candidacy Examination and no later than the end of the fifth year of graduate study. Candidates are allowed a maximum of two attempts to pass this examination. Students who do not pass the Dissertation Proposal Examination within two attempts and before the end of the fifth year will be dismissed from the program but may be eligible for an MS degree en route.
Dissertation Defense
0 Total Credits
- Upon completion of the student's research, the Dissertation Advisory Committee will schedule an oral defense of the dissertation. This defense serves as a formal presentation in which the student articulates their research findings, methodology, and contributions to the field. The successful completion of the dissertation defense marks the culmination of the student's academic journey, demonstrating their preparedness to contribute original research to the academic community and/or to take on leadership roles in industry, government, or other sectors where advanced mathematical thinking is essential.
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.
Time Limitation
0 Total Credits
- Most students are expected to complete their Ph.D. program within five years after joining the program. However, students are required to finalize their dissertation within a maximum of seven years after starting the program. This timeline ensures that students remain engaged with their research and maintain a trajectory towards timely graduation, while also providing adequate time to address any challenges that may arise during the dissertation process. Students who do not successfully defend their thesis by the end of the seventh year will be dismissed from the program but may be eligible for an MS degree en route. In exceptional cases, students may appeal for an extension to continue the program beyond the seventh year. For more information on UCF’s seven-year rule and the relevant forms, please refer to this link: https://graduate.ucf.edu/7-year-rule-completion-form/
Grand Total Credits: 72
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.
