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Math 5387 Regression Analysis, Modeling and Time Series 
Math 6330 Workshop in Statistics Consulting (This course can be taken more than once.) 
Math 6388 Advanced Statistical Methods for Research 
Math 7381 Mathematical Statistics I 
Math 7382 Mathematical Statistics II 
Math 5310 Probability 
Math 5792 Probabilistic Modeling 
Math 6380 Stochastic Processes 
One of the following two courses: 
Math 6131 Real Analysis 
Math 7381 Mathematical Statistics I 
i) Two foundations courses from the following list: 
Math 5446 Theory of Automata (same as C SC 5464) 
Math 5576 Mathematical Foundations of Artificial Intelligence I 
C SC 5451 Algorithms 
ii) One discrete computerrelated course from the following list: 
Math 5410 Modern Cryptology 
Math 5793 Discrete Math Modeling 
Math 6404 Applied Graph Theory 
Math 7409 Applied Combinatorics 
Math 7421 Projective Geometry 
iii) A continuous computerrelated math course from the following list: 
Math 5660 Numerical Analysis I 
Math 6653 Intro to Finite Element Methods 
Math 6595 Comput Methods/Nonlinear Programming 
Math 7663 Numerical Solution of PDE 
Math 7665 Numerical Linear Algebra 
iv) One additional computerrelated course from the following: 
One course from i, ii, iii (above), or a sequel to it 
Math 5674 Parallel Computing & Architectures (same as C SC 6551) 
Math 5779 Math Clinic, subject to approval 
Math 6380 Stochastic Processes 
Four of the following nine courses: 
Math 6404 Applied Graph Theory 
Math 5410 Modern Cryptology 
Math 5432 Computational Graph Theory 
Math 5490 Network Flows 
Math 5793 Discrete Math Modeling 
Math 7409 Applied Combinatorics 
Math 7410 Combinatorial Structures 
Math 7413 Modern Algebra I 
Math 7421 Projective Geometry 
Math 7821 Topics in Projective Geometry 
Math 7823 Topics in Discrete Math 
Other suggested courses: 
Math 5110 Number Theory 
Math 5593 Linear Programming 
Math 7414 Modern Algebra II 
Math 7594 Integer Programming 
Three of the following seven courses: 
Math 5387 Regression Analysis, Modeling and Times Series 
Math 5779 Math Clinic 
Math 5791 Continuous Modeling 
Math 5792 Probabilistic Modeling 
Math 5793 Discrete Math Modeling 
Math 5794 Optimization Modeling 
Math 6735 Continuum Mechanics 
Two of the following seven courses: 
Math 5660 Numerical Analysis I 
Math 5661 Numerical Analysis II 
Math 5733 Partial Differential Equations 
Math 6653 Intro to Finite Element Methods 
Math 7663 Numerical Solution of PDEs 
Math 7665 Numerical Linear Algebra 
Math 5660 Numerical Analysis I 
Math 5661 Numerical Analysis II 
Three of the following ten courses: 
Math 5593 Linear Programming 
Math 5733 Partial Differential Equations 
Math 6595 Computational Methods in Nonlinear Programming 
Math 6653 Intro to Finite Element Methods, formerly Math 7172 
Math 6735 Continuum Mechanics 
Math 7667 Intro to Approximation Theory 
Math 7663 Numerical Solution of PDEs 
Math 7665 Numerical Linear Algebra 
Math 8664 Iterative Methods in Numerical Linear Algebra 
Math 8660 Math Foundations of Finite Element Methods 
Students in this area are also encouraged to take graduatelevel computer science and/or parallel computing courses. 
Math 5593 Linear Programming 
Math 5792 Probabilistic Modeling or Math 6380 Stochastic Processes 
Two of the following courses: 
Math 5390 Game Theory 
Math 5490 Network Flows 
Math 5779 Math Clinic, with approval 
Math 5794 Optimization Modeling 
Math 6595 Computational Methods in Nonlinear Programming 
Math 7025 Topics in Optimization 
Math 7593 Advanced Linear Programming 
Math 7594 Integer Programming 
Math 7595 Advanced Nonlinear Programming 
Math 5396 Bayesian Statistics 
Math 5610 Computational Biology 
C SC 5451 Algorithms 
Biol 5099 Biology for Computer Scientists, Engineers and Mathematicians 
Math 5840 Independent Study  Complex Programming Project (subject to approval) 
Two graduate mathematics courses 
One elective preapproved courses listed below 
Math 5060 Exploratory Data Analysis 
Math 5593 Linear Programming 
Math 5791 Continuous Modeling 
Math 5576 Mathematical Foundations of Artificial Intelligence 
Math 6404 Applied Graph Theory 
Math 6595 Computational Methods in Nonlinear Programming 
C SC 5582 Artificial Intelligence 
C SC 5559 Database Systems 
Biol 5124 Molecular Biology 
Biol 5550 Cell Signaling 
Chem 5810 General Biochemistry I 
There are six phases to the Ph.D. program. A candidate must fulfill course requirements, pass the preliminary examinations, meet the academic residency and participation requirements, pass the comprehensive examination, give an oral thesis proposal, and write and defend a thesis.
Ph.D. students must complete 42 credit hours of nondissertation graduate courses. The following courses will not count toward a graduate degree in applied mathematics: Math 50005010, Math 50125015, Math 5017, Math 5198, and Math 5250.
By graduate school rules, courses taken more than five years prior to applying for candidacy (including transfer courses) must be validated by the graduate program director to ensure their content is still current. For mathematics courses taken within ten years of applying for candidacy, the validation request will be automatically approved. For courses taken more than ten years prior to applying for candidacy, the validation process will require an assessment of the student's knowledge of the subject matter.The following courses are required as a part of the formal course work:
The Readings Courses are onehour seminar courses that are announced prior to the start of each semester. All courses should be chosen in consultation with an advisor. Course replacements and equivalencies should be approved by the Graduate Committee.
Within the coursework requirement, students must satisfy a breadth requirement by completing six graduate math courses from the following categories. No more than three of these courses can come from any one category:
A maximum of 30 credit hours of graduate coursework (including courses applied to a Master's degree), may be transferred into the Ph.D. program. Up to 6 credit hours of this 30 may be awarded for a Master's thesis. Only graduate courses completed with a grade of B or better may be considered for transfer credit. Credit cannot be transferred until the student has established a satisfactory record of at least one term of fulltime enrollment at UC Denver with a minimum GPA of 3.0. All transfer courses must be approved by the graduate committee. Courses taken while registered as a nondegree student are considered transfer courses. By graduate school rules, courses taken while enrolled as a graduate student at any campus of the University of Colorado system is considered resident coursework. However, all courses taken outside of the Department of Mathematical and Statistical Sciences must be approved by the graduate committee to be applied toward the Ph.D. In addition, courses taken at other campuses prior to enrollment in the Ph.D. program will be counted as part of the 30 credit hour limit on transfer courses. Courses taken outside the Department of Mathematical and Statistical Sciences while enrolled in the Ph.D. program should be approved before they are taken in order to be counted toward the Ph.D. degree.
All Ph.D. course work must be completed with at least a 3.25 grade point average. Grades below a B are not acceptable for the Ph.D. A student who receives a grade of C+ or lower, or whose overall GPA as a doctoral student falls below 3.25 will be reviewed by the Graduate Committee and may be put on probation or suspended.
Each student must pass two preliminary exams: one in Applied Analysis and one in Applied Linear Algebra. These are fourhour written exams that cover material roughly at the level of firstyear graduate study. The exams are offered twice a year, from 10:00 am to 2:00 pm. The first set of exams are offered in January. The Applied Analysis preliminary exam will be held the Monday prior to the start of classes, and the Applied Linear Algebra exam is offered the Friday before the start of classes. The second exam offerings follow the spring semester. The Applied Linear Algebra exam is held the Friday before the start of the summer semester, and the Applied Analysis exam is offered the first Friday of the summer semester.
A student wishing to take a preliminary exam must sign up with the graduate program assistant at least one month prior to the administration of the exam. Students who cannot take the exam as planned must give notice of withdrawal at least one week prior to the exam.
Students have a maximum of three attempts to pass each exam (including attempts prior to enrolling in the Ph.D. program). The time limit clock starts the first semester a student is admitted into the program.
Each student must choose an advisor and, with the advisor's help, select other members of the Ph.D. advisory committee. This committee assumes the dual responsibility for advising and testing the student. In particular, under the direction of the advisor, each member of this commitee will:
The Ph.D. advisory committee consists of five graduate faculty members from the CU system, one of whom is the student's advisor. At least one committee member must have their primary and prior affiliations outside of the Department of Mathematical & Statistical Sciences. At least three committee members will be regular faculty members of the Department of Mathematical & Statistical Sciences. Exceptions to this must be approved by the graduate committee. Committee members may be from outside the CU system, but must be granted special membership on the graduate faculty for this purpose. The chair of the Ph.D. advisory committee must be a regular faculty member of the Department of Mathematical & Statistical Sciences. Under graduate school rules, the thesis advisor cannot be the chair of the committee. The thesis advisor may be affiliated with another university department, a government lab, or industry, subject to approval by the Graduate Committee, and subject to approval of a special appointment to the graduate faculty.
After the student has been admitted to candidacy and has presented the research proposal, the advisory committee will meet with the student at least once per year to assess progress. If progress is unsatisfactory, the committee will write a written report to the student and the graduate program director specifying steps to be taken to rectify the situation. If the student does not achieve an acceptable level of progress within six months after receipt of this report, the student will be terminated from the program.By Graduate School rules, all doctoral students must be enrolled for a minimum of six (6) semesters of full time scholarly work beyond the attainment of a bachelor's degree. Two of these six semesters may be replaced by a master's degree in mathematics from another institution; however, at least four (4) semesters of credit must be earned for work performed while enrolled at UC Denver. For this purpose a full course load is defined to be five semester hours of course work.
Ph.D. students are expected to participate in the life of the department by attending colloquia, seminars, orientations and other department activities and by spending time on campus interacting with other students and faculty outside of normal class hours. Students should discuss with their advisors how to fulfill the spirit of this requirement.
Application for candidacy to the Ph.D. program must be made at least two weeks before the comprehensive examination is taken. Candidacy will be granted after at least three semesters of residence have been earned, an advisory committee has been selected, all preliminary and comprehensive examinations have been passed, and essentially all course requirements (including the breadth requirement) have been satisfied.
The comprehensive exam has the following objectives: to determine mastery of graduate level mathematics, capacity to synthesize mathematical concepts, and ability to embark upon doctoral thesis research. The comprehensive exam has two parts:
1. The first part consists of a written exam of roughly (but not limited to) six hours. The written exam covers material from the student's intended area of research. The choice of area and the extent of coverage within that area will be determined by the student's advisory committee. The advisory committee will prepare and conduct the exam and determine the outcome. The written exam has three possible outcomes: pass, conditional pass, and failure.
2. Given a conditional pass on the written exam, the second part of the exam consists of an oral followup, not to exceed two hours in length. The student will be given a copy of the graded written exam, no later than seven days after that exam, and a list of topics in which the committee found the student deficient. The oral followup will cover questions on the written exam and topics on the list provided to the student. The oral exam is open to the public and must be scheduled and advertised through the graduate program assistant at least two weeks prior to the exam. The oral must be given within four weeks of returning the graded exam to the student, and it has two possible outcomes: pass or failure based on a majority vote of the committee.
"In the event of failure of either the written exam or oral followup, the student's advisory committee will determine the next step (dismissal from the program, retake of the oral followup and/or a retake of the written exam are possible outcomes). If a retake is allowed, the retake must be completed within 12 months. The Graduate Committee will hear grievances and appeals of the outcome of the comprehensive exam.
All members of the advisory committee must be present for the oral exam; however, a minority of members, but not the chairperson nor the student, may participate by interactive video. In the event of an emergency that prevents one committee member from attending the exam, the exam can proceed with the faculty who can attend. However, the student will need to meet with the absent committee member at an alternate time.
Admission to candidacy follows successful completion of the two parts of the comprehensive exam.
Within six months of successful completion of the written exam and/or oral followup, the student must give an oral research proposal before the advisory committee. The purpose of this presentation is to determine the feasibility of the student's proposed thesis topic. The research proposal is open to the public and must be scheduled and advertised through the graduate program assistant at least two weeks prior to the presentation. The student will be provided a detailed summary of the committee's assessment and recommendations. At the discretion of the advisory committee, a student may be asked to give a subsequent oral proposal at a later date.
Each student must complete at least 30 hours of thesis credit. Not more than 10 of these hours may be taken in any one semester. Not more than 10 thesis hours taken prior to the semester of the comprehensive examination may be applied to this requirement.
Candidates for the Ph.D. degree are required to write a dissertation containing original contributions and evidence of significant scholarship. The thesis is written under the guidance of an advisor who is a member of the graduate faculty of the University of Colorado. The thesis must comply in format with the specifications of the Graduate School and must be prepared in TeX, LaTeX, or AMSTeX (thesis templates available). Six weeks before the date of graduation, the Graduate School must be notified by the candidate of the dissertation title. Thirty days before the final thesis defense, the thesis must be available in written form. Eighteen days before the date of graduation, three complete copies of the thesis must be filed with the Graduate School. The student must provide the Department with a .pdf file of the thesis with an attached statement giving the Department the right to distribute the thesis.
At least thirty days before the date of graduation, the candidate must present and defend the dissertation before the student's advisory committee. The defense is open to the public and must be scheduled and advertised through the graduate program assistant at least two weeks in advance. The defense is open to the public and must be scheduled and advertised through the graduate program assistant at least two weeks prior to the exam. All members of the advisory committee must be present for the defense; however, a minority of members, but not the chairperson nor the student, may participate by interactive video. In the event of an emergency that prevents one committee member from attending the exam, the exam can proceed with the faculty who can attend. However, the student will need to meet with the absent committee member at an alternate time. The outcome of the defense can be "pass", "conditional pass", or "fail", as determined by a majority vote of the committee. If the student receives a conditional pass, the examining committee will define requirements that the student must satisfy to pass the defense. These requirements must be completed to the satisfaction of the committee within 60 days. Any extensions to this deadline requires a recommendation from the graduate committee and approval by the graduate school.
The committee may declare the thesis defense successful, but may request further minor changes in the thesis, and specify a deadline and the manner in which the revised thesis will be reviewed. In that case, the student does not need to register for additional thesis hours, but the requirements for the Ph.D. are not satisfied until the final version of the thesis is approved by the advisory committee and the Graduate School. If no member of the committee raises further questions or objections within 30 days after the revised thesis has been received by the advisor, the thesis will be considered approved by the advisory committee.
In the event of failure, by graduate school rules, the student will be dismissed from the program. Any exceptions to this will require approval from the Dean of the Graduate School.The University of Colorado permits each department to decide whether or not to implement a foreign language requirement. The Department of Mathematical & Statistical Sciences has approved the following policy:
Recognizing that the need of a foreign language to do research varies from no languages in many areas to one or two languages in a few areas, the department has no formal language requirement. Instead, the necessity for some level of proficiency in a foreign language is left to the discretion of each student's advisor, as is the case for other matters related to a student's preparation for research.
Students must select an advisor and a Ph.D. advisory committee by the end of the semester in which all preliminary exams have been passed. Students must pass the comprehensive examination by the end of the fourth year in the Ph.D. program. All requirements for the Ph.D. degree must be completed within four years of passing the comprehensive examination and within eight years of entering the Ph.D. program.
It is recognized that flexibility is necessary, especially for transfer and parttime students; hence petitions for exceptions will be considered by the Graduate Committee. The four year deadline for passing the comprehensive exam and the eightyear deadline for completing the Ph.D. is imposed by the Graduate School and exceptions require approval of the Dean of the Graduate School.
A student may request up to a oneyear leave of absence from the Ph.D. program. The student must be in good standing, indicate the return date, give justification for the leave of absence, and agree to contact his/her advisor and the Graduate Committee at least once per semester. Each petition must be approved by the Graduate Committee.
A leave of absence pauses the clock, but does not extend deadlines automatically; extension of deadlines requires a separate petition to the Graduate Committee. A leave of absence does allow the student to interrupt registration for thesis hours following the comprehensive exam. Students who leave a graduate program for more than three consecutive semesters must reapply for admission.
All current M.S. and Ph.D. students have the option of graduating under the current rules or under the rules in effect when they were admitted. See the Graduate Handbook Archive for earlier versions of the rules.
Unless otherwise stated, exceptions to these rules must be approved by the graduate committee. In cases where an exception can be made while still satisfying all requirements of the graduate school, the graduate committee may approve the request with consultation of the students advisory committee. In cases involving exceptions to graduate school rules, the graduate committee, if it approves the exception, will submit a petition to the graduate school, which will then either approve or deny the exception.
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