Advancement to Candidacy
Course Requirements
A three-course sequence in each of the three fields of algebra, analysis, and geometry-topology (manifolds) will be offered each year. Preliminary examinations will be given for each core sequence at the beginning and end of each academic year.
First-level passage of a preliminary examination satisfies the core sequence requirement for that field. Ph.D. students are required to complete the full core sequence in the field associated with the preliminary examination in which they do not achieve a first-level pass. The core sequences are as follows:
The following course:
Students are also required to complete six additional courses in mathematics.
No more than three courses may be independent study or thesis research courses. Sample courses include:
Foreign Language Requirements
Beginning with the cohort entering fall 2022, there is no longer a foreign language requirement for the Ph.D. in Mathematics. Although there is no longer a foreign language requirement, some students may need to read papers in unfamiliar languages to carry out their research. When this is the case, students are strongly encouraged to reach out to faculty for guidance.
Teaching Requirement
Ph.D. students must complete a minimum of three quarters as a teaching assistant (TA). All TAs are required to participate in the department's teaching assistant training program.
TA appointments are usually made at 50 percent time (an assigned workload of approximately 220 hours per quarter). TAs are under the supervision of the faculty member responsible for the course. TAs are covered by a collective bargaining agreement between the University of California and the United Auto Workers (UAW).
Instructors and their TA(s) will meet at the beginning of the quarter to complete the Notification of TA Duties form in order to identify the agreed upon tasks. The performance of these tasks will form the basis of the end-of-quarter performance evaluation and will use the following criteria: quality of work; accuracy and attention to detail; interaction with students, peers, and instructor; knowledge of subject; and dependability. The specific allocation of TA duties is subject to change, depending on enrollments and the number of teaching assistantships in the department allocation. The general duties vary, depending on the course assigned and level of the course.
Pre-Qualifying Requirements
Preliminary Examinations
Preliminary examinations are given for each core sequence in the fields of algebra, analysis, and geometry-topology at the beginning and end of each academic year. The exams will be designed and graded by recent instructors of each core sequence.
A first-level pass signifies that the student has the knowledge to start research with a thesis advisor in this particular area. A second-level pass signifies that the student has an understanding of the basic concepts, but not necessarily enough to conduct research.
Ph.D. students must obtain a first-level pass on at least one of the three written preliminary examinations and a second-level pass on at least one other. Students must complete the full three-course sequence in the field associated with the preliminary examination in which they did not achieve a first-level pass. Students may take the preliminary examinations as often as they wish.
Ph.D. students must complete the preliminary examinations and core sequence requirements by the end of spring quarter of their second year to maintain satisfactory progress. A student who may be experiencing academic difficulty will first develop an academic support plan with the department before being recommended for academic notice or disqualified from the program in alignment with the following timeline:
- If the preliminary exam requirements are not met after the spring preliminary exam offering of a student's second year, an academic support plan will be developed with the graduate vice chair and the preliminary exams in fall of the third year must be taken.
- If the requirements are not met after the fall preliminary exam offering of a student's third year, the student will be recommended for academic notice, and their academic support plan reassessed to follow through the spring of the third year.
- If the requirements are not met after the spring preliminary exam offering of the student's third year, the student may be disqualified from the program, and may transfer to a terminal master's program or appeal to the department to continue on the Ph.D. route.
Topics for the preliminary examinations include:
- Algebra
- Linear algebra
- Group theory
- Ring and module theory
- Field theory
- Galois theory
- Analysis
- Basic analysis
- General topology
- Metric spaces
- Measure and integration
- Complex analysis
- Functional analysis
- Geometry-topology (manifolds)
- Manifold and tangent bundle
- Differential forms and integration on manifolds
- Fundamental group and covering space
- (Co)homology
- Differential geometry
Qualifying Examination
Oral Qualifying Examination
By the beginning of the third year, a student should have a faculty advisor whose research aligns with their research interests. Students will work with their faculty advisor to develop a research plan which is proposed during their qualifying exam.
All graduate students in the Ph.D. program are required to take an oral examination, called the oral qualifying examination, for advancement to candidacy for the Ph.D. degree. Students will demonstrate that they have a sufficient understanding of their Ph.D. thesis problem by presenting their research plan.
Qualifying Examination Committee Composition
The examining committee consists of the student’s faculty advisor, at least two other faculty members from the Mathematics Department, and at least one outside tenured faculty member from either another discipline at UCSC or another academic institution (involved in research and graduate education of the same or different discipline). The student, in consultation with the student’s faculty advisor, selects the committee. The chair of the committee must be someone other than the student’s faculty advisor.
The Graduate Division must approve the committee. The Committee Nomination of Ph.D. Qualifying Examination Form must be completed and submitted at least one month prior to the requested exam date. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate advisor and program coordinator for review and submission to the Graduate Division.
The committee decides on the topics for the examination, which should be broad enough to encompass a substantial body of knowledge in the area of the student’s interest. The written list of topics to be included in the examination, along with a short bibliography, is prepared by the student. A copy is given to each committee member and a copy is put into the student’s permanent records.
Following the Qualifying Examination
The Report on Qualifying Examination Form must be filled out by the qualifying examination committee immediately following the examination. The form can be found on the Graduate Division website, or can be provided by the Mathematics Department. The form should be turned in to the graduate advisor and program coordinator for review and submission to the Graduate Division. The student may request to see a copy of the report.
The oral exam must be completed by the spring quarter of a student's third year to maintain satisfactory progress. An oral qualifying exam may be taken at most twice. When retaking the exam, the membership of the examining committee typically remains the same.
A student who may be experiencing academic difficulty will first develop an academic support plan with the department before being recommended for academic notice or disqualified from the program in alignment with the following timeline:
- If the oral exam is not taken by the end of spring quarter of the third year, or upon an exam that is taken and failed, a student will develop an academic support plan with the graduate vice chair and their faculty advisor to follow through the fall of the fourth year.
- If the oral exam is not completed before the end of fall of the fourth year, the student will be recommended for academic notice, and their academic support plan reassessed to follow through the spring of the fourth year.
- If the oral exam is not completed by the end of spring of the fourth year, or upon a second exam that is taken and failed, a student may be disqualified from the program and may transfer to a terminal master’s program or appeal to the department to continue on the Ph.D. route.
Post-Qualifying Requirements
Dissertation Reading Committee Composition
A Ph.D. student, in consultation with their faculty advisor, is responsible for selecting a dissertation reading committee. The committee consists of the student’s advisor and at least two other members of the mathematics faculty. In special circumstances, a committee member may be chosen from another department and/or from another institution. The student’s advisor is the chair of the committee.
The Graduate Division must approve the committee. The Nominations for Dissertation Reading Committee Form must be completed and submitted prior to advancement to candidacy. The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate advisor and program coordinator for review and submission to the Graduate Division.
A new form must be submitted for approval if changes to the dissertation reading committee must be made.
Advancement to Candidacy
Students must complete the following in order to advance to candidacy:
- Complete the preliminary examinations and core sequences in accordance with the requirements outlined above;
- Pass the qualifying examination;
- Have a dissertation reading committee approved by the Mathematics Department and the Graduate Division;
- Have no incomplete grades (I) on their record.
An advancement to candidacy fee will be billed to the student’s account. The student will be officially advanced the following term after all of these requirements are met.
Dissertation
Dissertation
Each graduate student in the Ph.D. program is required to write a Ph.D. dissertation or thesis on a research topic in mathematics. The Ph.D. dissertation should contain original research results that are publishable in a peer-reviewed journal. All members of the student’s dissertation committee must read and approve the dissertation.
More information about dissertation submission can be found at the Graduate Division website.
Dissertation Defense
After the dissertation has been approved, the student has an option of making a public oral presentation of the mathematical results contained in the dissertation—the “thesis defense.” A recommendation by the dissertation committee will be made to the Mathematics Department and to the Graduate Council on the granting of the Ph.D. degree.
Academic Progress
Ph.D. students are expected to adhere to the below degree timetable:
- Preliminary examinations and course sequence requirements
Completed by the end of the student’s second year
- Oral qualifying examination (and advancement to candidacy)
Completed no later than student’s fourth year
- Dissertation defense
Completed no later than the end of the sixth year
Annual meetings with the graduate vice chair and the graduate advisor and program coordinator are conducted with each student on a one-on-one basis. These meetings serve to notify the student of their current progress within the program and outline expectations for the continuation of normative progress toward the Ph.D. degree.
If the dissertation is not completed by the end of the sixth year, the student can either transfer to a terminal master’s program or appeal to the department to extend their Ph.D. program.
Applying for Graduation
Ph.D. students must complete the Application for the Doctor of Philosophy degree form by the appropriate quarter’s deadline listed in the current Academic and Administrative Calendar.
The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the graduate advisor and program coordinator for review and submission to the Graduate Division.