Doctoral
Program (Ph.D.)
Prerequisites: To enter the Ph.D. program a student
should hold at least a Bachelor's degree in mathematics. The
academic record of a student applying to the Ph.D. program should
contain substantial evidence that the student will succeed in the
doctoral program. In reviewing an applicant's folder, the
Graduate Committee gives substantial weight the applicant's transcripts,
letters of recommendation, and GRE scores.
Requirements: The Ph.D. degree has no rigid course
requirement beyond the residency requirement (however, breadth and
depth of knowledge are strongly encouraged). It does require
(1) knowledge of two ``languages'' as discussed below, (2) passing
written and oral preliminary examinations, (3) writing a dissertation
embodying the results of original research which is acceptable to
the student's dissertation committee, and (4) a final oral defense
of the dissertation. A student's progress towards the Ph.D.
degree is supervised by a five-person committee, formed at the beginning
of his or her graduate program. The student's faculty advisor
chooses this committee, and is its chair.
The Ph.D. Preliminary Examination System consists of two
parts. The first part consists of three Written Prelims and
the second consists of an Oral Prelim.
Qualifying Exams are offered in Algebra, Analysis, Topology,
Numerical Analysis and Probability. The 8000-level sequences
in each area are designed to prepare the student for Qualifying
Exams. Written Qualifying Exams are offered every year in
August, during the week before the start of classes, and in January.
Syllabi, and copies of old Exams, are available from the department
office for students to use in studying for Exams.
Each Ph.D. candidate is required to pass three Qualifying Exams,
including the Analysis Qualifying Exam and either the Topology or
Algebra Qualifying Exam.
The final determination of pass or fail on a written examination
lies with the student's committee. The committee may elect
to reverse the decision of the examiner (with four out of five votes)
or may administer its own examination in addition.
The Oral Prelim is based on the student's anticipated area
of specialization. In it, the student is expected to present material
from a
research paper and to answer general questions about his or her
area of specialization. It is to be taken within 9 months of the
time the student passes his or her third Written Prelim. (A
student who passes Written Prelims early will be allowed additional
time to pass the Oral Prelim.) To begin preparing for the
Oral Prelim, the student chooses a thesis advisor; and the
student, advisor, and committee agree upon a body of material which
the student will be responsible for. The student reads research
papers in the area: In general, in the examination, the student
presents a 30-minute lecture on those prepared papers, followed
by a question period of at least one hour on the paper and background
material.
Ph.D. Language Requirements: A student must either
demonstrate: a reading knowledge of two foreign languages
with significant mathematical literature, chosen from French,
German, and Russian; OR a reading knowledge of one
of the above languages, and sufficient competence with computers
to do mathematical research; OR a reading knowledge
of one of the above languages, together with sufficient improvement
in English, if the student is an international student whose English
is initially inadequate.
A student can satisfy the language requirements by passing an appropriate
course (French, German, or Russian 2001 or above) with a B
or better; OR by translating an unfamiliar mathematical
paper, using a dictionary, in a reasonable length of time (3-4 hours
for a 4 page paper), to the satisfaction of a qualified examiner
from the Mathematics faculty; OR by having native
proficiency in one of the languages above, as certified by a qualified
examiner; OR by passing Technical Writing in English with
a B or better.
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