MATHEMATICS 2007/2008
Why study mathematics at the University of Denver?
Our graduate programs provide a personalized, congenial and rewarding educational atmosphere. You will interact with our faculty from your first day at DU. The Department of Mathematics at the University of Denver offers an MA and MS in mathematics and a PhD in mathematics.
Your options for involvement in mathematical research are numerous and varied; our faculty is active in research and eager to partner with you. While the department is large enough to provide high quality research opportunities, it is small enough to guarantee close faculty-student interaction.
The faculty is highly distinguished, having received awards for excellence in both research and teaching. Most graduate-level classes are scheduled in the late afternoon and evening to accommodate working students.
The MS and MA degrees prepare the practitioner for careers in which mathematics plays a central role. The PhD is a research degree that prepares the recipient to advance the frontiers of knowledge within a specific area of interest. It is structured to provide the flexibility for students to incorporate interests in allied fields.
Our graduates are highly sought, not only for their knowledge of mathematics, but also for their ability to solve problems, to think abstractly and to see the big picture, and to articulate their ideas with clarity and precision. Our graduates have been successful in a remarkably diverse collection of careers, including industry, business, education and academia.
Department of Mathematics
John Greene Hall, Room 203
2360 S. Gaylord St.
Denver, CO. 80209
303-871-2911
www.math.du.edu
Application Requirements and Deadlines
| Program | Degrees Offered | Number of Credits | Full Time/Part Time | Tests Required—Min. Scores |
| Mathematics | MA | 45 | FT/PT |
GRE — varies TOEFL — 80/550 (IBT/PBT) |
| Mathematics | MS | 45 | FT/PT |
GRE — varies TOEFL — 80/550 (IBT/PBT) |
| Mathematics | PhD | 135 | FT/PT |
GRE — varies TOEFL — 80/550 (IBT/PBT) |
Additional Requirements:
Official transcripts
Essay
Letters of recommendation
Any graduate applicant whose native language is not English (including U.S. citizens and permanent residents) and who wishes to be considered for a GTA position must first demonstrate fluency in spoken English by taking the Internet-based TOEFL exam (iBT). The minimum qualifying score for a GTA on the iBT speaking section is 26. If the iBT is not available, a minimum score of 50 is required on the TSE.
Application Deadline:
Applications are accepted on a rolling basis, and students are admitted for the fall, winter or spring quarters. Students interested in competing for graduate teaching assistantships (GTAs) or financial awards need to submit their complete application by March 15 to ensure consideration for an appointment in fall quarter.
Prerequisite Courses/Degrees:
Degree candidates must have a BS or BA in mathematics or a related field. Specific course requirements and prerequisites for individual degree programs are subject to change.
Admission Forms:
International Applicants
For complete international applicant information, please visit the Office of Graduate Studies Web site. International applicants are strongly encouraged to submit a complete admission packet at least six weeks prior to the program’s application deadline.
Graduate Record Exam (GRE)
Applicants must request that Educational Testing Services forward results to the University of Denver, Office of Graduate Admission. The institution code for the University of Denver is R4842. For information concerning GRE registration, please visit www.gre.org or contact:
Graduate Record Examination
Educational Testing Service
P.O. Box 6000
Princeton, NJ 08541-6000
609-771-7670
Applicants should take the entrance exam well in advance of their intended application date. Please allow at least 14 business days for your general test scores and six weeks for your subject test scores to be received. Several departments and schools will not process applications until scores have been received. Entrance exam scores older than five years from the date of the application may not be acceptable for admission.
Application fees
There is a $60 nonrefundable application fee, which covers the cost of processing application materials. The application fee may be paid online with a credit card at the time of application submission. Otherwise, the application fee may be paid on a bank draft or personal check drawn on a U.S. bank and submitted with the supplemental application materials. Applications will not be forwarded to the department for review until this fee is paid. No waivers or deferrals are allowed with the exception of McNair and CORE scholars. A letter of verification needs to be included with the supplemental application materials.
Transcripts
Applicants are required to submit two official transcripts from each post-secondary institution they have attended or are presently attending where 2 quarter hours (or 1 semester hour) or more were completed. This includes transcripts for credit earned as transfer work or study abroad and college credit earned in high school.
An official transcript must include the original signature of the registrar and/or the seal of the issuing institution, and it must be enclosed in an envelope with the stamp or signature of the registrar across the sealed flap. Proof of a bachelor's and/or master's degree (if applicable) is required from a regionally accredited college or university. Requested transcripts should be mailed to the applicant and submitted to the University of Denver with supplemental admission materials. Please do not request that transcripts be mailed directly to the University of Denver from other institutions.
Applications will not be forwarded to the department for review until official transcripts have been received. The University is not responsible for obtaining an applicant's transcripts, including any record of work completed at the University of Denver.
All credentials submitted become property of the University of Denver and cannot be copied or returned to the student or any person(s).
Letters of recommendation
Three (3) letters of recommendation are required. All recommendations are to be included with application materials.
Essay
You should submit a personal statement of at least 300 words. Your essay should include information concerning your life, education, practical experience, special interests and specific purpose for applying to the University of Denver Department of Mathematics.
Mailing address
Mail all supplemental admission materials, including official transcripts, in one package to:
University of Denver
Office of Graduate Admission
University Hall, Room 216
2197 S. University Blvd.
Denver, CO 80208
Department of Mathematics
John Greene Hall, Room 203
2360 S. Gaylord St.
Denver, CO. 80209
303-871-2911
www.math.du.edu
Scholarship and Financial Aid
Scholarships & Financial Aid
Domestic students may be eligible for additional financial aid. A Free Application for Federal Student Aid (FAFSA) form must be filed for a student to be considered for federal and state-funded aid. The Office of Financial Aid will notify you if further paperwork is required. Students are encouraged to file the FAFSA by March 1.
Merit-Based Financial Assistance
A limited number of graduate teaching assistantships are available; they provide full tuition waivers, stipends of $16,000 (academic year 2008–2009) for the nine-month year and an optional health insurance plan at no cost to the student. These positions are merit-based and awarded on a competitive basis, with most awards made in late March.
Complete applications requesting teaching assistantships for fall admission, along with a personal statement, transcripts, recommendations and GRE test scores, must be received by the department in one package no later than March 15. There is no deadline for admission, but allow up to eight weeks to process an application.
Download Form
Application for Financial Assistance for Graduate Studies
Degree Requirements
Master of Arts in Mathemematics
This degree requires completion of 45 quarter hours of graduate-level courses, including 12 hours of approved mathematics courses at the 4000 level. At most, 10 hours of courses from another university may count toward the degree. No thesis is required.
Degree Requirements
Master of Science in Mathematics
This degree requires completion of 45 quarter hours of graduate-level courses, including 12 hours of approved mathematics courses at the 4000 level. Up to 15 hours may be in an approved cognate area. At most, 10 hours of courses from another university may count toward the degree.
One tool is required and may be chosen from among the following: proficiency in the use of a modern computing typesetting system; outside courses; laboratory experience; or reading competency in French, German or Russian. No thesis is required.
Degree Requirements
Doctor of Philosophy in Mathematics
This degree requires completion of at least 135 quarter hours (see following details) beyond the BA or BS degree; completion of a written dissertation that makes a significant contribution to the research literature in mathematics; and completion of a tool requirement. Although a master’s degree is not a prerequisite for acceptance into the PhD program, each student must have completed a master’s degree in mathematics before completing 80 hours of the PhD program.
Course Requirements
Of the 135 hours, at least 36 must be at the 4000 level. Up to 35 credits may be taken in other relevant disciplines, as approved by the mathematics department graduate committee. Courses should be chosen in consultation with, and are subject to the approval of, the student’s academic advisor.
Examinations
By the time a student has completed 80 hours of graduate credit, students should have successfully passed a written qualifying exam covering three topics from the following list. One of the examinations must be in real analysis.
• real analysis
• complex analysis
• linear algebra
• abstract algebra
• automata theory
• computational complexity
Tool Requirement
It is strongly recommended that students satisfy their tool requirement by demonstrating the ability to use a modern computer typesetting system. Other options include: reading competency in two languages selected from French, German and Russian; a series of outside courses in another discipline; or a significant laboratory experience involving mathematics.
Course Descriptions
Not every course is offered every quarter, and special topics courses change too often to be listed here.
Graduate credit cannot be earned in courses numbered below 3000. Courses numbered 4000 and above are open only to graduate students.
Some courses in abstract algebra have just been approved and may not show up in the catalog for a while, or they may be listed under old numbers.
MATH 3000 The Real World Seminar
Lectures by alumni and others on surviving culture shock after leaving the University and entering the job world. Open to all students regardless of major. Cross-listed as COMP 3000. 1 qtr. hr.
MATH 3050 Set Theory
Zermelo-Fraenkel axioms, axiom of choice, Zorn’s Lemma, ordinals, cardinals, cardinal arithmetic. Prerequisite: MATH 2200 or MATH 2050. 4 qtr. hrs.
MATH 3070 Math Modeling and Computer Simulation
Mathematical models in social, life and management sciences; models include growth processes, epidemics, queues, land usage, etc.; Markov chains, optimization, game theory, graphy theory, etc. Prerequisites: completion of one 3000-level mathematics course and COMP 1672. 4 qtr. hrs.
MATH 3080 Introduction to Probability
Basic probablity models, combinatorial methods, random variables, independence, conditional probability, probability laws, applications to classical problems. Prerequisite: MATH 1952. 4 qtr. hrs.
MATH 3090 Mathematical Probability
Limit theorems for independent random variables, multivariate distributions, generating functions, random walks and statistical techniques. Prerequisites: MATH 1953 or MATH 1963 and MATH 3080. 4 qtr. hrs.
MATH 3120 Introduction to Topology
Point set topology including topological spaces, connectedness, compactness and separation axioms; preparation for advanced courses in analysis. Prerequisite: junior standing. 4 qtr. hrs.
MATH 3151 Linear Algebra I
Vector spaces, linear mappings, matrices, inner product spaces, eigenvalues and eigenvectors. Prerequisite: MATH 2060. 4 qtr. hrs.
MATH 3152 Linear Algebra II
Linear operators on finite dimensional vector spaces, eigenvalues, eigenvectors, Jordan forms; special properties of self-adjoint and normal operators; special topics. Prerequisite: MATH 3151. 4 qtr. hrs.
MATH 3161 Introduction to Real Analysis I
A theoretical introduction to limits and continuity; sequences and series of numbers and functions; a theoretical introduction to the foundations of calculus. Prerequisite: MATH 2080. 4 qtr. hrs.
MATH 3162 Introduction to Real Analysis II
A theoretical introduction to limits and continuity; sequences and serices of numbers and functions; a theoretical introduction to the foundations of calculus. Prerequisite: MATH 2080. 4 qtr. hrs.
MATH 3170 Introduction Abstract Algebra
Examples of groups, permutations, subgroups, cosets, Lagrange theorem, normal subgroups, factor groups, homomorphisms, isomorphisms, rings, integral domains, quaternions, rings of polynomials, Euclid algorithm, ideals, factor rings, maximal ideals, principal ideals, fields and construction of finite fields. Prerequisite: MATH 2200 or one year of university-level mathematics. 4 qtr. hrs.
MATH 3180 Mathematical Statistics
Mathematical foundations of statistical theory, random sampling, theoretical distributions, estimation, test of hypotheses, limit theorems, correlation and regression, nonparametric statistics and decision theory. Students may not receive credit for both MATH 3180 and 3190. Prerequisites: MATH 1953 or MATH 1963 and MATH 3080. 4 qtr. hrs.
MATH 3190 Statistics for Social and Behavioral Sciences
Basic statistical techniques commonly used in quantitative analysis in the fields of economics, political science and sociology: probability, sampling theory, hypothesis testing, analysis of variance and correlation, and regression analysis. Students may not receive credit for both MATH 3190 and MATH 3180. 4 qtr. hrs.
MATH 3221 Automata and Formal Languages I
Introduction to computability, effective procedures, format languages, undecidability; finite automata and regular languages. Prerequisite: MATH 2200 or one year of university-level mathematics. 4 qtr. hrs.
MATH 3222 Automata and Formal Languages II
Pushdown automata and context-free languages; Turing machines; decidability, recursive and recursively enumerable sets. Prerequisite: MATH 3221. 4 qtr. hrs.
MATH 3311 Introduction to Operations Research I
Linear optimization models, simplex algorithm, sensitivity analysis and duality, network models, dynamic programming, applications to physical, social and management sciences. Prerequisite: MATH 2060. 4 qtr. hrs.
MATH 3312 Introduction to Operations Research II
Nonlinear and stochastic models, elementary queuing theory, integer programming, introduction to simulation; applications to physical, social and management sciences. Prerequisites: MATH 1953 or MATH 1963 and MATH 3311. 4 qtr. hrs.
MATH 3400 Introduction to Geometry
Specific geometrical systems including finite, Euclidean, non-Euclidean and projective geometries. Prerequisites: junior standing and one year of university-level mathematics. 4 qtr. hrs.
MATH 3451 Chaos, Dynamics and Fractals I
Introduction to one-dimensional dynamical systems, fractals; fixed and periodic points; sources and sinks; period doubling and tangent node bifurcations; chaotic dynamical systems; Sarkovskii’s Theorem. Prerequisites: MATH 2080 and instructor’s permission. 4 qtr. hrs.
MATH 3452 Chaos, Dynamics and Fractals II
Dynamical systems in two (or more) real variables or one complex variable; stable manifold theorem; Henon attractor; Julia sets; Mandelbrodt set. Prerequisite: MATH 3451 or instructor’s permission. 4 qtr. hrs.
MATH 3550 Introduction to Theory of Numbers
Concepts of nonanalytical number theory and its history; prime numbers, divisibility, continued fractions, modular arithmetic, Diophantine equations and unsolved conjectures. Prerequisites: MATH 2200 or MATH 2050. 4 qtr. hrs.
MATH 3571 Introduction to Numerical Methods I
Algorithmic approach to numerical problems, solution of systems of linear equations, interpolation and approximation, numerical integration, and the numerical solution of ordinary and partial differential equations. Cross-listed as COMP 3571. Prerequisites: MATH 1952 or MATH 1962, and MATH 2060 and COMP 1672. 4 qtr. hrs.
MATH 3572 Introduction to Numerical Methods II
Algorithmic approach to numerical problems, solution of systems of linear equations, interpolation and approximation, numerical integration, and the numerical solution of ordinary and partial differential equations. Cross-listed as COMP 3572. Prerequisite: MATH 3571 or COMP 3571. 4 qtr. hrs.
MATH 3651 Differential Equations and Applied Math I
Modeling of phenomena by ordinary differential equations; techniques of analysis and solution of such equations; oscillation theory and boundary value problems, power series methods, special functions, Laplace transforms and difference equations. Prerequisites: MATH 2060 and MATH 2070. 4 qtr. hrs.
MATH 3652 Differential Equations and Applied Math II
Modeling of phenomena by ordinary and partial differential equations; classification of second order partial differential equations; separation of variables, transform methods, special functions, method of characteristics. Prerequisite: MATH 3651. 4 qtr. hrs.
MATH 3701 Combinatorics
The principle of inclusion and exclusion, elementary counting techniques, systems of distinct representatives, partitions, recursions and generating functions, Latin squares, designs and projective planes. Prerequisite: MATH 2200. 4 qtr. hrs.
MATH 3705 Topics in Mathematics
4 qtr. hrs.
MATH 3706 Introduction to Computer Algebra
Introduction to computer algebra, the algorithmic solution of mathematical problems; use of computer algebra software (MAPLE or MATHEMATICA); algorithms for analysis and manipulation of polynomial, algebraic and trigonometric expressions; algorithms for differentiation and integration; applications to calculus and differential equations. Cross-listed as COMP 3706. Prerequisites: MATH 2070 and either COMP 1672 or COMP 1680. 4 qtr. hrs.
MATH 3707 Math Methods Computer Algebra
Mathematical theory and algorithms used to design modern computer algebra systems. Includes selected topics from integer algorithms, greatest common divisor algorithms for polynomials, polynomial factorization algorithm, resultant computation and applications, polynomial decomposition and the Risch integration algorithm. Cross-listed as COMP 3707. Prerequisite: MATH 3706, COMP 3706 or 1680. 4 qtr. hrs.
MATH 3730 Vector Analysis
Differential vector calculus, gradient, divergence, curl; introduction to differential geometry, integration, Stokes’ theorem in several variables. Prerequisite: MATH 2080. 4 qtr. hrs.
MATH 3851 Functions Complex Variable I
Complex numbers, analytic functions, complex integration, series expansions, residue theory, conformal maps, advanced topics and applications. Prerequisite: MATH 2080. 4 qtr. hrs.
MATH 3852 Functions Complex Variable II
Advanced topics in complex analysis with applications. Prerequisite: MATH 3851. 4 qtr. hrs.
MATH 3950 Undergraduate Research Seminar
Opportunity to conduct mathematics research; bridges the gap between homework exercises and research problems with directed readings and challenging projects. Prerequisite: instructor’s permission. 0–4 qtr. hrs.
MATH 3991 Independent Study
Cannot be arranged for any course that appears in regular course schedule for that particular year. 0–4 qtr. hrs.
Note: Enrollment in courses listed at the 4000-level or above are limited to graduate students.
MATH 4091 Mathematical Probability
Limit theorems for independent random variable, multivariate distributions, generating functions, random walks, statistical techniques. Prerequisites: MATH 1953 or equivalent, and MATH 3080. 3 qtr. hrs.
MATH 4092 Stochastic Processes
Stochastic models in biology, physics, economics; Markov, Poisson, renewal branching, birth-death, queuing processes. Prerequisite: MATH 4091. 3 qtr. hrs.
MATH 4120 Algebraic Topology
Fundamental groups, simplicial homology, Euler characteristic, classification of surfaces, manifolds. Prerequisites: MATH 3170 and MATH 3120. 3 qtr. hrs.
MATH 4161 Group Theory
Groups and homomorphisms, isomorphism theorems, symmetric groups and G-sets, the Sylow theorems, normal series, fundamental theorem of finitely generated abelian groups. Prerequisite:
MATH 3170.
3 qtr. hrs.
MATH 4162 Rings and Modules
Ideals, UFDs, PIDs, minimal polynomials, field extensions, ruler and compass constructions, modules, free modules, projective modules, tensor products, Noetherian and Artinian rings. Prerequisite: MATH 3170. 3 qtr. hrs.
MATH 4163 Universal Algebra and Categories
Universal algebras, congruences, lattices, distributive lattices, modular lattices, Boolean algebras, subdirectly irreducible algebras, Mal’cev theorems, varieties, Birkhoff theorem, objects, morphisms, categories, functors. Prerequisite: MATH 3170. 3 qtr. hrs.
MATH 4221 Automata and Formal Languages I
Computability, effective procedures, formal languages, undecidability; finite automata, regular languages. Prerequisites: MATH 2200. 3 qtr. hrs.
MATH 4222 Automata and Formal Languages II
Pushdown automata, context-free languages; Turing machines; computational complexity. Prerequisites: MATH 4221. 3 qtr. hrs.
MATH 4251,4252, 4253 Modern Analysis
Metric spaces, point set topology, Hilbert and Banach spaces, symmetric compact operators; Lebesgue integral, LP spaces, measure theory. Prerequisite(s): MATH 3152 and MATH 3161. 3 qtr. hrs.
MATH 4501, 4502, 4503 Functional Analysis
Advanced topics in structure of linear spaces, generalized functions; spectral theory of operators in Hilbert spaces; applications to partial differential equations, semigroups, stochastic processes.
Prerequisites: MATH 4253. 3 qtr. hrs.
MATH 4701 Combinatorial Algorithms
Basic enumeration techniques; representations of combinatorial objects; algorithms for searching, sorting, generating combinatorial objects; graph algorithms. 3 qtr. hrs.
MATH 4705 Special Topics in Mathematics
Varying selected advanced topics in mathematics, depending on student demand. Possible alternatives include calculus of variations, partial differential equations, algebraic topology, differential manifolds or special functions. 3 qtr. hrs.
MATH 4707 Mathematical Methods of Computer Algebra
Mathematical theory and algorithms used to design modern computer algebra systems; includes selected topics from: integer algorithms, greatest common divisor algorithms for polynomials, polynomial factorization algorithm, resultant computation and applications, polynomial decomposition, and the Risch integration algorithm. Cross-listed as MATH 3707 and COMP 4707. Prerequisites: MATH 3706 or COMP 3706. 3 qtr. hrs.
MATH 4851, 4852 Introduction to Functions of a Complex Variable
Concepts of complex numbers, functions of a complex variable, continuity and differentiability in complex plane; advanced topics, applications. Prerequisite: MATH 2080. 3 qtr. hrs.
MATH 4991 Independent Study (MA/MS)
Cannot be arranged for any course that appears in regular course schedule for that particular year. arr.
MATH 4995 Independent Research (MA/MS)
Research projects undertaken in conjunction with a faculty member. arr.
MATH 5000 Doctoral Seminar
Techniques and methods used in mathematical and computing research. Includes proofs, bibliographic searching, writing styles and what constitutes an acceptable dissertation. arr.
MATH 5991 Doctoral Independent Study
Cannot be arranged for any course that appears in the regular course schedule for that particular year. arr.
MATH 5995 Independent Research
Research leading to a dissertation. arr.
Faculty
Alvaro Arias
Associate professor and chair
PhD, Texas A&M University
Areas of interest: functional analysis, operator algebras, probability and convex geometry
aarias@math.du.edu
Richard N. Ball
Professor
PhD, University of Wisconsin
Areas of interest: ordered algebra, general topology, topological dynamics and universal algebra
rball@du.edu
Natasha Dobrinen
Assistant professor
PhD, University of Minnesota
Areas of interest: set theory, Boolean algebras, mathematical logic and foundations of mathematics
ndobrine@du.edu
Nikolaos Galatos
Assistant professor
PhD, Vanderbilt University
Areas of interest: lattice theory, ordered algebras and mathematical logic
ngalatos@du.edu
Stanley Gudder
Professor
PhD, University of Illinois
Areas of interest: functional analysis, mathematical physics, foundations of quantum mechanics and probability theory
sgudder@du.edu
James N. Hagler
Professor
PhD, University of California, Berkeley
Areas of interest: functional analysis, topological dynamics and hereditarily indecomposable continua
jhagler@du.edu
Mario Lopez
Professor
PhD, University of Minnesota
Areas of interest: computational geometry, computer graphics and spatio-temperal databases
mlopez@cs.du.edu
Michael Kinyon
Assistant professor
PhD, University of Utah
Areas of interest: nonassociative algebra, loops and quasigroups, differential equations and Leibniz algebras
mkinyon@math.du.edu
Frédéric Latrémolière
Assistant professor
PhD, University of California, Berkeley
Areas of interest: C*-algebras and C*-dynamics
flatremo@du.edu
Nicholas S. Ormes
Associate professor
PhD, University of Maryland, College Park
Areas of interest: topological dynamics, tiling problems and ordered algebra
normes@math.du.edu
Petr Vojtechovsky
Associate professor
PhD, Iowa State University
PhD, Charles University, Prague
Areas of interest: nonassociative algebra (loops), combinatorics and universal algebra
petr@math.du.edu