# PhD Program

**Prerequisites**

The Department of Mathematics offers 2 PhD degrees, one in Mathematics
and one in Applied Mathematics. Applicants for admission to either PhD
program are expected to have preparation comparable to the undergraduate
major at Berkeley in Mathematics or in Applied Mathematics. These
majors consist of 2 full years of lower-division work (covering
calculus, linear algebra, differential equations, and multivariable
calculus), followed by 8 one-semester courses including real analysis,
complex analysis, abstract algebra, and linear algebra. These eight
courses may include some mathematically based courses in other
departments, e.g., physics, engineering, computer science, or economics.

Applicants for admission are considered by the Department's Graduate
Admissions and Appointments Committees. The number of students that can
be admitted each year is determined by the Graduate Division and by
departmental resources. In making admissions decisions, the Committee
considers, among other things, grades in mathematics courses, level of
mathematical preparation, letters of recommendation, and GRE scores.

Experience has shown that the score on the Mathematics Subject GRE is a
partial indicator of preparation for Berkeley's PhD program. A score
below the 80th percentile suggests inadequate preparation and must be
balanced by other evidence if a favorable admission decision is to be
reached.

**Degree Requirements**

In outline, to qualify for the PhD in either Mathematics or Applied
Mathematics, the candidate must meet the following requirements.

- During the first year in the PhD program:

- take at least 4 courses, 2 or more of which are graduate courses in mathematics;

- and pass the six-hour written Preliminary Examination
covering primarily undergraduate material. (The exam is given just
before the beginning of each semester, and the student must pass it
within their first 3 semesters.)

- take at least 4 courses, 2 or more of which are graduate courses in mathematics;
- Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted
*within two years*of entering the program.

- Complete a seminar, giving a talk of at least one hour duration.

- Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee.

- Meet the University residence requirement of two years or four semesters.

The detailed regulations of the PhD program are as follows:

**Course Requirements**

During the first year in the PhD. program, the student must enroll in at
least 4 courses. At least 2 of these must be graduate courses in
mathematics. Exceptions can be granted by the student's Graduate
Advisor.

**Preliminary Examination**

The Preliminary Examination consists of 6 hours of written work given
over a two-day period. Most of the examination covers material, mainly
in analysis and algebra, that should be part of a well-prepared
student's undergraduate training. The Preliminary Examination is
offered twice a year---during the week before classes start in both the
Fall and Spring Semesters. A student may repeat the examination twice.
A student who does not pass the Preliminary Examination within 13
months of the date of entry into the PhD program will not be permitted
to remain in the program past the third semester. In exceptional cases,
a fourth try may be granted upon appeal to Committee Omega.

**Qualifying Examination**

To arrange for the Qualifying Examination, a student must first settle
on an area of concentration, and a prospective Dissertation Supervisor,
someone who agrees to supervise the dissertation if the examination is
passed. With the aid of the prospective supervisor, the student forms
an examination committee of 4 members, at least 2 of which must be
members of the Department. The Graduate Division requires that at least
one committee member be from outside the Department and that the
committee chair be someone other than the Dissertation Supervisor. The
syllabus of the examination is to be worked out jointly by the
committee and the student, but before final approval it is to be
circulated to all faculty members of the appropriate Sections. The
Qualifying Examination must cover material falling in at least 3
subject areas and these must be listed on the application to take the
examination. Moreover, the material covered must fall within more than
one Section of the Department. Sample syllabi can be seen in 910 Evans
Hall.

The student must attempt the Qualifying
Examination within twenty-five months of entering the PhD program. If a
student does not pass on the first attempt, then, on the
recommendation of the student's examining committee, and subject to the
approval of the Graduate Division, the student may repeat the
examination once. The examining committee must be the same, and the
re-examination must be held within thirty months of the student's
entrance into the PhD program.

For a student to pass the Qualifying Examination, at least one
identified member of the subject area group must be willing to accept
the candidate as a dissertation student, if asked. The student must
obtain an official Dissertation Supervisor within one semester after
passing the Qualifying Examination or leave the PhD program. For more
detailed rules and advice concerning the Qualifying Examination, consult
the Graduate Assistant in Room 910 Evans Hall.