Doctor of Philosophy in Mathematics

- Completion of all required coursework
- Completion of all written examinations, Comprehensive and Qualifying
- Completion of Advancement to Candidacy Oral Examination
- Completion of Teaching Experience
- Submission of Doctoral Dissertation

When accepted into the doctoral program, the student embarks on a program of formal courses, seminars, and individual study courses to prepare for the Ph.D. Qualifying Examinations, Advancement to Candidacy Examination, and Dissertation.

All students will take Advisory Examinations in Algebra and Analysis upon entrance to the graduate program. The Advisory Examination in Algebra is based on the courses Mathematics 120A and 121A-B, plus some advanced topics in group theory and linear algebra; students who do not pass this examination will be asked to take the Mathematics 206A-B-C sequence. The Advisory Examination in Analysis is based on material covered in Mathematics 140A-B-C-D; students who do not pass this examination will be asked to take the Mathematics 205A-B-C sequence. All students who take Mathematics 205A-B-C (Analysis) and 206A-B-C (Algebra) must pass the corresponding Comprehensive Examination, which covers the material of the Advisory Examination plus Mathematics 205 or 206, respectively. The Comprehensive Examination will be given in spring. If the exam(s) are not passed in spring, the Comprehensive exam will be taken in fall and must be passed by the beginning of the second year in the graduate program.

Each student must choose at least two series of the following three series of courses-Mathematics 210A-B-C (Real Analysis), 220A-B-C (Complex Analysis), or 230A-B-C (Algebra)-and pass two written Qualifying Examinations from these courses before the end of their third year. Each examination may be taken twice. A student who passes the examination prior to taking the corresponding course will be exempted from taking the course. The Department will offer the Qualifying Examinations twice each year, during orientation week before the fall quarter and at the end of spring quarter.

By the end of their second year, students must declare a major specialization from the following areas: Algebra, Analysis, Applied and Computational Mathematics, Geometry and Topology, Logic, or Probability. Students are required to take two series of courses from their chosen area. (Students who later decide to change their area must also take two series of courses from the new area.) Additionally, all students must take two series of courses outside their declared major area of specialization. Special topics courses within certain areas of specialization and courses counted toward the M.S. degree, other than Mathematics 205A-B-C and 206A-B-C, will count toward the fulfillment of the major specialization requirement.

By the beginning of their third year, students must have an advisor specializing in their major area. With the advisor's aid, the student forms a committee for the Advancement to Candidacy oral examination. This committee will be approved by the Department on behalf of the Dean of Graduate Studies and the Graduate Council and will have five faculty members. At least one, and at most two, of the members must be faculty from outside the Department. Before the end of the third year, students must have a written proposal, approved by their committee, for the Advancement to Candidacy oral examination. The proposal should explain the role of at least two series of courses from the student's major area of specialization that will be used to satisfy the Advancement to Candidacy requirements. The proposal should also explain the role of additional research reading material as well as providing a plan for investigating specific topics under the direction of the student's advisor(s). The courses Mathematics 210A-B-C, 220A-B-C, and 230A-B-C cannot count for both Qualifying Examinations and the course requirement for Advancement to Candidacy Examinations. After the student meets the requirements, the Graduate Studies Committee recommends to the Dean of Graduate Studies the advancement to candidacy for the Ph.D. degree. Students should advance to candidacy by the beginning of their fourth year.
After advancing to candidacy, a student is expected to be fully involved in research toward writing his or her Ph.D. dissertation. Ideally, a student should keep in steady contact/interaction with his or her Doctoral committee.
Teaching experience and training is an integral part of the Ph.D. program. All doctoral students are expected to participate in the Department's teaching program.

The candidate must demonstrate independent, creative research in Mathematics by writing and defending a dissertation that makes a new and valuable contribution to mathematics in the candidate's area of concentration. Upon advancement to candidacy a student must form a Thesis Committee, a subcommittee of the Advancement Examination Committee, consisting of at least three faculty members and chaired by the student's advisor. The committee guides and supervises the candidate's research, study, and writing of the dissertation; conducts an oral defense of the dissertation; and recommends that the Ph.D. be conferred upon approval of the doctoral dissertation. The normal time for completion of the Ph.D. is five years, and the maximum time permitted is seven years.

Areas of Specialization and Their Corresponding Advancement to Candidacy Courses

Ph.D. students will choose from one of six areas of specialization in the Mathematics Department, which determines coursework requirements. Each area of specialization will have a core course, which the Department will do its best to offer each year. The Department will offer other courses every other year, or more frequently depending on student demands and other Department priorities.
Students are required to take two series of courses from their chosen area and take two series of courses outside their declared major area of specialization. Special topics courses within certain areas of specialization and courses counted toward the M.S. degree, other than Mathematics 205A-B-C and 206A-B-C, will count toward the fulfillment of the major specialization requirement.

Algebra: Math 230ABC (core), Math 232ABC, Math 233ABC, 234ABC, 235ABC, 239ABC

Analysis: Math 210ABC(core), Math 220ABC(core), Math 211ABC, Math 260ABC, Math 295ABC, Math 296

Applied and Computational Mathematics: Math 290ABC (core), Math 225ABC, Math 226ABC, Math 227AB, Math 291ABC, Math 295ABC

Geometry and Topology: Math 218ABC(core), Math 222ABC, Math 240ABC, Math 245ABC, Math 250ABC

Logic: Math 280ABC (core), Math 281ABC, Math 282ABC, Math 285ABC

Probability: Math 210ABC, Math 211ABC, Math 270ABC, Math 271ABC, Math 272ABC, Math 274

PhD Requirements Summarized

- By the beginning of the 2nd year: Pass Comprehensive exams in Algebra and Analysis.
- By end of the 2nd year: Declare a major specialization.
- By the beginning of the 3rd year: Select an advisor specialist in the major area and form a committee for the Advancement to Candidacy oral exam.
- Before the end of the 3rd year: Have a written proposal, approved by the committee, for the Advancement to Candidacy Examination. Qualifying exams must be passed before the end of the 3rd year.
- By the beginning of the 4th year: Students should have advanced to Candidacy.
Upon Advancement to Candidacy: Form a Thesis Committee, a subcommittee of the Advancement Examination Committee.
- Completion of the PhD: Average completion time is five years; maximum time permitted is seven years. The Department will not financially support students past their sixth year in the PhD program.

Graduate Program in Mathematical and Computation Biology

The graduate program in Mathematical and Computational Biology (MCB) is a one-year "gateway" program designed to function in concert with selected department programs, including the Ph.D. in Mathematics. Detailed information is available online at here.