The Ph.D. program in Mathematics has the following requirements (details follow):

• Area Requirements: A basic full-year graduate course sequence and comprehensive exam must be passed in each of three areas of mathematics.
• Course Requirements: 72 total units of graduate work offered by the Mathematics Department, including a fourth full-year sequence. (Courses are four units each.)
• Preliminary Oral Examination and Advancement to Candidacy: An oral examination administered by the student's Doctoral Committee on topics set by the committee that are close to the area of the student's research. Doctoral students are required to advance to candidacy within four years of beginning their graduate studies at UCSB.
• Dissertation: A dissertation satisfactory to the Doctoral Committee must be written and successfully defended

Area and Course Requirements

The student must satisfy three area requirements. To fulfill an area requirement, typically the student will pass an examination on (primarily) undergraduate material and complete a designated graduate-level sequence with good grades. Detailed descriptions of the area requirements and exams can be found at http://www.math.ucsb.edu/graduate/requirements . Completion of the three area requirements will ensure the student has passed three full-year graduate course sequences in Mathematics. Two pre-approved options are: Algebra, Analysis, and any other, or Analysis, Applied Math, and Geometry/Topology. Other sets of three areas may be proposed by a student, in consultation with his/her advisor, but they require approval by the Graduate Committee.

Students must pass a total of 72 units of 200 level graduate courses offered by the Mathematics Department, with a grade of a least B or S in each course; courses fulfilling area requirements normally require an average grade of at least A- for the sequence. The required 72 units of graduate course work must include a fourth complete full-year graduate sequence. S/U grading is not allowed in the four full-year sequences (48 of the 72 total units). Mathematics 596 may not be used to satisfy any portion of the 72 unit requirement. Students should obtain approval of their faculty advisor before using Mathematics 260 courses to satisfy the 72 unit requirement.

Doctoral Committee

The student should ask a faculty member of the Mathematics Department to serve as his or her Dissertation Advisor. Choosing a Dissertation Advisor is not a triviality, but it is usually a natural process. A faculty member is likely to accept a thesis student who has done well in graduate courses related to that faculty member's field of research. Often the student will take a reading course (596) with the faculty member first in order to ensure mutually acceptable compatibility of interests and expectations. The Dissertation Advisor will help find other members of the student's Doctoral Committee and will supervise the student's dissertation research.

Oral Examination Requirements

After fulfilling the three area requirements, each Ph.D. student shall take and pass the Preliminary Oral Examination given by the student's Doctoral Committee before advancing to candidacy. Students may advance to candidacy before completing the 72 unit requirement, with the condition that they submit a course proposal of how and when the missing units will be completed. The purpose of this examination is to ensure that the student has: (A) gained breadth of knowledge on advanced topics in the areas surrounding his/her proposed dissertation topic and (B) engaged in sufficient preliminary reading and research on the proposed dissertation topic to convince his/her Doctoral Committee that he/she has a reasonably good chance to write an acceptable dissertation. The content of the examination meeting criterion (A) shall be agreed upon in writing by the student and the members of his/her committee at least three months prior to the proposed date of the examination, and will normally be equivalent to the material in about five quarter courses beyond the first year graduate level. Criterion (B) requires a satisfactory presentation by the student of his/her proposed research topic and the results of his/her preliminary reading and research on the topic. The examination will normally be scheduled to last two hours.

Dissertation

After advancing to candidacy, students complete their original research, write their dissertation on a topic agreed upon with their Dissertation Advisor, and take any other appropriate courses.

Students complete the requirements for the Ph.D. with the oral dissertation defense to the Doctoral Committee, and the filing of their dissertation with the Graduate Division Office. Students must make the arrangements for their dissertation defense. In anything other than exceptional circumstances, these arrangements must be in place at least one month prior to the date of the defense. Such arrangements include, but are not limited to: (i) making sure all of the committee members are available on the date in question, (ii) providing copies of the dissertation in a reasonably final form to each committee member, and (iii) informing the Staff Graduate Advisor.

Normal Progress for Ph.D. Students

First Two Years: Most students who have not yet begun a research program should take three courses per quarter, although in some cases two or four courses may constitute a reasonable program. The program should be planned in close consultation with a faculty advisor. Ideally, course selections in the first two years should enable the student to complete the area requirements in three areas, as well as a fourth full-year course sequence. It is common for students entering without any previous graduate work to take one or more upper division undergraduate courses in their first year to fill gaps in their preparation. Students entering with a strong Master's degree will be expected to move through the program more rapidly, and should complete area requirements within one year

Third Year: By the end of this year the student should know the area in which he or she wants to do research and should have secured a Dissertation Advisor. Ideally, research will begin during this year and the student will advance to candidacy during the year or soon thereafter. Doctoral students are required to advance to candidacy within four years of beginning their graduate studies at UCSB.

Fourth Year and Beyond: If all goes well, the student will complete the Ph.D. research, write up the results, successfully defend the dissertation and receive the degree. The departmental normative time to the Ph.D. degree is 6 years, although many students complete the degree in 5 years. These figures are slightly more than average median times to degree over the last 10 years.