**Doctor of Philosophy in Mathematics**

- Completion of all required coursework
- Completion of required written examinations
- 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. written examinations, advancement to candidacy oral examination, and dissertation.

Upon entering the program, students are expected to take Math 210, Math 220 and Math 230, which must be passed with a grade of B or better. Students must complete these sequences by the end of the second year.

By the start of the second year, students must achieve at least two passes at the M.S. level among three Exams in Real Analysis, Complex Analysis and Algebra. By the start of the third year, students must achieve two Ph.D. level passes among three Exams in Real Analysis, Complex Analysis and Algebra.

To satisfy the exam requirements, students may take the Core Assessment Exams (offered in the Spring of every year) or the Qualifying Exams (offered before the start of the fall quarter) in these areas. Students may not attempt to take an exam in a particular subject area more than 3 times. A student who passes a Qualifying examination prior to taking the corresponding course will be exempted from taking the course. . *Please Note: corresponding qualifying exam coursework (210,220,230) cannot be used to satisfy both exam and coursework requirements.*

Some students may require additional background prior to entering Math 210 and 230. This will be determined by assessment prior to the start of the students’ first year by the Vice Chair for Graduate Studies, upon consultation with the graduate studies committee. Such students will be directed into Math 205 and/or Math 206, or equivalent, during their first year. These students may pass one Comprehensive Exam in the areas of Algebra or Analysis in lieu of achieving a M.S. pass on one Core Assessment or Qualifying Exam that must be obtained prior to the start of the students’ second year. Comprehensive Exams in Analysis and Algebra will be offered once per year in the 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). ** Only one of the courses Mathematics 210A-B-C, 220A-B-C, and 230A-B-C may count for 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 six years, and the maximum time permitted is seven years. Completion of the Ph.D. degree must occur within 9 quarters of Advancement to Ph.D candidacy.

**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 at the Master’s level of proficiency two exams in real analysis, complex analysis or algebra.

**By end of the 2nd year: **Declare a major specialization. Complete the course series 210A-B-C, 220A-B-C, 230A-B-C.

**By the beginning of the 3rd year:** Demonstrate Ph.D.-level proficiency on Qualifying exams in two of the following three areas: Real analysis, Complex analysis and Algebra. 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.

**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 5.8 years; maximum time permitted is seven years. The Department will not financially support students past their sixth year in the PhD program. Completion of the Ph.D. degree must occur within 9 quarters of Advancement to Ph.D candidacy.

**Graduate Program in Mathematical and Computation Biology**

The graduate program in Mathematical, Computational Systems Biology (MCSB) is designed to meet to meet the interdisciplinary training challenges of modern biology and function in concert with selected department programs, including the Ph.D. in Mathematics.

Detailed information is available online at ** HERE [1]**.