Mathematics (PhD)

Program Overview

The University of Central Florida's Mathematics PhD program offers a comprehensive foundation in pure, applied, and industrial mathematics. Through rigorous coursework, specialization options, and research opportunities, the program aims to equip students with the skills to excel in their chosen research fields. The program provides flexibility and supports students' interests and career aspirations.

Program Outline

Degree Overview:

The Mathematics PhD program at the University of Central Florida (UCF) is designed to provide students with a comprehensive foundation in pure, applied, and industrial mathematics. The program aims to produce graduates who excel in their chosen research fields. To achieve this, the program includes required core courses and a selection of electives that offer cross-disciplinary subjects. The program offers specialization options in various areas 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
• Tomography and Medical Imaging
The program provides flexibility in choosing core courses and the candidacy examination to accommodate diverse student interests. It offers opportunities for research across various disciplines.

Outline:

The Mathematics PhD program requires 75 credit hours of coursework beyond the bachelor's degree. The program structure is as follows:

• Required Courses (12 credit hours):
• MAA 5237 - Mathematical Analysis (3)
• MAT 5712 - Scientific Computing (3)
• MAS 5145 - Advanced Linear Algebra and Matrix Theory (3)
• MAP 6385 - Applied Numerical Mathematics (3)
• MAA 6306 - Real Analysis (3)
• One of the following:
• MAA 6405 - Complex Variables (3)
• MAP 5336 - Ordinary Differential Equations and Applications (3)
• MAA 6506 - Functional Analysis (3)
• MAP 6111 - Mathematical Statistics (3)
• Restricted Electives (12 credit hours):
• Two 2-semester sequences, each consisting of two related courses.
• Sequences must be approved by the dissertation advisor, dissertation committee, and the graduate program director.
• Examples of acceptable sequences include:
• MAA 6405 - Complex Variables (3) and MAA 6404 - Complex Analysis (3)
• MAD 5205 - Graph Theory I (3) and MAD 5205 - Graph Theory I (3)
• MAP 5336 - Ordinary Differential Equations and Applications (3) and MAP 6356 - Partial Differential Equations (3)
• MAA 6238 - Measure and Probability I (3) and MAA 6245 - Measure and Probability II (3)
• MAP 6111 - Mathematical Statistics (3) and MAA 7239 - Asymptotic Methods in Mathematical Statistics (3)
• MAA 6306 - Real Analysis (3) and MAA 6506 - Functional Analysis (3)
• MAS 5311 - Algebra I (3) and MAS 6312 - Algebra II (3)
• MAP 6195 - Mathematical Foundations for Massive Data Modeling and Analysis (3) and MAP 6197 - Mathematical Introduction to Deep Learning (3)
• MTG 5256 - Differential Geometry (3) and MTG 6345 - Algebraic Topology (3)
• Unrestricted Electives (30 credit hours):
• Chosen in consultation with the student's advisory committee.
• May be selected from areas such as:
• Discrete Mathematics
• General Applied Mathematics
• Mathematical Computer Tomography
• Image Processing and Computer Graphics
• Mathematical Finance
• Mathematical Physics
• Pure Mathematics
• Mathematical Statistics
• Courses taken outside the Mathematics department must be approved by the advisor and graduate program director.
• Dissertation (15 credit hours):
• XXXX 7980 Dissertation Research (15 credit hours minimum)

Assessment:

The program includes the following examinations:

• Qualifying Examination:
• Based on core coursework (MAA 5237 and MAS 5145 - Advanced Linear Algebra and Matrix Theory).
• Students must pass the examination at the PhD level.
• Two attempts are permitted.
• Administered twice a year: Fall and Spring semesters.
• Students must have earned a "B" or better in each core course and a minimum GPA of 3.0 in the program, or obtain permission from the graduate program director.
• Normally taken after completing the core courses and expected to be passed by the end of the second year of study, unless a postponement is approved.
• Candidacy Examination:
• Written examination based on two selected 2-semester sequence courses.
• A committee prepares and grades the examinations.
• Can be attempted after passing the qualifying examination.
• Must be completed within three years of passing the qualifying examination.
• Two attempts are permitted.
• Dissertation Proposal Examination:
• Oral presentation of the dissertation proposal to the dissertation advisory committee.
• Includes a description of research performed and planned research.
• Must be passed within 18 months of passing the candidacy examination and no later than the end of the sixth year of graduate study.
• Two attempts are permitted.
• Dissertation Defense:

Other:

• Program Prerequisites:
• Bachelor's degree in a related field.
• Working knowledge of undergraduate calculus, differential equations, linear algebra (or matrix theory), and advanced calculus (at the level of MAA 4226).
• Master en route:
• Permitted and encouraged.
• Total Credit Hours Required:
• 75 credit hours minimum beyond the bachelor's degree.
• Minimum Formal Coursework:
• 45 hours exclusive of independent study, directed research, and internship.
• Financial Assistance:
• Fellowships, assistantships, tuition support, or loans.
• Fellowships:
• Awarded based on academic merit to highly qualified students.
• No work obligation.
• Graduate Teaching Assistantships:
• Over 20 offered annually on a competitive basis.
• Graduate Research Assistantships:
• Available for qualified students.

Other:

• The program is comprehensive with opportunities for students to pursue research in a variety of disciplines.
• Students are expected to complete the dissertation in no more than seven years from the date of admission to the program.
• 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.