Graduate Chairman - Robert Little

The Mathematics Department is located in Keller Hall, near the East--West Center and Hamilton Library, the main research library. The Student Center, Bookstore, Post Office and Undergraduate Library are about five minutes on foot from Keller Hall.

The Mathematics Department offers work leading to the B.A., B.S., M.A., and Ph.D. degrees. The faculty of the Mathematics Department presently consists of 28 permanent members, whose interests span the entire range of modern mathematical research.

Beginning courses in mathematics are usually taught in sections of no more than 30 students. Advanced undergraduate and graduate classes are small (5 to 15 students) and the mathematics major or graduate student may be assured of individual attention and help.

Graduate assistantships are available for a few of the best qualified applicants. About half of our students are supported in this way. The main duty is to teach one course per semester. The stipend begins at $14,382 payable in twelve monthly installments. Tuition is waived for graduate assistants, but they are not exempt from any fees listed in the Graduate Bulletin. Applications for graduate assistantships may be obtained from the department and should be returned to the Mathematics Graduate Chair before February 1 (for Fall) or August 1 (for Spring). Each application must be accompanied by three letters of recommendation from persons familiar with the student's mathematical ability.

All applicants for graduate assistantships must first apply to and be accepted by the Graduate Division for admission as a degree candidate. Application forms should be obtained from the Mathematics Department well in advance of the application deadline. The Graduate Division application deadline for foreign students for the Fall semester is January 15 and for the Spring semester, August 1. The Graduate Division accepts all other applications for the Fall semester from November 1 to March 1 and for the Spring semester from May 1 to September 1. Foreign applicants are reminded that a TOEFL score of at least 500 is required to be admitted by the Graduate Division and a score of 600 or above is required to be considered for a teaching assistantship in mathematics.

All applicants must submit current GRE scores and must receive a score of at least 650 on the Quantitative section of the GRE to be admitted as a regular student. It is highly recommended that applicants submit the Subject test score in mathematics as well, especially when applying for a graduate assistantship.


Incoming graduate students will be required to take a diagnostic examination in undergraduate mathematics. This exam, given by the Mathematics Department, consists of two written parts -- linear and abstract algebra, and calculus and elementary real analysis (not including measure theory). The exam is used to help plan the student's graduate program, and is given twice a year, in August and in January, during the week before registration.


GRADUATE DIVISION REQUIREMENTS: (See current Graduate Information Bulletin for additional information).

The department does not have a thesis option (Plan A) for the M.A. degree. Most M.A. candidates will take Plan B, which requires 30 credit hours, at least 18 of which must be in graduate courses (600--800 level and one seminar course). An exceptional student may be admitted to Plan C (Examination) at the discretion of the Graduate Chair. All Plan B candidates must meet the following requirements.

Each M.A. candidate must form a two-member Master's Committee. The committee will normally be formed after completion of the first semester graduate classes. We suggest that the student form their committee no later than their fourth semester in the program.

In addition to general Graduate Division requirements, the student is required to pass an examination on a topic that meets the approval of the student's Master's Committee and the Graduate Chair. A list of potential topics is available. The student will also have the option of proposing their own topic. It is expected that the student would often take a Math 699 class (reading/research) in the area as well. The topic will generally involve reading a paper or a (non--introductory) chapter of a book. The exam will consist of a one hour presentation on the chosen topic followed by a one hour oral exam. Prior to the examination, the student must provide a written version of the presentation which will be made available to the Graduate Faculty in Mathematics by posting it online. The written version of the presentation must demonstrate significant mathematical growth beyond beginning graduate level. This can be demonstrated, for example, by including in the paper one or more of the following: a) new results; b) new proofs of old results; c) new exposition or synthesis of an area; d) new applications or examples. The voting on whether the student has passed the examination will be done by the student's Master's Committee and the Graduate Committee, both of which are expected to attend the examination. Here is an example, in PDF form, of a written presentation.

Among the required 30 credit hours of course work, the student must take Math 611-612 (MODERN ALGEBRA), Math 631 (FUNCTIONS OF A REAL VARIABLE) and Math 644 (ANALYTIC FUNCTION THEORY).



GRADUATE DIVISION REQUIREMENTS: (See current Graduate Information Bulletin for additional information).

The normal requirement for admission to the Ph.D. program is satisfactory completion of a standard undergraduate program in mathematics. The candidate will be expected to know linear algebra, the elements of abstract algebra, and elementary real analysis. A student whose degree has been awarded in some other field may be considered if he/she has had the appropriate background courses.

Candidates for the Ph.D. must show proficiency as described below in two of French, German, Russian or a computer language.

No examination taken by a Ph.D. candidate may be administered by a faculty member who is the candidate's thesis advisor. This does not preclude this faculty member from subsequently becoming the candidate's thesis advisor.

To show proficiency in a foreign language, the candidate may either:

(i) Take an examination, lasting at most 3 hours, during which the candidate will translate into written English two selections of text in the foreign language. Each selection will be one or two pages long and will be from a fairly recent publication. One selection will be from a publication specified in advance by the examiner and no dictionary will be permitted for that translation. The other selection will be from a publication not specified in advance, but the candidate may use a dictionary for that translation.

(ii) Take 2 semesters of the foreign language at UH and earn an average grade of B or better.

To show proficiency in a computer language, the candidate must demonstrate the ability to write and then code a moderately difficult and interesting algorithm. Documentation must be provided, plus user instructions. The problem and the permitted languages or "packages" will be agreed upon in advance by the examiner and the candidate. An example of a "medium--sized" problem would be: Write a program to play tic--tac--toe.

In order to develop the skills necessary for interaction with students, teaching experience is required of all Ph.D. candidates.

Effective Fall 1997, all new students in the Ph.D. program shall complete a minimum of five Mathematics Department courses numbered between 600 and 690, other than 611, 612, 621, 631, 632, 644, 645, 649. These five courses may be taken under the CR/NC option. Exceptions: Up to two three--credit 649 (alpha) seminars (meeting three hours/week) may be substituted for (up to) two of these required five courses, with the written approval of the Graduate Chair. Also, with the written approval of the Graduate Chair, credit may be given for equivalent courses taken in another mathematics department or for graduate level courses taken in another department which are recommended by the student's thesis advisor and directly related to the dissertation topic; such credit for graduate courses taken in another department is limited to a total of no more than two courses.

No particular course is required of a Ph.D. candidate, nor is there a credit--hour requirement, except for the six credit--hour per semester minimum required by the Graduate Division for Graduate Assistants. The Ph.D. Comprehensive Examination consists of two written examinations covering (i) linear and abstract algebra, and (ii) real analysis and the basic facts of complex analysis and general topology, followed by (iii) a written or oral examination on a subject chosen by the student together with his advisor and with the approval of the Graduate Chair. Finally, there is an oral examination administered by the student's thesis committee. The student may repeat any failed examination no more than once.

Although no definite time limit is set on passing the examinations, students are urged to take them soon after completing their basic course work and before specializing.

A Ph.D. candidate typically will have fulfilled the requirements for the M.A. before embarking on a dissertation, and hence is eligible to receive this degree. However, the M.A. is not a requirement for the Ph.D.


The first two comprehensive examinations for the Ph.D. include the following material:


A. Linear Algebra

Vector spaces, linear equations, algebra of matrices and linear transformations. Dual space of a finite dimensional vector space, transpose of a linear transformation. Determinants. Similarity, diagonalizability, minimal and characteristic polynomials, rational and Jordan canonical forms. Real and complex inner product spaces, orthogonal bases, adjoint of a linear transformation.

B. Abstract Algebra.

Groups and subgroups. Homomorphisms and factor groups. Normal series, Jordan-Holder theorem. Direct products. Cyclic groups, solvable groups, p-groups, Sylow theorems. Fundamental theorem of abelian groups. Transformation and permutation groups.

Rings, subrings, homomorphisms, ideals, factor rings. Modules, submodules, homomorphisms, factor modules. Hom, tensor product, projective and injective modules. Field of quotients of an integral domain. Principal ideal domains. Unique factorization domains. Finitely generated modules over a principal ideal domain. Fields, subfields, extension fields, degree of a finite field extension. Separable and purely inseparable algebraic extension fields. Splitting field of a polynomial, theorem of the primitive element. Roots of unity. Normal extension fields. Galois theory. Finite fields. Cyclic extensions. Solution of equations by radicals.

1. Halmos, Finite Dimensional Vector Spaces.
2. Hoffman and Kunze, Linear Algebra.
3. Hungerford, Algebra.
4. Jacobson, Basic Algebra.
5. Lang, Algebra.


A . Real Analysis.

Elementary topology of Rn, continuous functions in Rn, uniform continuity, uniform convergence, differentiability and implicit function theorem, differentiation under an integral sign. Stone-Weierstrass theorem on the real line, measure spaces, Lebesgue measure and integral, convergence theorems for the Lebesgue integral, types of convergence for sequences of functions, absolute continuity and the derivative, product measures and Fubini's theorem, Lp spaces and the Riesz representation theorem, Radon-Nikodym theorem.

1. T. Apostol, Mathematical Analysis.
2. M. Rosenlicht, Introduction to Analysis.
3. W. Rudin, Principles of Mathematical Analysis.
4. P. Halmos, Measure Theory.
5. H. Royden. Real Analysis.

B. Complex Analysis

The complex derivative; Cauchy Riemann equations; Cauchy's theorems; calculus of residues with applications to definite integrals. Infinite series; the classification of isolated singularities. Definition of elementary functions. Mapping propertes of fractional linear transformations and other elementary functions. Approximation by rational functions; normal families, the Riemann mapping theorem. Zeros of holomorphic functions.

1. W. Rudin, Real and Complex Analysis.
2. K. Knopp, Theory of Functions, Vol. 1.
3. Z. Nehari, Conformal Mapping.
4. L. Alfors, Complex Analysis.
5. Conway, Functions of One Complex Variable.

C. Topology

For Ph.D.'s, concepts such as separation axioms and product and quotient topologies may also be included.

1. J. Hocking and A. Young, Topology, Addison-Wesley.
2. J. L. Kelley, General Topology, Van Nostrand.
3. G. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill.



The University of Hawaii is on a two--semester system with two summer sessions. Fall semester starts in late August or early September and Spring in mid--January; first summer session in late May and second summer session in late June. (Note: There is no summer admission to advanced degree programs in the Graduate Division.) The Mathematics Department usually offers only undergraduate courses during the summer sessions, although informal seminars and reading courses are given frequently.

II. TUITION AND FEES FOR GRADUATE STUDENTS (subject to change without notice)

Tuition and fees are charged according to the number of credit hours carried by the student; auditors (those enrolled in a course for no credit) pay the same fees as students enrolled for credit. For tuition purposes only, a full--time student is any student enrolled for 8 or more credit hours.


The General and Graduate Information Catalog may be purchased from the University of Hawaii Bookstore, 2465 Campus Road, Honolulu, HI 96822.


Almost all room assignments to on-campus residence halls go to Hawaii residents who have priority. There are limited facilities on campus for married students.

For applications and more information, please call or write to the following address:

Student Housing Office
University of Hawaii
Johnson Hall A
2555 Dole Street
Honolulu, HI  96822-2381 
(808) 956-8177