The Doctorate in Mathematics is a degree that certifies both a high level of scholarship and the ability to make original contributions in one's own field. Students must take several advanced courses, pass certain qualifying examinations, and write a dissertation containing original research.

Students in the Ph.D. program may obtain a Master's degree by fulfilling the course and basic examination requirements. This degree may be awarded during any quarter in which the requirements have been satisfied.

Entering students are assigned a faculty advisor and are expected to consult with that advisor at the beginning of the fall and spring quarters concerning their course selection. Students are free to change advisors, as long as the new advisor approves. The course of study falls naturally into three stages, although in some cases, these stages may be passed over or merged together.

There is a structured curriculum for students in their first two years who have not yet passed all their Area Exams. The course sequences listed below help prepare students for the Area Exams, and provide other essential background. First year students are required to take at least two of these sequences, unless they have already passed some of the Area Exams, in which case the corresponding sequences can automatically be counted toward this requirement. Starting in 2012-13, second year students who have not yet passed all their Area Exams are required to take at least one of the sequences. Any exceptions require approval from the Graduate Vice-Chair at the beginning of the year.

Failure to complete all parts of the required sequences is a violation of Satisfactory Progress and may impact student funding.

Students should consult with their faculty advisor and/or the Graduate Vice-Chair on course selection. It is important for students who want to work in a particular field to plan for taking the corresponding Area Exam(s), since potential thesis supervisors would expect that later on.

Core Sequences for students in Pure Mathematics:

• Algebra 210ABC; these help prepare students for the Algebra Area Exam.
• Real Analysis 245ABC and Complex Analysis 246AB; these help prepare students for the Analysis Area Exam.
• Differentiable Topology/Differential Geometry/Algebraic Topology 225ABC; these help prepare students for the Geometry/Topology Area Exam.
• Mathematical Logic and Set Theory 220ABC; these help prepare students for the Logic Area Exam. 

Core Sequences for students in Applied Mathematics:

• Advanced Numerical Analysis 269ABC; these help prepare students for the Numerical Analysis Area Exam.
• Applied Ordinary/Partial Differential Equations 266ABC; these help prepare students for the Applied Differential Equations Area Exam.
• Real Analysis 245ABC and Complex Analysis 246AB.

First Stage:  Students take foundational courses, and prepare,  independently and in groups, for qualifying examinations. Preparation consists of study of the topics of the exam syllabi, as well as study of old exam problems (which are posted online.) Students may attempt Area Exams any time they are offered.  Depending on background preparation, and rate of learning, students may attempt an Area Exam in September of their year of entry, or in March of their year of entry. All students should attempt an Area exam by September of their second year at the latest to avoid the risk of failing to maintain Satisfactory Progress. In the second year or earlier, students may begin to explore special topics, or to specialize in their area of interest by taking appropriate courses. Normally,  by the end the second year  a student will have completed the Area Exam requirement and  will  have a definite idea of the area of mathematics in which they wish to write their dissertation.  As a general rule, students should take advanced courses as soon as they can, but only with proper preparation and without compromising on the mastery of foundational material.  Special participation courses, numbered 290 and 296, are offered to help students to become acquainted with research areas and current research.

Middle Stage: Once the Area Exams have been passed and a general area of research interest has been found, it is time to begin (or complete) the search for a thesis advisor and specific research problems. This process involves taking several advanced courses, participating in seminars(especially the 290 and 296 courses), enrolling in reading courses with individual faculty members, especially possible advisers, etc. Most successful students have a good idea of their research area and problem within a year of passing their qualifying examinations or by the end of the third year. However, this is not always the rule. Some advisers pose problems directly to students, some pose preliminary research problems to get the student started, and some expect that a good problem will be found by the student as they make a guided study of current literature. During this middle stage, students in Pure Mathematics should also fulfill the language requirement. This must be done prior to Advancement to Candidacy, and, while never a serious hurdle, can take some effort.

The Dissertation Stage: This last stage  begins when the student shifts attention rather fully from mastery of  courses and texts to study and problem-solving necessitated by thesis problems. While this period officially commences with the first oral qualifying examination and Advancement to Doctoral Candidacy (ATC), many students enter this phase considerably before they ATC. It ends with the Final Oral Exam, if required, and the submission of the thesis to the University. The length of time needed varies, but typically students devote at one or two years to a small area of mathematics, and sometimes to a single difficult problem. Most students continue to take advanced courses and seminars in their field while working on their thesis.

Pure Mathematics - Must complete at least 12 approved graduate Mathematics courses numbered from 205 to 285, not including 210AB, 245AB or 246AB, with a grade of B or better. A maximum of three 285 courses may be applied toward the 12 course requirement. 596/599 courses may not be applied toward the 12-course requirement. Students who have received a Master's degree from the UCLA Department of Mathematics may receive credit for the graduate courses taken during the Master's program (excluding Math 210AB, 245AB, and 246AB).

In addition to the 12-course requirement, students must complete two Participating Seminars from Math 290 or 296. In these advanced seminars students give 2, 1-1/2 hour lectures. It is advisable to take the seminars after passing the qualifying exams. Students are responsible for picking up credit forms from the graduate office and returning them with the professor's signature after completion of the seminar. These seminars do not apply towards the 12-course requirement.

Applied Mathematics - Must complete at least 18 approved graduate courses, including at least 12 Mathematics courses numbered from 205 to 285, with a grade of B or better. A maximum of three 285 courses may be applied toward the 18 course requirement. 596/599 courses may not be applied toward the 18-course requirement.

Prior to taking the oral qualifying examination for advancement to candidacy, students in the pure program must fulfill the foreign language requirement.

Students are required to pass one written examination in French, German, or Russian. Permission to take another language must be approved by the GVC. This is granted only in exceptional circumstances with clear scientific justification. The language exam must be passed prior to nominating a doctoral committee, and taking the first oral exam. Examinations are given fall and spring quarters, and are graded on a Pass/Fail basis. A student whose native language is one of the above, may petition the GVC for an exemption or an oral examination. The exam can be repeated as often as necessary. Two references, including a dictionary can be used during the exam.

Students in the applied program are not required to fulfill the foreign language requirement.

The University normative time to degree is the number of quarters established for students to complete requirements for the doctorate degree. The normative time for the Math department is 15 quarters. The maximum time to complete the math Ph.D. degree is 18 quarters.

Satisfactory progress towards the Ph.D. degree is defined as maintaining at least a 3.0 GPA, completing core courses as described in the section on Initial Curriculum, and passing the qualifying exams according to the following schedule: the basic exam must be passed by the end of the first year, one Area exam must be passed by the beginning of the sixth quarter, and all qualifying exams must be passed by the beginning of the seventh quarter. NOTE! A leave of absence does not delay this schedule.

Students for the Ph.D. must pass the Basic Examination (preferably upon entrance to the graduate program) and two Ph.D., qualifying examinations chosen from among the following six areas: algebra, analysis, applied differential equations, geometry/topology, logic, numerical analysis.

A minimum four member Doctoral Committee administers this exam. The purpose of this exam is to test the depth of knowledge in the research area of the proposed dissertation. It often involves a discussion of the specific problems involved in the dissertation.

Students must consult with their advisors for recommendations on committee members. Students complete a Nomination of Doctoral Committee form available in the graduate office. Students should then choose a date/time and request a room for their exam from a staff member in the graduate office. The Nomination of Doctoral Committee form must be filed with the math graduate office two weeks prior to the exam. It takes two weeks for the Graduate Division to approve the nomination of the doctoral committee. Approval from the Graduate Division must be received prior to taking exam.

After passing the first oral exam, students are advanced to candidacy. There is a small advancement fee ($90 as of January 2011, subject to periodic change).

Students must prepare a dissertation containing original mathematical research. To meet the University standards for thesis preparation and filing, students should obtain the Graduate Division publication, "Policies and Procedures for Thesis and Dissertation Preparation and Filing". This publication can be obtained from 1255 Murphy Hall or 390A Powell Library, or online at http://www.gdnet.ucla.edu/publications.asp.

It is highly recommended that students attend a workshop on manuscript preparation and filing procedures. The schedule is annonced quarterly. (See http://www.gdnet.ucla.edu/gasaa/library/thesismtg.htm where the quarterly filing deadlines are also given.)

When student and advisor agree that the thesis is near completion (i.e. all key results proven and written down, and only several months to go before the thesis is fully finished), it is time to take the Final Oral Examination. The student must then notify his doctoral committee members and arrange a date and time for the exam. Students must also notify the Math Graduate Office to arrange for a room and to ensure that the necessary paperwork will be available on exam day. This exam is usually taken several weeks before the actual filing of the dissertaion. Students are expected to distribute preliminary versions of their thesis, or extensive written summary of their work, to the members of their committee at least one week before the exam. Students should not delay this exam since it can be very hard to get the committee together for the Exam on short notice.

In the exam, the student gives a lecture on the dissertation, its relation to the mathematical research area to which it contributes, and then answers questions, both during and after the exam, about the work.