Foreign Language
exams
Two foreign language exams (or one foreign
language and one computer programming exam) are required for the PhD degree.
A PhD student is expected to pass one language or computer exam by the
end of Summer quarter preceeding the third year, and the second by the
end of Summer quarter preceeding the fourth year. Students are encouraged
to look at the Recommended
Programs of Study or contact relevant faculty members for advice concerning
which language or computer exams are most appropriate for the field(s)
of study they are considering specializing in.
To pass a language exam, a student must
demonstrate the ability to translate mathematical literature from the foreign
language into English. It is customary for the examiner and the student
to agree ahead of time on a book from which the translation is to be made.
It is up to the examiner to set the format of the exam, choosing for instance
whether it should be oral or written, and to what extent a dictionary or
word list is permitted. The following faculty serve as examiners:
| French |
German |
Russian |
| Gerald Folland |
Monty McGovern |
Krzystof Burdzy |
| Monty McGovern |
Steffen Rohde |
James King |
| Patrick Perkins |
|
Neal Koblitz |
| Hart Smith |
|
Isabella Novik |
| Tatiana Toro |
|
|
| Ginger Warfield |
Computer Programming
Exam
PhD students may choose to pass a computer
programming exam instead of one of the required two foreign languages.
A PhD student is expected to pass one language or computer exam by the
end of Summer quarter preceeding the third year, and the second by the
end of Summer quarter preceeding the fourth year. Students are encouraged
to look at the Recommended
Programs of Study or contact relevant faculty members for advice concerning
which language or computer exams are most appropriate for the field(s)
of study they are considering specializing in.
To pass the computer programming exam,
a student will be required to demonstrate the ability to design, implement,
debug, and explain a functioning computer program. A student may
fulfill the computer programming requirement by doing one of the following:
-
Obtaining a grade of 3.0 or higher in CSE
142 or AMath 301.
-
Presenting a working computer program (see
below).
-
Presenting other evidence of programming proficiency,
if approved by the examining committee.
To fulfill the requirement by means of option
#2 above, a student will be expected to present a working program that
he or she has designed, written, and debugged. The program may be one the
student has previously created for a course or a job, or may be a new one
specifically written for this purpose. If a new program is to be
written, the student should obtain prior approval of the planned project
from a member of the examining committee. (The examiner can suggest
a project if the student prefers.)
A program presented to fulfill this requirement
must consist of at least 100 lines (not including comments, blank lines,
or other fillers) of well documented, structured code, and must include
most of the following features: floating-point computations, symbolic manipulation,
iteration, input/output, conditional execution, and subroutines.
Any of the following languages are acceptable: C, C++, FORTRAN, Java, Lisp,
Mathematica, Maple, MATLAB, Pascal, and Python. Other languages may be approved
by the examiners if they are sufficiently powerful to express programs
meeting all of the above criteria. The student must demonstrate a
working version of the program, and must present the source code for examination.
The examiner may ask the student to explain the functioning of any part
of the program.
To fulfill the computer programming exam,
contact one of the members of the examining committee below. When
you have fulfilled the requirement, have the examiner sign the Computer
Programming Exam Record Sheet and submit it to the Student Services
Office, Padelford C-36.
Computer Programming Examining Committee
The languages shown in parentheses are
the ones in which each examiner prefers to evaluate programs. Individual
examiners might be willing to evaluate other languages by special arrangement
with the student.
-
Sara Billey (any)
-
Ken Bube (FORTRAN, MATLAB)
-
James Burke (C, C++, FORTRAN, MATLAB, Lisp)
-
Thomas Duchamp (C, C++, Mathematica, Pascal)
-
Chris Hoffman (Mathematica or C)
-
John Lee (FORTRAN, Mathematica, Pascal)
-
John Palmieri (Lisp, Python, or SAGE)
-
Don Marshall (FORTRAN)
-
William Stein (C, C++, Python, Magma, PARI or SAGE)
-
Paul Tseng (FORTRAN)
Preliminary exams
The preliminary exams, or "prelims," are the
examinations required by the department for admission to official candidacy
for the PhD degree.
These exams, four hours in
length, are offered every September in the mathematical subjects treated by
the designated core courses. Currently, these subjects are algebra, real
analysis, complex analysis, topology and geometry of manifolds, and linear
analysis.
A student may substitute completion of a full three-quarter sequence of a
designated core course, in which grades of 3.8 or above are received each
quarter, for the passing of the corresponding preliminary exam. Only one
such exam can be replaced in this manner.
First and second year students
who have satisfied the requirements for two preliminary exams, who have an
established research direction, and who wish to prepare to work under the
direction of a specific faculty member have an additional option, the Oral Prelim
Option.
The written preliminary exams are given
once a year, during the week in September that precedes the start of Autumn
Quarter by two weeks. Each exam is four hours long, starting at 9:30
a.m. each day of the given week. The exams are written and graded by a
committee, which gives the results to the Graduate Program Committee. The
Graduate Program Committee meets sometime during the week after exams to
make decisions about student performances; each student receives a note
describing his or her performance by the end of that week.
The prelims are intended to serve as an
objective measure of a student's mastery of basic graduate level mathematics.
A student is expected to pass three of the five exams by the September
beginning the student's third year in the PhD program. A student who fails
to do so, or who gives clear evidence earlier in his or her career that
failure is likely, will be advised to discontinue his or her studies.
Normally, students take several exams in
September of year two, completing them at that time or in September of
year three, but students are welcome to attempt exams in September of year
one. There is no limit on the number of times a student may attempt a given
exam (other than the obvious bound of 3).
There are five exams:
-
Algebra: Topics at the level of 402-3-4
and 504-5-6.
-
Real Analysis: Topics at the level
of 424-5-6 and 524-5-6.
-
Complex Analysis: Topics at the level
of 534-5-6.
-
Manifolds: Topics at the level of 544-5-6.
-
Linear Analysis: Topics at the level
of 554-5-6.
Each syllabus below lists certain topics that
have appeared on the exams. This list is advisory only -- it is intended
to suggest the level of the exams, not to prescribe exactly the material
that will appear. Past exams can be a useful source of practice questions,
but a student need not master all material that has been covered on these
exams. A student who knows the material in the syllabus and who has spent
some time solving problems should do well on the exams.
Algebra
Topics: Linear algebra
(canonical forms for matrices, bilinear forms, spectral theorems), commutative
rings (PIDs, UFDs, modules over PIDs, prime and maximal ideals, noetherian
rings, Hilbert basis theorem), groups (solvability and simplicity, composition
series, Sylow theorems, group actions, permutation groups, and linear groups),
fields (roots of polynomials, finite and algebraic extensions, algebraic
closure, splitting fields and normal extensions, Galois groups and Galois
correspondence, solvability of equations).
References: Dummit and Foote,
Abstract
Algebra, second edition; Lang, Algebra; MacLane and Birkhoff,
Algebra;
Herstein,
Topics
in Algebra; van der Waerden, Modern Algebra; Hungerford,
Algebra.
Real Analysis
Topics: Elementary set
theory, elementary general topology, connectedness, compactness, metric
spaces, completeness. General measure theory, Lebesgue integral, convergence
theorems, Lp spaces, absolute continuity.
References: Hewitt and Stromberg,
Real
and Abstract Analysis; Rudin, Real and Complex Analysis; Royden,
Real
Analysis; Folland, Real Analysis.
Complex Analysis
Topics: Cauchy theory and
applications. Series and product expansions of holomorphic and meromorphic
functions. Classification of isolated singularities. Theory and applications
of normal families. Riemann mapping theorem; mappings defined by elementary
functions; construction of explicit conformal maps. Runge's theorem and
applications. Picard's theorems and applications. Harmonic functions; the
Poisson integral; the Dirichlet problem. Analytic continuation and the
monodromy theorem. The reflection principle.
References: Ahlfors, Complex
Analysis; Conway, Functions of One Complex Variable, vol. 1;
Rudin, Real and Complex Analysis (the chapters devoted to complex
analysis).
Manifolds
Topics: Elementary manifold
theory; the fundamental group and covering spaces; submanifolds, the inverse
and implicit function theorems, immersions and submersions; the tangent
bundle, vector fields and flows, Lie brackets and Lie derivatives, the
Frobenius theorem, tensors, Riemannian metrics, differential forms, Stokes's
theorem, the Poincaré lemma, deRham cohomology; elementary properties
of Lie groups and Lie algebras, group actions on manifolds, the exponential
map.
References: Lee, Introduction
to Topological Manifolds (Chapters 1-12) and Introduction to Smooth
Manifolds (all but Chapter 16); Massey, Algebraic Topology: An Introduction
or A Basic Course in Algebraic Topology (Chapters 1-5); Boothby,
An
Introduction to Differentiable Manifolds and Riemannian Geometry (Chapters
1-6); and Warner, Foundations of Differentiable Manifolds and Lie Groups
(Chapters 1-4).
Linear Analysis
Topics: Linear algebra
(spectral theory and resolvents, canonical forms and factorization theorems
for matrices), ordinary differential equations (existence and uniqueness
theory, linear systems, numerical approximations), Fourier analysis (Fourier
series and transforms, convolutions, applications to PDE), functional analysis
(theory and examples of Banach and Hilbert spaces and linear operators,
spectral theory of compact operators, distribution theory).
References: Kato, A Short Introduction
to Perturbation Theory for Linear Operators; Horn and Johnson, Matrix
Analysis; Birkhoff and Rota, Ordinary Differential Equations;
Coddington and Levinson, Theory of Ordinary Differential Equations;
Lambert, Numerical Methods for Ordinary Differential Systems; Dym
and McKean, Fourier Series and Integrals; Folland, Fourier Analysis
and its Applications; Jones, Lebesgue Integration on Euclidean Space;
Riesz and Nagy, Functional Analysis; Retherford, Hilbert Space:
Compact Operators and the Trace Theorem; Schechter, Principles of
Functional Analysis; Friedlander, Introduction to the Theory of
Distributions.
The Oral Prelim Option
First and second year students who have satisfied the requirements for
two preliminary exams, who have an established research direction,
and who wish to prepare to work under the direction of a specific faculty
member, may replace their third exam by the Oral Prelim Option:
Such students may propose a two-quarter program of reading
under the direction of a member of the graduate Mathematics
faculty. The reading program culminates in an Oral Preliminary
Examination based on a document written by the student. Upon approval
of the Graduate Program Committee and provided that the faculty member
becomes the official doctoral advisor of the student, the student is
judged to have completed prelims.
Additional Remarks:
-
To qualify for the Oral Prelim Option, the student, in consultation
with a prospective advisor, must submit a short (one or two page)
written proposal by November 28 of the academic year in which reading
is to be done. The proposal should be roughly comparable in scope and
level to two quarters of a core graduate course and must not duplicate
the material treated in a concurrent course. The proposal should
include descriptions of the reading to be done and of the scope of the
written document to be produced, and it must be signed by both the
student and her/his prospective advisor.
The proposal must be approved by a vote of the Graduate Program
Committee, which will meet in early December. In its deliberations, the
Committee shall take the student's entire academic record into
consideration. Upon approval of the Committee, the student's
prospective advisor becomes her/his preliminary advisor.
- The reading must be done in the Winter and Spring Quarters following
approval of the proposal, and the student must sign up for 5 credits of
Math 600 in each of these quarters.
- The written document, between 10 and 20 pages in length, must
be submitted by May 1 of Spring Quarter.
- The Oral Examination consists of an oral presentation of approximately
50 minutes, followed by a question and answer session. The exam is to
be administered by a Reading Committee headed by the student's advisor
and including two additional members of the graduate faculty. The
Reading Committee must be formed by May 1 and must be approved by the
Graduate Program Coordinator. The Oral Preliminary Examination must be
held on or before May 15.
- Following the exam, the Reading Committee shall meet to come to a
consensus regarding the outcome of the Oral Preliminary Exam. The
student's advisor shall report the recommendations of the Reading
Committee to the Graduate Program Committee at its annual renewal
meeting, which is held during the third or fourth week of May. The
final decision regarding the outcome of the Oral Preliminary Exam rests
with the Graduate Program Committee.
General exam
By the end of Winter Quarter of the fourth
year, a PhD student must choose an area of specialization and obtain the
provisional agreement of a faculty member to be his or her PhD advisor.
In consultation with the advisor, the student should choose other members
of the faculty to serve on his or her supervisory committee. This also includes finding a GSR (Graduate School Representative). Faculty members with primary, joint, or affiliate appointments in our department are not eligible to serve as the GSR. The student
can then ask those faculty if they are willing to serve and provide the
information to the Student Services Office. Four months must elapse between
the formation of the committee and the taking of
the general exam.
The general exam must be taken no later
than the Winter Quarter of a student's fourth year, unless an extension
is granted by the Graduate Program Committee. Prior to the general exam,
through a combination of lecture courses and reading courses under the
guidance of the thesis advisor, a student is expected to obtain some depth
of knowledge in the chosen field, extensive enough to have an understanding
of some outstanding contemporary problems in that field and the methods
that currently exist to attack such problems.
The purpose of the general exam is for
the student to demonstrate to the satisfaction of the supervisory committee
that he or she has attained the desired understanding of a problem or problems
in the field. The exam has two components, the preparation of a written
document called the General Paper that must be given to the members
of the Supervisory Committee at least two weeks before the date of the
oral exam, and the oral exam itself. The exact format of each of these
components is up to the supervisory committee, but the Graduate Program
Committee envisions the following as typical:
-
In the General Paper, the student gives a
10-20 page expository account of his or her research area, culminating
in a problem or list of problems to be studied, together with a discussion
of some of the relevant literature.
-
In the oral exam, the student gives a 40-50
minute lecture including a discussion of the background to the problem
or problems to be studied, a discussion of recent work related to the problems,
and a discussion of methods available to approach the problems. Following
the lecture, the committee members ask the student questions about the
problems or their background.
Final exam
The supervisory committee also serves as the student's
examining committee for the final exam, the traditional PhD thesis
defense. One full quarter must elapse between the quarter in which
the general exam is taken and the quarter in which the final exam
is taken. |
