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.
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
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.
For option #2, 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.)
The program 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 the languages 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 examiners listed below. When you have completed the requirement to the examiner's satisfaction, have the examiner sign the Computer Programming Exam Record Sheet and submit it to the Student Services Office, Padelford C-36.
Computer Programming Examiners
The languages shown in parentheses are the ones in which each examiner prefers to evaluate programs, but 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)
Preliminary exams
The preliminary exams, or "prelims," are the examinations required by the department for admission to official candidacy for the PhD degree. Old prelims are available on-line.
These exams, four hours in length, are offered every September in the mathematical subjects treated by the designated core courses: 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, described below.
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. The exams are written and graded by a committee, which gives the results to the Graduate Program Committee. The Graduate Program Committee meets 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, 2nd ed. (Chapters 1-12) and Introduction to Smooth Manifolds, 2nd ed. (all but Chapters 18 and 22); 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 the 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, typically 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 is taken.
Graduate school requirements
Of course, to obtain a Ph.D., you need to satisfy the requirements spelled out by the University of Washington Graduate School. The main ones of these are the general exam and the final exam, but see this page for full details. (So, for example, prelims and the language exams are internal math department requirements.)
