Ph.D. Program
Ph.D. Program Requirements
The following requirements are required and must be completed for students to complete their program and obtain their degrees. This page was last updated on March 19th, 2025. For the latest program requirements, please see the Mathematics Ph.D. Program Catalog.
- Complete course requirements, including MATH 288A + 6 graduate courses in mathematics.
- Complete the Preliminary Exam Requirements
- Complete the Oral Qualifying Exam and Advance to Candidacy
- Complete three quarters as a Teaching Assistant
- Complete a Ph.D. Dissertation
These requirements each have specific timelines for completion. Requirement not noted below do not have a timeline requirement.
- Preliminary Exam Requirements should be completed by the end of a student’s 2nd year.
- Oral Qualifying Exams should be completed by the end of a student’s 3rd year.
- Dissertations should be completed by the end of a student’s 6th year.
We are dedicated to supporting students experiencing academic difficulty. You may read more about academic difficulty on the Graduate Division website regarding academic progress. When these requirements are not met in the defined timeline, student will be encourage to develop an academic support plan in collaboration with the the graduate vice chair, the graduate advisor, and their faculty advisor if applicable.
Course Requirements
Students are required to complete MATH 288A – Pedagogy of Mathematics plus 6 additional graduate courses in Mathematics. Graduate courses that qualify for this requirement can be found on the Mathematics Graduate Course Catalog. Current course offering can be found on the Schedule of Classes.
No more than three quarters of Independent Study (MATH 297) may count towards the additional course requirement, and must be approved by the graduate vice chair. To submit a course for approval towards this requirement, please complete the Independent Study Course Requirement Application Form.
Preliminary Exams
Preliminary exams are comprehensive exams in the subject areas of Algebra, Analysis, and Geometry / Topology. Students are required to 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. The core sequence of any topic that did not receive a first-level pass must also be completed.
UCSC affiliates can visit the preliminary exam archive for to access previous exams as study material. UC Santa Cruz’s chapter of Association of Women in Mathematics (AWM) facilitates preliminary exam study sessions in the Spring and Summer quarters. For a list of preliminary exam topics, please see the preliminary exam overview page.
The preliminary exam requirements should be completed by the end of a Ph.D. students 2nd year.
Oral Qualifying Exam + Advancement to Candidacy
A key milestone in the Ph.D. program is Advancement to Candidacy, which relies on the passing of the oral examination. This examination is an oral exam, usually accompanied by a visual presentation that demonstrates that they have a sufficient understanding of their Ph.D. Dissertation Problem. Students will work with their faculty advisor on establishing their Dissertation Problem and can provide guidance and expectations for the exam.
Before the exam can be taken, an oral qualifying exam committee must be established and approved. The committee must be submitted for approval by the graduate division at least 30 days before the examination occurs. The committee should consist of four members, where at least one member is outside of the Mathematics Department. Outside committee members may be recommended by your faculty advisor or others in the mathematics department.
Once the exam committee is approved, the exam can be taken. Once the exam is complete, the Report on Qualifying Examination Form must be completed by the committee members and submitted to the graduate advisor immediately after the examination. A copy of your report will be shared with you.
The oral qualifying examination may only be failed and retaken once. If the exam is re-taken, the membership of the exam committee should remain the same. If the oral qualifying examination is passed, you will qualify for advancement to candidacy upon the completion of the following conditions and materials:
- Completed Preliminary Exam Requirements
- Report on Qualifying Examination
- Dissertation Reading Committee Form
- Have no Incomplete Grades
Advancement to Candidacy should occur by the end of a Ph.D. students 3rd year.
Teaching Requirements
Students are required to complete three quarters as a Teaching Assistant. Students that do not have alternative forms of financial support will be automatically considered for Teaching Assistant positions. MATH 288A is required to be taken before or during a students’ first quarter as a teaching assistant. For more information regarding Teaching Assistantships, Academic Student Employment, and related resources, please see the the Academic Student Employment page.
Dissertation and Defense
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.
The Dissertation Reading Committee should consist 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 committee is submitted and approved at the time that a student applies to advance to candidacy, but can be revised at any time prior to submission by submitted a dissertation reading committee revision form.
All members of the student’s dissertation committee must read and approve the dissertation. To apply for graduation upon the completion and approval of the Dissertation, the Application for the Doctor of Philosophy Degree form must be submitted to the graduate advisor prior to the deadline noted in the current Academic calendar for the quarter the student intends to graduate.
Graduate Program Contacts
Longzhi Lin, Graduate Vice Chair
lzlin@ucsc.edu | 831-459-4180
Lyss Melton, Graduate Advisor and Program Coordinator
mathgradadvising@ucsc.edu | 831-459-5461