Electrical And Computer Engineering 20132014
Degree Requirements
MASTER OF SCIENCE
The Department of ECE offers both parttime and fulltime programs. The Department recognizes that a student may be employed fulltime while studying for a degree. Therefore, most courses are offered at times and on days that will permit a student to complete the program by taking courses either late in the day or outside normal business hours.. The MS degree programs can generally be completed in about four years if one course is taken each quarter, but it is usually possible to take two courses per quarter, bringing completion time closer to the more common duration of two years. Also, students who select the oneyear nonthesis will be able to graduate within 12 months, four academic quarters. For parttime students who are working in industry positions and who have chosen the thesis option, a topic related to the job function may be acceptable as the thesis research topic. Furthermore, a qualified staff member at the place of employment may be approved to serve as an adjunct faculty on the thesis committee.
Students not interested in pursuing a degree but interested in taking an occasional course may register as special status students by following an abbreviated admissions process. However, only 15 QH earned as a special status student may be applied toward a MS or PhD degree.
Minimum Credit Requirements
Every candidate for the MS degree must complete 45 QH of credit, at least 36 of which must be completed at the University of Denver.
Admission to the Master's Programs
A Bachelor of Science (BS) degree in Computer Engineering (BSCpE), Electrical Engineering (BSEE), or closely related field is required for admission to the programs. Those students whose backgrounds differ significantly from EAC/ABETaccredited BS Computer or Electrical Engineering programs may be required to complete prerequisite undergraduate courses. Such courses are not considered part of the 45 QH requirements for the degree. A competency examination may be required of candidates who do not possess a 3.0 GPA or a BS in Electrical, Electronic or Computer Engineering from an EAC/ABET accredited program. Students with BS degrees in Physics, Mathematics, Computer Science, Engineering Science, Electrical Engineering Technology, Engineering Physics or similar BS degrees may also be admitted. However, these students should be able to demonstrate competency in the following basic subjects by passing an appropriate competency examination:
MSCpE
 Circuits and Electronics
 Digital Systems
 Computer Organization
 A high or lowlevel computer language
MSEE
 Digital Design Methods
 Physical Electronics
 Introductory Electromagnetics
 Signals and Systems
 Principles of Communications
 Circuits and Electronics
MSE
 Controls
 Robotics
 Signals and Systems
 Circuits and Electronics
 Digital Design Methods
 Mechanics
 Electromagnetics
Students may be admitted provisionally while they take the appropriate prerequisite courses should it be determined from the competency examination or from the prior academic records that certain needed skills are lacking. Through the satisfactory completion of the stated provisional requirements, the student's status will be changed from provisional to regular status.
MASTER OF SCIENCE IN COMPUTER ENGINEERING, ELECTRICAL ENGINEERING, MECHATRONIC SYSTEMS ENGINEERING, OR ENGINEERING
The Master of Science in Computer Engineering (MSCpE), Electrical Engineering (MSEE), or Mechatronic Systems Engineering (MSMSE) is designed to advance the student's knowledge in several areas of engineering. Each degree provides breadth while permitting the student to achieve depth in one of several specialization areas. These specialization areas, with thematic sequences of courses, have been selected to coincide with those of high current interest as well as those emerging technologies that hold promise of increasing importance for the future. The purpose of these programs is to serve the profession of engineering and the Colorado community through advanced study in computer engineering, electrical engineering, and other related fields. Each program prepares the student for academic and industrial advancement. All programs offer a thesis and a nonthesis option.
Program Structure
Candidates may elect either the thesis or nonthesis option. This choice may be made at any time, although a delay in declaration may impact the completion date. Students who are GTAs or who receive financial support from a University research grant, such as GRAs, are required to elect the thesis option. These programs are designed to be completed in about six quarters if two courses (usually 8 QH) are taken each quarter.
NonThesis Option
The nonthesis option is the more flexible of the two options. This program is designed with the working professional in mind. For this option, a grade of B or better must be obtained in each course in order for that course to count toward the requirement of 45 QH. An overall minimum GPA of 3.0 is also required for graduation. Students may only take up to 8 quarter hours of independent study to be counted toward the degree. Each student must take a minimum of 24 quarter hours at the 4000level.
One Year (four quarters) – Nonthesis Option
Effective with the Academic Year 201314, the Department of Electrical and Computer Engineering (ECE) offers a oneyear, nonthesis option. Students who select the oneyear program will be able to graduate within 12 months, four academic quarters, as there are enough courses offered in each specialization to meet the 20 QH depth requirement. The breadth requirement (14 QH) is fulfilled by taking courses offered in other specializations. In addition, every year courses that satisfy the mathematics requirement (3 QH) are offered. The MS nonthesis structure is shown below. QH in each category denote minimum requirements that must be satisfied. Any changes in the student's plan of study must be approved aprior by the student's advisor.
The basic structure of the minimum 45 QH for the nonthesis option is as follows:
Core Requirement Based on Degree  Minimum QH 
Each degree program has two required classes, please see program for specifics.  8 QH 
Depth Requirement  Specialization Area  20 QH 
Mathematics Requirement Requires one approved course at the 3000level or higher 
3 QH (*) 
Breadth Requirement  14QH 
Total QH, nonthesis option  45 QH 
(*) This indicates minimum number of quarter hours. Any credits over the required 3 QH from the mathematics courses will count toward the breadth requirement.
Thesis Option
A thesis permits a candidate to obtain depth in an area of study and it is especially useful for individuals who seek to pursue a subsequent degree, for example, a PhD degree. Thesis candidates work closely with a thesis advisor. The thesis option is required for all GRAs and GTAs. For this option, a grade of C or better must be obtained in each course in order for that course to count toward the 45 QH hour requirements. An overall minimum GPA of 3.0 is also required for graduation. Students may only take up to 8 quarter hours of independent study to be counted toward the degree. Each student must take a minimum of 16 quarter hours at the 4000level.The basic structure of the minimum 45 QH for the thesis option is as follows:
Core Requirement based on Degree  Minimum QH 
Each degree program has two required classes. Pease see program for specifics. 
8 QH 
Depth Requirement Specialization Area  16 QH 
Breadth Requirement  6 QH (*) 
Thesis  15 QH 
Total QH, thesis option  45 QH 
(*) The breadth requirement must be preapproved by the student's advisor.
It is strongly recommended that students choose math related courses to satisfy the breadth requirement
If a student who has elected to pursue a thesis option, then at any time thereafter elects to change to a nonthesis option, all requirements for the nonthesis must be met. Any independent research taken may be forfeited and students must adhere to the grade requirements of the nonthesis option.
Breadth Requirement (NonThesis and Thesis Option)
Breadth Requirement courses (each with not less than 3 QH of credit) may be chosen from courses offered in other specialization areas. A course that appears in more than one specialization area may only be counted toward either the specialization requirement or the breadth requirement. The remaining courses are chosen from appropriate courses numbered 3000 or higher, offered by the Department Mechanical & Materials Engineering, Department of Computer Science or NSM (Natural Sciences and Mathematics). Prior approval by the student's advisor is required.
MASTER OF SCIENCE IN COMPUTER ENGINEERING (mS Cpe)
The MSCpE program offers one area of specialization:
• Intelligent Information System Design
The student's degree program will be a combination of the core courses, specialization areas (depth requirement) and the breadth requirement. Each student is required to complete the 2 core courses. Students may choose from any of the courses from their area of specialization but should keep in mind the 4000level requirement of the degree.
Core courses for all Computer Engineering Students
The following courses are required for all computer engineering students:
ENCE 4110 Modern Digital Systems Design
ENGR 3620 Advanced Engineering Mathematics
Specialization in Intelligent Information System Design
This area of specialization prepares students with fundamental and working knowledge of methods for analysis, design, and implementation of intelligent systems (IS). Particular attention is given to signal and information processing in IS, design of IS, and implementation of IS using stateoftheart technology. This is accomplished through several theoretical courses and applied courses. Students must choose from the following courses:
ENCE 3321 Network Design
ENCE 4231 Embedded Systems Programming
ENCE 4250 Advanced Hardware Description Language (HDL) Modeling and Synthesis
ENCE 4501 Advanced VSLI Design
ENCE 4620 Advanced Computer Vision
ENCE 4900 Machine Learning
ENEE 3670 Introduction to Digital Signal Processing
master of science in electrical engineering (ms enee)
The MSEE program offers four areas of specialization:
 Control System Analysis and Synthesis
 Electric Power and Energy Systems
 Optics/Optoelectronics
 Modern Communications System Design
Each student must choose an area of specialization. The student's degree program will be a combination of the core courses, specialization areas (depth requirement) and the breadth requirement. Each student is required to complete the 2 core courses. Students may choose from any of the courses from their area of specialization but should keep in mind the 4000level requirement of the degree.
Core courses for all Electrical Engineering Students
The following courses are required for all electrical engineering students, regardless of area of specialization:
ENEE 4640 Electromagnetic Compatibility
ENGR 3620 Advanced Engineering Mathematics
Specialization in Control System Analysis and Synthesis
This area of specialization prepares students for basic and applied research and development of complex systems, including, electrical, mechanical, bioinspired, mechatronic, and robotic systems, as well as unmanned systems. This is accomplished through several theoretical courses and applied courses. Students must choose from the following courses:
ENCE 4231 Embedded Systems Programming
ENEE 3670 Introduction to Digital Signal Processing
ENEE 4630 Optical Networking
ENGR 3721/3722 Controls and Controls Systems Laboratory
ENGR 4730 Introduction to Robotics
*ENGR 4350 Reliability
*ENGR 4620 Optimization
ENGR 4735 Linear Systems
*This course may count toward the specialization with advisors preapproval. This course may not or may not be offered on a regular basis.
Specialization in Electric Power and Energy
This area of specialization prepares students with the basic foundation and advanced knowledge, required for the research and development in the area of power systems, renewable energy systems, and power electronic devices. This is accomplished through several theoretical courses and applied courses. Students must choose from the following courses:
ENGR 3510 Renewable and Efficient Power and Energy Systems
ENGR 3540 Electric Power Systems
ENGR 3721/3722 Controls and Controls Systems Laboratory
ENGR 4530 Introduction to Power and Energy
ENGR 4545 Electric Power Economy
ENGR 4560 Power Generation Operation and Control
ENGR 4735 Linear Systems
Specialization in Optoelectronics & Photonics
This area of specialization prepares students for research, development, and design of devices and systems operating based on wave theory; focusing on laser, optics, light wave devises, and systems.
ENEE 4030 Optoelectronics
ENEE 4035 Nanophotonics
ENGR 4200 Introduction to Nanotechnology
ENCE 4250 Advanced Hardware Description Language (HDL) Modeling and Synthesis
ENEE 4630 Optical Networking
ENGR 4735 Linear Systems
Specialization in Modern Communications Systems Design
This area of specialization prepares students for research, development and design of devices and systems that operate using wave theory: laser, optics and light wave devices, electromagnetic theory, waveguides and antennas. This is accomplished through several theoretical courses and applied courses. Students must choose from the following courses:
ENCE 3321 Network DesignENEE 3111 Signals & Systems
ENEE 3141 Digital Communications
ENEE 4030 Optoelectronics
ENEE 4035 Nanophotonics
ENEE 4620 Advanced Optical Fiber Communication
ENEE 4625 Radio over Fiber Communication
ENEE 4630 Optical Networking
master of science in mechatronic systems engineering (mS Mse)
This area of specialization is designed to meet the needs of industry and federal research laboratories for engineers with multidisciplinary experience and ability to design and integrate complex systems requiring knowledge from diverse engineering disciplines. Said differently, mechatronic systems involves integration of mechanical, electrical, and computer engineering to design complex systems that perform realworld tasks. This program includes a broad set of common course requirements along with a selection of appropriate technical electives providing both breadth and depth of knowledge in a student's area of interest.
The MSE program offers one area of specialization:
• Mechatronic System Design
This program is a combination of core courses, specialization areas (depth requirement) and technical electives (breadth requirement). Each student is required to complete two core courses. Students, along with their advisor, should develop a plan of study choosing thematic sequences of courses from different specialization areas, but they should keep in mind the 4000level requirement of the degree.
Core courses for all Mechatronic Systems Engineering Students
The following courses are required for all mechatronic systems engineering students regardless of area of specialization:
ENEE 4641 Electromagnetic Compatibility OR
ENCE 4110 Modern Digital Systems Design
ENGR 3620 Advanced Engineering Mathematics
Specialization in Mechatronic System Design
ENCE 4620 Advanced Computer Vision
ENCE 4250 Advanced Hardware Description Language (HDL) Modeling and Synthesis
ENCE 4231 Embedded Systems Programming
ENCE 4900 Machine Learning
*ENGR 3350 Reliability
*ENGR 3630 Finite Element Methods
*ENGR 4620 Optimization
ENGR 4730 Introduction to Robotics
ENGR 4735 Linear Systems
*ENME 4020 Advanced Finite Element Analysis
ENMT 4220 Mechatronics II
*This course may count toward the specialization with advisors preapproval. This course may not or may not be offered on a regular basis.
engineering  concentration in engineering management
This program allows students to pursue a concentration in engineering management. The concentration in engineering management is designed to meet the increasing needs of students to enhance their career opportunities as managers or as entrepreneurs by supplementing advanced engineering knowledge with a fundamental understanding of business principles within the context of technology enterprises. Drawing upon the strengths of both the Ritchie School of Engineering and Computer Science (RSECS) and the Daniels College of Business (DCB), the program provides relevant content for graduates to lead technology enterprises. There is only a nonthesis option available for this program. The program structure is as follows:
Engineering Focus (depth requirement)  Minimum QH 

24 QH 
Engineering Mathematics Requirement Must be at the 4000level and have advisors approval 
3 QH 
Management Focus Students may choose any 4000level management 
18 QH 
Total QH  45 QH 
The engineering focus must consist of a set of coordinated and related engineering courses designed to give the student depth in a particular engineering discipline (CpE, EE. MSE or ME). Students must choose and follow an area of specialization from a given discipline (the specializations are listed above), but this requirement may be waived with approval by the student's engineering advisor if the student wishes to pursue a coherent interdisciplinary curriculum.
Additional Mathematics Requirement:
All MS nonthesis candidates must complete an advanced mathematics requirement. Students may choose from the following courses:
ENGR 3620 Advanced Engineering Mathematics
ENGR 4620 Optimization
ENEE 3670/4800 Introduction to DSP
ENGR 3630 Finite Element Methods
ENGR 3800 Topics in Engineering: Numerical Methods
ENGR 4350 Reliability
ENGR 4735 Linear Systems
ENGR 4740 Principles of Adaptive and Optimal Control Systems
ENGR 4745 Advanced NonLinear Control System
ENME 4020 Advanced Finite Element Analysis
MATH 3120 Introduction to Topology
MATH 3151 Advanced Linear Algebra
MATH 3152 Linear Algebra II
MATH 3161 Introduction to Real Analysis
MATH 3166 Group Theory
MATH 3400 Introduction to Geometry
MATH 3705 Topics in Mathematics
MATH 3720 Coding Theory
MATH 3851 Functions Complex Variable I
MATH 3852 Functions Complex Variable II
doctor of philosophy
The objective of the PhD in ECE or MSE degree programs are to provide an educational environment that encourages students to develop the ability to contribute to the advancement of science, engineering and technology, through independent research. The PhD students of the 21st century may pursue academic, research, entrepreneurial, and/or industrial careers. We offer opportunities to develop individualized plans of study based on the students' previous experience and desired research areas. The plan of study allows students to work on interdisciplinary research, while also satisfying the PhD in ECE or MSE degree requirements.
Admission to the PhD Program
Students with a MS in CpE, MS in MSE, MS in EE, MS in ME, or closely related areas may apply for admission to the PhD in ECE or PhD in MSE programs. Admission with only a BS in this field is also possible, but students with only a BS degree are strongly encouraged to enroll first in the MS (CpE, EE, MSE) programs. Admission to the PhD programs is based on review of the application and associated references. Normally, a GPA of at least 3.0 is required. All graduate engineering courses presuppose mastery of the subject matter of a modern ABETaccredited curriculum in engineering. Students with a BS in other engineering or related science fields and students with a BSCpE, BSEE, or BSME who have not taken graduate academic work for some time may be required to complete preparatory courses that are prerequisites for the core courses of the engineering concentrations on which the qualifying exams are based. These courses carry no credit toward the graduate degree.
Program Structure
Research requires an indepth study of engineering problems with a broad knowledge base in science and engineering. Therefore, advanced courses are offered to strengthen the fundamentals and to broaden the engineering and science perspective. The minimum credit requirements are different for individuals entering a program with a closely related master's degree and for those entering with a bachelor's only. All requirements for the degree must be completed within seven years (eight years without a master's degree) from admission to candidacy. A grade of C or better must be obtained in each course in order for that course to count toward the credit hour requirements. An overall minimum GPA of 3.0 is also required for graduation.
PhD in ECE
The PhD in ECE is appealing to students because it offers the much needed specialization component and the 'degree identity' required to be competitive in the job market. Graduates from this program will be well equipped to follow academic careers, or be hired in federal laboratories, industry and the private sector.
PhD in MSE
The PhD in MSE is at the forefront and intersection of the coupled disciplines of Electrical, Mechanical, Computer Engineering, and Computer Science. This unique degree is appealing to students because they will acquire the knowledge and ability to deal with and solve highly complex problems where integration is a key component. This degree provides a holistic approach to graduate education focusing on the ability to cover both breadth and depth of knowledge. Graduates of this program will lay the foundation for the modern engineering departments of the future, where 'integration' will be the key ingredient of studies.
Program requirements
All PhD students who have been admitted to the PhD in ECE or PhD in MSE programs must successfully complete three milestones before the PhD degree can be conferred. These milestones refer to:
 Demonstrating that the student is qualified to begin PhD studies
 Demonstrating that the student may identify and formulate a research problem
 Demonstrating that the student can defend her/his thesis
These three milestones are referred to as the "PhD Qualifying Exam", the "Comprehensive Exam" (also known as the "PhD Proposal"), and the "Thesis Defense".
If a student is entering the PhD program without a relevant master's degree, the student should work with their advisor in order to meet the degree requirements for a master's degree. All requirements for the given master's degree must be met in order for the students to receive the degree.
Qualifying Examination
Each student must demonstrate sufficient breadth and depth of basic engineering knowledge relevant to electrical and computer engineering. Each student must demonstrate ability to organize and present her/his thoughts in a convincing manner. The PhD Qualifying Exam achieves this through three components: a written Common Exam of basic engineering knowledge (breadth), two written Specific Area Exams (depth), and an oral Design Exam (breadth, depth, organization and presentation). Failure to pass any component of the PhD Qualifying Exam will prevent the student from continuing in the PhD program.
All PhD students who are admitted into the Department of Electrical and Computer Engineering must pass the PhD Qualifying Exam. There are three components of the PhD Qualifying Exam consisting of four test subject areas. The three components are
PhD Common Exam This is a common twohour written exam. Each student, with advice from his/her advisor must choose one of the three subject areas. The Common Exam will be graded as pass/fail; with 70% constituting as passing grade.
 Engineering Mathematics (Calculus, Engineering Analysis, Linear Algebra)
 Circuits and Electronics
 Digital Design, Computer Organization, and HDL
PhD Specific Area ExamThis part of the exam will consist of two written subject area texts lasting two hours each. Students must pick two specific subject areas and cannot be the same subject area as the topic chosen for the PhD Common Exam. The Specific Area Exam will be graded as pass/fail; with 70% constituting as passing grade.
 Digital Design, Computer Organization, and HDL (only if NOT taken for the common component)
 Circuits and Electronics (only if NOT taken for the common component)
 Microprocessors
 Data Structures, Algorithms, & Operating Systems
 Control, Signals & Systems
 Electromagnetics
 Power & Energy Systems
 Optoelectronics
 Optical Fiber Communication
 Communication & DSP
 Robotics
 Image Processing & Computer Vision
 Pattern Recognition
PhD Specialization Design Exam This portion of the exam will be an openended design problem developed by the candidate's advisor. The Design Exam will be based on the candidate's area of research. This exam will be evaluated by a committee of three faculty (one must be the candidate's advisor) in both written and oral form. The candidate will have five days to complete the written report and an additional two days to prepare the presentation. If the written design exam is turned in late for any reason, it will be considered as an automatic fail. The Specialization Design Exam will be graded on a pass/fail scale.
The purpose of this design examination is to demonstrate the candidate's ability to follow good design procedures at the level of an undergraduate senior design course and to explain/justify the tradeoffs that are tantamount to design. For the purposes of this examination, "good design procedures" shall include the following steps as outlined in the DU ENGR 3313/3323/3333 sequence in the undergraduate curriculum:
 Conceptual Design
 Requirements Analysis
 Specification
 Brainstorming
 Architectural Design
 Detailed Design
 Test and Validation
 Maintenance and Support
In general, it will not be possible to fully execute a design solution to an openended design problem in one week. The emphasis of this exam is on the candidate's ability to complete the above steps to some convincing level of detail, the candidate's ability to explain/justify the tradeoffs considered at each step of the process, and the candidate's ability to communicate the completeness of the design exercise to the committee at a level appropriate for a beginning graduate student in engineering. As part of this exercise, comparisons to the existing body of literature are expected.
If a student is unable to pass the PhD Common Exam and/or any of the PhD Specific Area Exams, the student must take the same exam(s) during the second attempt; the student is not allowed to switch subject areas.
All PhD students must attempt the PhD Qualifying Exam by the end of their first year. If a student is unsuccessful at passing all four test areas, the student will be given an additional year to pass the PhD Qualifying Exam. All students must take and pass the PhD Qualifying Exam by the end of their second year. A student shall be considered to have passed the PhD Qualifying Exam only after all four test areas have been successfully completed within the given time constraints identified.
Comprehensive Examination
The purpose of the Comprehensive Examination is to ascertain the potential of the student for PhD quality research. At least two quarters prior to the final defense, the student shall schedule and take the Comprehensive Examination. This oral and written examination will be attended by a minimum of three faculty members, the same faculty who will attend the student's final dissertation defense. The Comprehensive Exam may be open to other students based on the requirements of the student's advisor. The student is expected to make a 30 to 40 minute concise presentation on her/his dissertation topic. The oral and written presentation will highlight previous work in this area, demonstrate a need for the given research, and explain how the given research will contribute to the advancement of the area. The student will also present completed work and results, anticipated work and results, and a detailed plan for project completion. In addition, the student will be expected to answer general fundamental questions in the area of her/his concentration and detailed questions in the area of the student's graduate course work.
The PhD Qualifying Examination must be taken and passed prior to the student taking the Comprehensive Examination. The Comprehensive Examination can be taken at most 2 times. If the student does not pass the Comprehensive Exam on the second try, the student will be terminated from the program. The comprehensive exam will be graded on a pass/fail system, revisions maybe required.
Dissertation
The student is required to complete and defend a dissertation of publishable quality based on the student's original research. The dissertation must be completed in written form in accordance with the University's Graduate School guidelines. A summary of the dissertation must be presented in a public seminar and subsequently defended by the student in the final oral examination. The examining committee will consist of the student's entire PhD committee.
Residence Requirement
Enrollment in at least six quarters (four semesters), including at least two consecutive quarters (one semester) of fulltime attendance is required for graduation.
PhD Committee
The PhD committee should consist of at least four faculty members. Three faculty members must be from within the student's specialty area; these can include the student's advisor, other faculty in that degree program and, if necessary, offcampus experts. Finally, for the final oral defense of the thesis, an oral defense chair, who must be a tenured faculty member outside the Department of Electrical and Computer Engineering and Mechanical and Materials Engineering, needs to be identified in consultation with the DU Graduate Studies Office. The PhD committee must approve the student's plan of study and research plan and must be in place before the PhD comprehensive exam.
Minimum credit requirements Students with a Bachelor of Science in Engineering/Science
For students admitted to the PhD program with a bachelor's degree, 90 QH are required, 72 of which must be completed at the University of Denver. A minimum of 48 QH must be at the 4000level or higher and may include as many dissertation research hours (Independent Research and Independent Study) as considered appropriate by the advisor. The student with his/her advisor will develop an appropriate plan of study with core requirements, an area of specialization (depth requirement), breadth requirement and advanced mathematics. The core will consist of 8 QH of coursework. The area of specialization will consist of 16 QH of coursework. An additional 6 QH of coursework (excluding independent research) is required as related breadth requirement. The student must complete a minimum of 16 HQ at the 4000level courses, excluding independent research. Prior to completion of the comprehensive exam, the plan of study must be approved by the student's PhD committee.
Students with a Master of Science in Engineering/Science
If a student is admitted with a closely related master's degree, up to 45 hours may be transferred and applied to the doctorate degree. A minimum of 45 quarter hours is required at the University of Denver. The student with his or her advisor will develop an appropriate program consisting of a minimum of 45 quarter hours at the 4000level, which may include as many dissertation research hours (Independent Research and Independent Study) as considered appropriate by the advisor. The student with his or her advisor will develop an appropriate plan of study with an area of specialization, breadth requirements and advanced mathematics. Prior to completion of the comprehensive exam, the student's plan of study must be approved by the student's PhD committee.