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Department of Mathematics – University of ÃÛÌÇÖ±²¥ Boulder

You can find a list of faculty research interests from here.

1. Diagnostic exam
   • The diagnostic exam is offered in the summer before a student starts and consists of two parts: algebra and analysis, both of which are based on the material in a typical first-semester undergraduate course. The goal is to establish how prepared students from widely varying backgrounds are for particular graduate courses.
   • On the basis of the exam results, the Graduate Committee will recommend that the student either register for the graduate courses or that the student take a lower-level course to make up for deficiencies. Typical recommendations are Math 5140 or Math 5150 (for algebra) and Math 5001 or AppM 5440 (for analysis). Although the exam is mandatory, the recommendations are only advisory, and the student may choose not to follow them.
   • Undergraduate courses recommended by the Graduate Committee on the basis of diagnostic exam results will count toward the student's graduate credit requirements; however no other undergraduate-level courses will count without explicit permission from the Graduate Committee.
2. Credit hours each semester
   • In each of years one and two, all funded students must take 15 credits of graduate study. These will typically be math department courses or in fields related to math unless approved by the Graduate Committee. Math 5905 does not count toward this requirement.
   • In year three and beyond, all funded students must take 12 credits of graduate study (6 credits in each semester), up to and including the semester in which they pass their comprehensive exams.
   • After the comprehensive exam is passed (for doctoral students), all students must take between five and ten hours of dissertation thesis credit for each semester up to and including the semester in which they defend the thesis. (Students who are off-campus and not using campus facilities may instead take a minimum of three hours of dissertation credit if they apply for off-campus status.) The Graduate Assistant handles dissertation hour registration.
   • All TA funded students (except those in their last year) must register for at least one mathematics department course in every semester; for students who have passed the comprehensive exam, this course may be taken pass/fail. Courses taken for no credit will not be covered by the tuition waiver; hence funded students should only take courses for credit.  Grant funded students (Research Assistants) must consult with their Advisor on an appropriate plan of study for the term. This plan may include registering only in dissertation credits, for students who have advanced to candidacy.
   • Students in their last year are exempt from the course-for-credit registration requirement for two semesters, provided their advisor submits to the Graduate Chair a letter stating that the student is likely to graduate in that year. Students may not take this exemption for more than two semesters even if they do not graduate on time.
   • Students may not register for more than 15 credit hours in any single semester.
   • Undergraduate courses, or graduate courses in fields not directly related to mathematics, may be taken with the approval of the Graduate Committee (via petition). They generally will not count toward the 15-credit minimum requirement; the only exception is undergraduate courses taken to make up for deficiencies measured by the diagnostic exams.
   • Students must be registered full-time for at least six semesters beyond a bachelor's degree, or four semesters beyond a master's degree. Students must be registered continuously unless they obtain permission otherwise (such as the Time Out program).
3. Required courses
   • All students must pass Algebra 1 (Math 6130), Algebra 2 (Math 6140), Analysis 1 (Math 6310), Analysis 2 (Math 6320), Topology 1 (Math 6210), Differential Geometry 1 (Math 6230), and Complex Analysis 1 (Math 6350) with a grade of B- or better in each. The only exceptions are for students who unconditionally pass the corresponding preliminary exam (Algebra, Analysis, or Topology/Geometry); students who pass conditionally must complete the full pillar sequence. Students who have taken similar courses elsewhere may petition the Graduate Committee to obtain an exemption.
   • All students must complete at least 30 hours of graduate (5000 level or above) mathematics department courses, including those listed above, with a grade of at least B- in each course. At most six hours of seminars and independent study may be used toward this requirement.
   • All students must take a total of 30 hours of dissertation credit. Students must take a minimum of five hours and a maximum of ten hours in each semester after the comprehensive exam is passed (unless the student is claiming part-time status). At most 10 of these in all semesters before the comprehensive exam is passed can count toward the requirement.
   • First- and second-year students should consult with their advisor/mentor each semester and get a course-registration form signed and submitted to the Graduate Assistant before registration holds are lifted.
4. Course timeline (for funded students)
   • The three "pillar sequences" are Algebra 1 and 2, Analysis 1 and 2, and Topology 1/Geometry 1. At least one pillar sequence must be completed by the end of the first year. We recommend completing at least two by the end of the first year.
   • All three pillar sequences must be completed by the end of the second year.
   • Complex Analysis (Math 6350) must be passed by the end of the semester in which the student passes the Comprehensive Exam, which must happen by the end of the seventh semester.
5. Transfer credits
   • Ph.D. students may transfer up to 21 hours of graduate course credit from another institution, including work done towards a Master's degree. Master's students may transfer up to 9 hours. However students may not transfer credits already used to obtain a degree at the same level (for example a student who obtained a Master's degree elsewhere cannot transfer those courses to obtain a second Master's degree from ÃÛÌÇÖ±²¥ Boulder).
   • Students may request transfer credit after completing 6 credits at ÃÛÌÇÖ±²¥ Boulder with a grade of 3.0 or better. Students must have earned at least a B- in transferred courses, and the transferred courses must be comparable to graduate courses offered at ÃÛÌÇÖ±²¥ Boulder. Students should provide copies of the syllabus and other relevant material to assist the Graduate Committee in making this determination.
6. GPA requirements
   • The Graduate School requires a minimum cumulative GPA of 3.0 in each semester, regardless of funding.
   • The Mathematics Department requires a minimum cumulative GPA of 3.3 in the first semester and 3.5 cumulative GPA in all subsequent semesters in order to maintain funding.
   • All courses must be passed with a grade of B- or better to count toward the Ph.D. degree.
   • Students who receive a grade below B- in a course may repeat the course once, provided the course has not previously been applied toward a degree. Written recommendation of the Graduate Chair and approval by the Graduate Dean is required. Only the latter grade is used in GPA requirement calculations, although all grades received will appear on the transcript.
   • Any courses used toward the 30 credit course requirement may not be taken pass/fail.

• Funded Master's degree students must take at least 15 credits of graduate study in the first and second years, and at least 6 credits of graduate study in the fifth and final semester.
• Master's students must register for at least two semesters.
• All Master's degree students must take 30 hours of graduate-level (5000-level or above) courses. Six of these credits must come from a two-semester sequences in mathematics (e.g., Algebra 1 and Algebra 2); in case of ambiguity the Graduate Committee will decide via petition whether two mathematics graduate courses will be considered a sequence. Courses outside the mathematics department must be approved by the Mathematics Graduate Committee.
• Master's students who choose to write and defend a thesis may count between 4 and 6 credits of dissertation hours toward the 30 hour requirement.
• Up to 6 hours of seminar and independent study credit may be applied toward the 30 hour requirement.
• M.S. (Master of Science) students have the following additional requirements. At least 18 of the 30 required hours must consist of mathematics department graduate courses. Math 5600 and Math 5610 (Numerical Analysis 1 and 2) are required. One additional two-semester sequence in mathematics is required. M.S. students can count a minimum of 6 and a maximum of 12 hours of credits from an area of application of mathematics (such as applied mathematics, physics, biology, economics, etc., with the student's advisor approving the area and the courses). At least two of these application courses must form a one-year sequence in the other department.
• Students may not register for more than 15 credit hours in any single semester.
• Courses taken for no credit will not be covered by the tuition waiver. Hence funded students should only take courses for credit.
• Students who choose to write and defend a thesis must take between 4 and 6 hours of dissertation credit, which will replace the credit hours in the 30-hour requirement.
• Students must obtain at least a C (2.0) in every course applied to the Master's program requirements, and must maintain a minimum 3.0 cumulative in all semesters to remain in the program. Funded Master's students must maintain a 3.3 GPA in the first semester and a 3.5 cumulative GPA in all subsequent semesters.
• All course requirements must be completed in five semesters. Students who switch from the Ph.D. program to the Master's program must petition the Graduate Committee to adjust this timeline.

• There are three preliminary exams: Algebra, Analysis, and Geometry/Topology. The syllabi for the preliminary exams in each subject are based on the syllabi for the corresponding courses; see preliminary exam materials. Students taking the exams are responsible for all material on the exam syllabi regardless of what was covered in the courses, although generally all such material is expected to be covered in the corresponding courses.
• Preliminary exams are offered twice every year, in January and August. The exams are written and graded by a committee of at least two graduate faculty members; both the exams and the grades are subject to the approval of the Graduate Committee.
• All Ph.D. students must pass two out of the three preliminary exams. Funded students must pass at least one exam by the end of the third semester, and at least two exams by the end of the fifth semester.
• There are three grades on a preliminary exam: fail, conditional pass (usually a score of at least 55%), and unconditional pass (usually a score of at least 70%). The Graduate Committee ultimately decides what the passing scores are for each exam.
• Students may take any preliminary exam any number of times, subject to the timeline requirement for funded students above. Students may take an exam again after passing conditionally in order to try passing unconditionally; only the highest grade will be counted.
• Ph.D. students may take a preliminary exam in a subject without having taken all of the requisite courses. If the student passes the exam unconditionally, the student need not take the remaining courses in the corresponding pillar sequence.
• If a student passes an exam conditionally, the student is typically required to take a non-required course at the 6000-level or above in the corresponding subject, and pass with a grade of A- or above. If the student completes the condition within one year of taking the exam, the student is considered to have satisfied the timeline requirement for passage of the preliminary exam (for example a student who conditionally passes a second exam at the end of the fifth semester and gets an A- in a conditional course at the end of the seventh semester has satisfied the preliminary exam timeline requirement). Students who pass conditionally must complete the full corresponding pillar sequence in addition to the conditional course.
• Students taking a preliminary exam may not bring into the exam room any of the following: notebooks, paper, electronic devices, reference materials, backpacks, or bags. No questions are allowed, except for inquiries concerning the correctness of a problem.

• Funded Ph.D. students must pass the comprehensive exam by the end of the seventh semester.
• Ph.D. students should choose an advisor after passing two preliminary exams; the advisor must be a member of the graduate faculty. This advisor is typically but not necessarily the student's anticipated dissertation advisor. The advisor and the student should write a syllabus for the comprehensive exam, which will typically consist of an amount of work equivalent to three graduate courses. (For example, three courses studied independently, or two courses and a research paper presentation.) The actual syllabus must be approved by the Graduate Committee. The syllabus of the exam must not overlap with any of the pillar sequence courses.
• The Ph.D. comprehensive exam committee consists of five faculty members, one of whom is the student's advisor. The exam is generally given orally: the student may give a presentation, and will then be asked questions based on the exam syllabus and the presentation. A majority of the committee must vote affirmatively for the student to pass.
• If a student fails the comprehensive examination on the first attempt, the student may attempt the exam once more after a period of time determined by the exam committee.
• Ph.D. students who do not already have a Master's may use the comprehensive exam to earn a Master's degree "en passant." Students who are qualified to advance to candidacy will also qualify for a Master's degree, and may obtain it by filling out the appropriate forms obtained from the Graduate Assistant.
• Students must file for advancement to candidacy at least two weeks before the comprehensive exam is attempted.

• Ph.D. students must take between five and ten dissertation hours each semester after passing the comprehensive exam, up to and including the semester in which the defense occurs. The only exception is for students who are not using any campus resources, who may apply for part-time status and take three dissertation hours each semester (but the student must still take five dissertation hours in the semester when the thesis is defended, even if in the summer semester). Grades for dissertation hours are assigned as "in progress" until the defense is completed.
• The dissertation committee consists of five members. The chair of the committee is the student's advisor. One of the members must be from outside the ÃÛÌÇÖ±²¥ Boulder mathematics department (either in another department at ÃÛÌÇÖ±²¥ Boulder or at another university). The chair and outside member must have regular or tenured graduate faculty appointments; the other members must have regular or special graduate faculty appointments. (Regular appointments usually include tenured or tenure-track faculty, while special appointments are granted for faculty qualified to teach graduate courses who are not on the tenure track.)
• The dissertation defense must consist of an oral defense which is open to the public. The dissertation must represent the equivalent of at least 30 semester hours of work. The student is responsible for notifying the Graduate School of the title on or before the posted deadline in the semester of graduation. It must comply with the specifications for theses and dissertations available in the Graduate School.
• To pass the dissertation defense, the student must have approval of all committee members (up to at most one dissenter). If the dissertation defense is not passed on the first attempt, the student may attempt the defense once more after a period of time determined by the dissertation committee. If it is not passed on the second attempt, the student must leave the program.
• For funded students, the Ph.D. dissertation defense must take place at most six years from the student's entrance, unless the student petitions the Graduate School for a seventh year and the student's advisor certifies that the student is likely to finish in the seventh year. No Ph.D. student will be funded by the Mathematics Department for more than seven years.
• The Ph.D. thesis must consist of original research; a purely expository thesis is not acceptable, and a historical thesis is not acceptable unless it consists of original research in the history of mathematics.

• The thesis is optional. If a student chooses to write a thesis, the student must complete between four and six hours of thesis credit (which may be used to replace course work in the 30 credit requirement). The thesis is usually of an expository nature and may be in any field of mathematics.
• Whether or not the student chooses to write a thesis, the student must give a presentation on the thesis or on some mathematical topic, approved by the Graduate Committee, to a committee which includes the advisor and two other graduate faculty members. The student will then answer questions on the thesis or the topic presented.
• The examination should be completed after the other requirements for the degree have been substantially completed; it is typically near the end of the student's last semester. It may be taken while required courses are still being taken, provided satisfactory progress is being made.
• The committee is composed of three members of the graduate faculty. The chair must have a regular or tenured graduate faculty appointment.
• The student must have an affirmative vote from a majority of the committee members to pass. A student who fails the Master's defense may attempt it once more after a period of time determined by the examining committee.

• Exceptions in timing will be made (via petition to the Graduate Committee) for students who fall behind because of extenuating circumstances beyond their control that prevent them from completing the requirements within the normal time frame.
• In general, a student who fails to satisfy one of the requirements of funded students (such as taking the minimum number of courses, maintaining a 3.50 GPA, passing preliminary exams on time, passing the comprehensive exam on time, or defending the dissertation on time) may request one extra semester of funding as a "parachute" (however, the department cannot promise this funding). During the parachute semester, the student is expected to do one of the following:
  1. get back on track to making satisfactory progress toward the Ph.D.;
  2. seek alternative funding for the Ph.D. program;
  3. apply to other graduate programs;
  4. consider other career options;
  5. get a Master's degree.
• For students who fail to take enough courses: if a student drops below the minimum number of courses mid-semester, the student's funding will be terminated at the end of the semester the drop occurs, and the department will not consider the student again for funding until the deficit is made up. The student will then have to compete with new students for such funding.
• For students whose GPA does not meet department standards (but do not fall under the previous case): students are typically given one additional semester to bring their GPA back up to the department standard. Funding will be cut off immediately unless the student submits to the Graduate Committee a plan to bring the student's GPA back up to the department standards in a semester (or within one year if the student is making use of course forgiveness for a single course). If approved by the Graduate Committee, the student will be given the required number of semesters to complete the plan before funding is cut off. Once cut off, a student will not be considered for additional funding until the student's GPA meets the standard, and the student will have to compete with new students for such funding.
• For students who do not pass the first preliminary exam on time (but do not fall under previous cases): if a student does not pass at least one preliminary exam by the end of the third semester, the fourth semester will be the last semester of the student's funding. The student may be advised to pursue a Master's degree at this point. The student can get back on track by passing two preliminary exams at the end of the fourth or fifth semester. Even if the student does so, the student will not be funded for the immediately subsequent semester, and would have to compete with new students for funding in the semester after that. The student's advisor must also certify that the student is likely to pass the comprehensive exam by the end of the seventh semester.
• For students who do not pass the second preliminary exam on time (but do not fall under previous cases): if a student does not pass both preliminary exams by the end of the fifth semester, the sixth semester will be the last semester of funding. The student may be advised to pursue a Master's degree at this point. The student can get back on track by passing the second preliminary exam at the end of the sixth semester. Even if the student does so, the student will not be funded for the seventh semester, and must compete with new students for funding for the eighth semester. The student's advisor must also certify that the student is likely to pass the comprehensive exam by the end of the seventh semester.
• For students who do not pass the comprehensive exam on time (but do not fall under previous cases): if a student does not pass the comprehensive exam by the end of the seventh semester (but has not yet earned a Master's degree), if the resources are available, the department will fund the student with a teaching assistantship for at most one additional semester if the student chooses to enter the Master's program, commits to taking the full load of required courses, and prepares a presentation or thesis as required. Alternatively, if there is availability, and depending on the record of the student, the Graduate Committee may recommend to the Chair to seek alternative funding for the student for at most one semester in the guise of recommending the student be hired by the School of Continuing Education. (Note that Continuing Education pays less for teaching a single course than a standard teaching assistantship, and that Continuing Education does not fund a student's tuition waiver.) Such students can get back on track by completing the comprehensive exam at the student's own expense, and will then compete with new students for additional funding.
• For students who do not finish the doctorate within seven years (but do not fall under previous cases): if a student has not yet met the requirements for a Master's degree and has not earned one from another institution, if the resources are available, the department will fund the student with a teaching assistantship for at most one additional semester if the student chooses to enter the Master's program, commits to taking the full load of required courses, and prepares a presentation or thesis as required. Alternatively, if there is availability, and depending on the record of the student, the Graduate Committee may recommend to the Chair to seek alternative funding for the student for at most one semester in the guise of recommending the student be hired by the School of Continuing Education. (Note that Continuing Education pays less for teaching a single course than a standard teaching assistantship, and that Continuing Education does not fund a student's tuition waiver.) Such students must complete the doctorate at the student's own expense and will not be funded again by the department.
• Options for funding when a student is studying at the student's own expense include taking out student loans, seeking other employment such as teaching at neighboring institutions, or teaching for other departments at the University of ÃÛÌÇÖ±²¥, but does not include going on "time out" and lecturing for the mathematics department except as specified above.

• Students with full-time teaching assistantships are granted a tuition waiver which covers up to 15 credits each semester, in addition to health insurance.
• Students funded as teaching assistants are expected to work 20 hours per week, a combination of class time, grading, office hours, help lab hours, preparation, and coursewide meetings. The specific combination of hours depends on the position and the course but will be determined by the course coordinator.
• Students must be available on campus the week before classes start in order to attend course coordination meetings and prepare for classes.
• All teaching assistants are responsible for having their classes, office hours, and Undergraduate Resource Center hours covered by others in case of illness or other unavailability. Students who do not show up for such hours without finding a substitute are subject to immediate termination of their funding.
• Teaching assistantships are conditional upon doing an acceptable job teaching. Failure to show up for classes or recitations is grounds for dismissal.
• Domestic graduate students must obtain ÃÛÌÇÖ±²¥ residency by their second year of employment in order to continue to qualify for a tuition waiver.
• All teaching assistantship awards are conditional on satisfying the Mathematics Department's criteria for normal progress (including taking at least 15 credits of graduate work, maintaining a 3.50 GPA, passing preliminary and comprehensive exams on time, etc., as described above).

• Graduate students who have passed their comprehensive exams are eligible for summer research fellowships including the Thron, Stribic, and University fellowships. Funding for such fellowships is dependent upon availability and is chosen based on performance, standing, and advisor's recommendation. Students awarded a summer research fellowship are expected to pursue dissertation research exclusively and are not permitted any other form of employment.
• The department also offers summer teaching positions, generally consisting of one summer course shared with another graduate student (each student teaching the first or second half of the course). Funding for such positions is dependent upon availability.
• The department offers funding for Ph.D. students who have passed the comprehensive exam to travel to conferences, generally $300 for domestic travel and $500 for international travel. Students requesting such funding should ask their advisor to send a brief note of support to the Graduate Chair.
• Students may wish to tutor undergraduates. Any such tutoring arrangements are private, between the tutor and the student tutored. Graduate students may never tutor students in any section of a course in which they are assigned as teaching assistants or lecturers, due to potential conflict of interest (even if the graduate student does not have direct authority over the tutored student's grades).

• Year 1: Complete all pillar courses (three each semester) and possibly electives. Take Math 5905 (teacher training) for two semesters. Maintain a 3.30 GPA in the first semester and a cumulative 3.50 GPA in all subsequent semesters. Meet with mentor weekly. Attend all Slow Pitch talks, Kempner Colloquia as desired, and at least one seminar talk. Attempt at least one preliminary exam mid-year; attempt all preliminary exams at the end of the first year.
• Year 2: Take Complex Analysis, and continue taking at least 15 credits of graduate mathematics classes. Take a preparatory course beyond the pillars in case of not passing a preliminary exam. Maintain 3.50 GPA. Attend all Slow Pitch and all Kempner Colloquia. Join a seminar group and attend regularly. At least one preliminary exam must be passed by mid-year. Upon passing two preliminary exams, begin preparing for comprehensive exam and take two courses per semester.
• Year 3: Find an advisor, pass the comprehensive exam and begin research. Register for 5-10 dissertation credits each semester. Take one course (for credit, pass/fail if desired) each semester thereafter. Give a talk in the Slow Pitch.
• Year 4: Take one course each semester (for credit, pass/fail). Register for 5-10 dissertation hours each semester. Give a talk in a seminar. Attend the Slow Pitch, the Kempner Colloquium, and a seminar. Attend a conference. Begin work on a publishable paper. Apply for graduate student travel grants.
• Year 5: Take one course each semester. Register for 5-10 dissertation hours each semester. Attend the Slow Pitch, the Kempner Colloquium, and a seminar. Present research at a conference. Apply for travel grants. Submit a paper for publication. Help to organize a seminar.
• Year 6: Prepare to graduate. Student is exempt from course requirement if advisor certifies that the student will graduate this year. Register for 5-10 dissertation hours each semester. Present research at a conference. Apply for grants and jobs. Defend dissertation and finish.

• Year 1: Take two pillar sequences if diagnostic exams passed; otherwise take one pillar sequence. Take at least five graduate courses this year, in addition to two semesters of Math 5905. Meet with advisor once or twice each semester. Maintain a 3.30 GPA in the first semester and a 3.50 GPA each semester thereafter. Attend all Slow Pitch talks, and at least one seminar talk. Take preliminary exams in August based on pillar courses passed.
• Year 2: Must take at least five graduate courses this year. Must pass one preliminary exam by mid-year. If passed conditionally, take required course within one year. Complete pillars (all must be passed by end of second year). Take electives. Attend all Slow Pitch talks, and attend Kempner Colloquium and seminar talks as desired.
• Year 3: Must take two graduate courses each semester until comprehensive exam passed. Must pass second preliminary exam by mid-year. If passed conditionally, take required course within one year. Take Complex Analysis. Find advisor for comprehensive exam and write syllabus. Attend all Slow Pitch talks and Kempner Colloquium talks and advisor's seminar.
• Year 4: Must take two graduate courses each semester until comprehensive exam passed. Must pass comprehensive exam by mid-year. Begin dissertation research; register for 5-10 dissertation hours each semester. After passing comprehensive exam, take one course each semester for credit (may be taken pass/fail). Attend all Slow Pitch talks, Kempner Colloquium talks, and a research seminar.
• Year 5-6: Take one course per semester for credit (may be pass/fail). Do dissertation research; register for 5-10 dissertation hours each semester. Begin work on papers, grants, conferences, and/or professional training for employment prospects. Attend seminars. If graduating this year, students are exempt from course requirement.
• Year 7: A seventh year requires special approval from the Graduate Committee and the Graduate School. Students graduating this year are exempt from the course requirement. Apply for jobs and continue building CV with papers, grants, conferences, and/or professional training. Defend dissertation and finish: no student may remain beyond seven years.