Mathematics
Office: C.M. Knudson Hall, Room 300
Mailing Address: 2390 S. York St, Denver, Colorado 80208
Phone: 303-871-2911
Fax: 303-871-3173
Email: math-info@du.edu
Web Site: http://science.du.edu/mathematics
The Department of Mathematics at the University of Denver offers MA, MS and PhD degrees in mathematics. Our graduate programs enroll about 30 students and provide a personalized, congenial and rewarding educational atmosphere where you will interact with faculty from the start. We have an extraordinarily active faculty of accomplished teachers and researchers with expertise in algebraic logic, computational geometry, dynamical systems, functional analysis, nonassociative mathematics, ordered structures, probabilistic combinatorics, quantum structures, and set theory.
The MS and MA degrees prepare students for careers in which mathematics plays a central role. The PhD is a research degree that prepares students to advance the frontiers of knowledge within a specific area of mathematics.
Our graduates are highly sought, not only for their knowledge of mathematics, but also for their ability to solve problems, to think abstractly, 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.
Program Learning Outcomes
Doctor of Philosophy in Mathematics
- Formulate rigorous arguments that involve multiple areas of mathematics.
- Communicate mathematics via oral presentations at a level suitable for the audience, exhibiting strong logical organization and awareness of problem context.
- Conduct original mathematics research and report the results in written form of a quality equivalent to research math journals.
Master of Arts in Mathematics
- Formulate rigorous arguments that involve multiple areas of mathematics.
- Communicate mathematics via oral presentations at a level suitable for the audience, exhibiting strong logical organization and awareness of problem context.
Master of Science in Mathematics
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Formulate rigorous arguments that involve multiple areas of mathematics.
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Communicate mathematics via oral presentations at a level suitable for the audience, exhibiting strong logical organization and awareness of problem context.
Doctor of Philosophy in Mathematics
Degree and GPA Requirements
Bachelor’s degree: All graduate applicants must hold an earned baccalaureate from a regionally accredited college or university or the recognized equivalent from an international institution.
University GPA requirement: The minimum grade point average for admission consideration for graduate study at the University of Denver must meet one of the following criteria:
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A cumulative 2.5 on a 4.0 scale for the baccalaureate degree.
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A cumulative 2.5 on a 4.0 scale for the last 60 semester credits or 90 quarter credits (approximately two years of work) for the baccalaureate degree.
-
An earned master’s degree or higher from a regionally accredited institution or the recognized equivalent from an international institution supersedes the minimum GPA requirement for the baccalaureate.
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A cumulative GPA of 3.0 on a 4.0 scale for all graduate coursework completed for applicants who have not earned a master’s degree or higher.
Standardized Test Scores
- GRE scores are optional for admission to this program. Applications submitted without scores will receive full consideration. Every application undergoes a comprehensive evaluation, including a careful review of all application materials.
If you choose to submit test scores, you may upload your Test Taker Score Report PDF, which is considered unofficial. Official scores must be received directly from the appropriate testing agency upon admission to the University of Denver. The ETS institution code to submit GRE scores to the University of Denver is 4842.
English Language Proficiency Test Score Requirements
The minimum TOEFL/IELTS/C1 Advanced/Duolingo English Test score requirements for this degree program are:
- Minimum TOEFL Score (Internet-based test): 80 for exams taken prior to January 21, 2026; 4.0 for exams taken on or after January 21, 2026
- Minimum IELTS Score: 6.5
- Minimum C1 Advanced Score: 176
- Minimum Duolingo English Test Score: 115
Master of Arts in Mathematics
Degree and GPA Requirements
Bachelor’s degree: All graduate applicants must hold an earned baccalaureate from a regionally accredited college or university or the recognized equivalent from an international institution.
University GPA requirement: The minimum grade point average for admission consideration for graduate study at the University of Denver must meet one of the following criteria:
-
A cumulative 2.5 on a 4.0 scale for the baccalaureate degree.
-
A cumulative 2.5 on a 4.0 scale for the last 60 semester credits or 90 quarter credits (approximately two years of work) for the baccalaureate degree.
-
An earned master’s degree or higher from a regionally accredited institution or the recognized equivalent from an international institution supersedes the minimum GPA requirement for the baccalaureate.
-
A cumulative GPA of 3.0 on a 4.0 scale for all graduate coursework completed for applicants who have not earned a master’s degree or higher.
Standardized Test Scores
- GRE scores are optional for admission to this program. Applications submitted without scores will receive full consideration. Every application undergoes a comprehensive evaluation, including a careful review of all application materials.
If you choose to submit test scores, you may upload your Test Taker Score Report PDF, which is considered unofficial. Official scores must be received directly from the appropriate testing agency upon admission to the University of Denver. The ETS institution code to submit GRE scores to the University of Denver is 4842.
English Language Proficiency Test Score Requirements
The minimum TOEFL/IELTS/C1 Advanced/Duolingo English Test score requirements for this degree program are:
- Minimum TOEFL Score (Internet-based test): 80 for exams taken prior to January 21, 2026; 4.0 for exams taken on or after January 21, 2026
- Minimum IELTS Score: 6.5
- Minimum C1 Advanced Score: 176
- Minimum Duolingo English Test Score: 115
Master of Science in Mathematics
Degree and GPA Requirements
Bachelor’s degree: All graduate applicants must hold an earned baccalaureate from a regionally accredited college or university or the recognized equivalent from an international institution.
University GPA requirement: The minimum grade point average for admission consideration for graduate study at the University of Denver must meet one of the following criteria:
-
A cumulative 2.5 on a 4.0 scale for the baccalaureate degree.
-
A cumulative 2.5 on a 4.0 scale for the last 60 semester credits or 90 quarter credits (approximately two years of work) for the baccalaureate degree.
-
An earned master’s degree or higher from a regionally accredited institution or the recognized equivalent from an international institution supersedes the minimum GPA requirement for the baccalaureate.
-
A cumulative GPA of 3.0 on a 4.0 scale for all graduate coursework completed for applicants who have not earned a master’s degree or higher.
Standardized Test Scores
- GRE scores are optional for admission to this program. Applications submitted without scores will receive full consideration. Every application undergoes a comprehensive evaluation, including a careful review of all application materials.
If you choose to submit test scores, you may upload your Test Taker Score Report PDF, which is considered unofficial. Official scores must be received directly from the appropriate testing agency upon admission to the University of Denver. The ETS institution code to submit GRE scores to the University of Denver is 4842.
English Language Proficiency Test Score Requirements
The minimum TOEFL/IELTS/C1 Advanced/Duolingo English Test score requirements for this degree program are:
- Minimum TOEFL Score (Internet-based test): 80 for exams taken prior to January 21, 2026; 4.0 for exams taken on or after January 21, 2026
- Minimum IELTS Score: 6.5
- Minimum C1 Advanced Score: 176
- Minimum Duolingo English Test Score: 115
Doctoral Program
Doctor of Philosophy in Mathematics
This degree requires completion of at least 117 graduate-level credits beyond the BA or BS degree; passing of preliminary examinations; completion of a tool requirement; and completion of a written dissertation.
Although a master’s degree is not a prerequisite for acceptance into the PhD program, each student is required to obtain a master’s degree in mathematics before completing 80 credits in the PhD program.
Degree Requirements
Course Requirements
Students are required to pass both of the sequences MATH 3161, MATH 4165, MATH 4110 and MATH 3170, MATH 4166, MATH 4176 in their first year. Based on performance in preliminary examinations or other considerations, students may be allowed to replace some courses in a sequence by other offerings.
Every student’s course of study must be approved in consultation with a designated departmental advisor.
Minimum credits required for degree: 117
| Code | Title | Credits |
|---|---|---|
| MATH 4XXX courses (minimum of 36 credits) | ||
| Additional Coursework (Chosen in consultation with the student's academic advisor.) | ||
Up to 35 credits may be taken in other relevant disciplines, as approved by the mathematics department graduate committee. | ||
| Total Credits | 117 | |
Non-coursework Requirements
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Preliminary Examinations
Every student admitted to the PhD program is expected to pass a written preliminary examination in analysis and a written preliminary examination in algebra. Both preliminary examinations are designed to test whether students in the PhD program have the adequate undergraduate preparation to continue in the program with a reasonable chance of success.
Both examinations are offered twice per year: during the week immediately preceding the first week of the fall quarter, and during the first week of the winter quarter. A student must pass both exams by no later than the end of the winter quarter of his/her second year in the program unless the graduate committee grants an extension of this deadline for exceptional and documented reasons.
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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; a significant laboratory experience involving mathematics.
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Dissertation and Oral Defense
The dissertation must make a significant contribution to the research literature in mathematics.
After the dissertation has been completed, the student must defend it in a final examination, as specified by the Office of Graduate Education.
Master's Programs
Master of Arts in Mathematics
Degree Requirements
Coursework Requirements
Every student’s course of study must be approved in consultation with a designated departmental advisor.
| Code | Title | Credits |
|---|---|---|
| Approved MATH 4XXX courses (minimum 12 credits) | ||
| Approved cognate area (up to 15 credits) | ||
| Additional graduate-level MATH courses | ||
| Total Credits | 45 | |
Minimum credits required for degree: 45
This degree requires completion of 45 credits of graduate-level MATH courses, including at least 12 credits of approved MATH courses at the 4000 level. Students are required to pass at least one of the sequences MATH 3161, MATH 4165, MATH 4110 or MATH 3170, MATH 4166, MATH 4176.
Up to 15 credits may be in an approved cognate area. At most 10 credits from another university may count toward the degree, and such credits must be approved in writing by an advisor from the Mathematics faculty.
Non-coursework Requirements
- Students are required to give an oral presentation in mathematics. This will typically occur after the completion of at least 24 credits of coursework, and should be scheduled after approval from a Mathematics faculty advisor.
No thesis is required.
Master of Science in Mathematics
Degree Requirements
Coursework Requirements
Every student’s course of study must be approved in consultation with a designated departmental advisor.
| Code | Title | Credits |
|---|---|---|
| Approved MATH 4XXX courses (minimum 12 credits) | ||
| Approved cognate area (up to 15 credits) | ||
| Additional graduate-level MATH courses | ||
| Total Credits | 45 | |
Minimum credits required for degree: 45
This degree requires completion of 45 credits of graduate-level MATH courses, including at least 12 credits of approved MATH courses at the 4000 level. Students are required to pass both of the sequences MATH 3161, MATH 4165, MATH 4110 and MATH 3170, MATH 4166, MATH 4176.
Up to 15 credits may be in an approved cognate area. At most 10 credits from another university may count toward the degree, and such credits must be approved in writing by an advisor from the Mathematics faculty.
Non-coursework Requirements
- Tool requirement- Demonstrated competency in a tool is required and may be chosen from among the following: proficiency in the use of a modern computing typesetting system; approved outside courses; laboratory experience; or reading competency in French, German or Russian.
- Students are required to give an oral presentation in mathematics. This will typically occur after the completion of at least 24 credits of coursework, and should be scheduled after approval from a Mathematics faculty advisor.
No thesis is required.
MATH 4040 Lattices and Order (4 Credits)
Ordered sets, lattices as relational and as algebraic structures, ideals and filters, complete lattices, distributive and modular lattices, Boolean algebras, duality for finite distributive lattices.
MATH 4050 Set Theory (4 Credits)
Zermelo-Fraenkel axioms, axiom of choice, Zorn's Lemma, ordinals, cardinals, cardinal arithmetic.
MATH 4060 Descriptive Set Theory (4 Credits)
Descriptive Set Theory is one of the main branches of modern set theory. Set theory provides techniques for the precise study of real analysis. This course covers trees as tools for analyzing sets of real numbers, Polish spaces, the Borel hierarchy, Baire-measurability, extensions of continuous functions, separation theorems, and more. Prerequisite: MATH 3050.
MATH 4070 Proof Theory (4 Credits)
Hilbert-style systems, Natural deduction, (simply typed) lambda calculus, combinatory logic, the Curry-Howard correspondence, normalization, cartesian cloased categories, Sequent calculi, cut elimination and applications, structural rules; logical systems: classical, intuitionistic, relevance, linear; algebraic semantics. Recommended prerequisite: MATH 2200.
MATH 4080 Algebraic Logic (4 Credits)
Elements of universal algebra, lattice theory and first-order logic; elements of abstract algebraic logic (deductive systems, algebraization, deduction filters, deduction theorems, matrix semantics); sequent calculi for substructural logics, residuated lattices, structure theory for congruences and deductive filters; subvariety lattices (atomic varieties, axiomatizations of joins, translations); algebraic cut elimination; (un)decidability and finite model property. Prerequisites: MATH 3170 and either MATH 3040 or MATH 3060.
MATH 4090 Mathematical Probability (4 Credits)
Limit theorems for independent random variables, multivariate distributions, generating functions.
MATH 4110 Topology (4 Credits)
Point set topology including topological spaces, connectedness, compactness and separate axioms; preparation for advanced courses in analysis. Prerequisite: MATH 3161. Cross listed with MATH 3110.
MATH 4120 Algebraic Topology (4 Credits)
Fundamental groups, simplicial homology, Euler characteristic classification of surfaces, manifolds. Prerequisites: MATH 3170 and MATH 3110/4110.
MATH 4151 Advanced Linear Algebra (4 Credits)
Vector spaces, linear mappings, matrices, inner product spaces, eigenvalues and eigenvectors.
MATH 4161 Introduction to Real Analysis (4 Credits)
A theoretical introduction to the structure of real numbers, to convergence of sequences and series, and to the topology of the real line, including limits and continuity.
MATH 4162 Rings and Modules (4 Credits)
Ideals, left and right R-modules, simple modules, totally decomposable modules, Wedderburn-Artin theorems, Artinian and Noetherian rings and modules, Hopkins theorem, Hilbert basis theorem, free modules, projective and injective modules, Kaplanski theorem. Prerequisites: MATH 3176 or MATH 4176.
MATH 4163 Universal Algebra (4 Credits)
Universal algebras, congruences, lattices, distributive lattices, modular lattices, Boolean algebras, subdirectly irreducible algebras, Mal'cev theorems, varieties, Birkhoff theorem. Prerequisites: MATH 3170 and either MATH 3040 or MATH 3060.
MATH 4164 Galois Theory (4 Credits)
The fundamental theorem of algebra, field extensions, ruler and compass constructions, normal and separable extensions, field automorphisms, Galois correspondence, solvability and simplicity, calculating Galois groups. Prerequisite: MATH 3176/MATH 4176 and MATH 3166/MATH 4166.
MATH 4165 Introduction to Real Analysis II (4 Credits)
A rigorous introduction to the analysis of functions of a real variable, including differentiation, Riemann integration, and the notions of pointwise and uniform convergence for sequences of functions. Prerequisites: MATH 3161.
MATH 4166 Group Theory (4 Credits)
Groups and homomorphisms, isomorphism theorems, symmetric groups and G-sets, the Sylow theorems, normal series, fundamental theorem of finitely generated abelian groups. Cross listed with MATH 3166. Prerequisite: MATH 3170.
MATH 4168 Lie Groups and Lie Algebras (4 Credits)
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semisimple Lie groups; classification of semisimple Lie algebras; representation theory of compact and semisimple Lie algebras and Lie groups. Additional topics as time permits: universal enveloping algebras, symmetric spaces. Prerequisites: MATH 3161 and MATH 3170.
MATH 4170 Introduction to Abstract Algebra (4 Credits)
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, construction of finite fields.
MATH 4176 Rings and Fields (4 Credits)
Rings, domains, fields; ideals, quotient rings, polynomials; PIDs, UFDs, Euclidean domains; maximal and prime ideals, chain conditions; extensions of fields, splitting fields, algebraic and transcendental extensions; brief introduction to Galois theory. Cross listed with MATH 3176. Prerequisite: MATH 3170 or equivalent.
MATH 4181 Loop Theory (4 Credits)
Quasigroups, loops, latin squares, 3-nets, isotopy, multiplication groups, inner mapping groups, nuclei, commutant, center, associator subloop, inverse properties, power-associative loops, Bruck loops, Bol loops, Moufang loops, octonions. Prerequisites: MATH 3166 or MATH 4166.
MATH 4270 Hilbert Spaces (4 Credits)
Schwarz and triangle inequalities, Reisz lemma, subspaces and othogonal projections, orthonormal bases, spectrum of bounded linear operators, compact, self-adjoint, normal and unitary operators, spectral theorem and, if time permits, unbounded operators. Also, if time permits, applications to partial differential equations, physics and engineering. Prerequisites: MATH 3260 or MATH 4260 or MATH 3110 or MATH 4110.
MATH 4280 Measure Theory and Applications (4 Credits)
Definition of Measure spaces; Lebesgue measure; limit theorems; Raydon-Nikodym Theorem; introduction to L_p spaces.
Prerequisite: (MATH 3260 with a minimum grade of D- or MATH 4260 with a minimum grade of C-) or (MATH 3110 with a minimum grade of D- or MATH 4110 with a minimum grade of C-).
MATH 4290 Dynamical Systems (4 Credits)
Topological and measure theoretic dynamical systems; properties and invariants of systems; symbolic dynamics; Ergodic Theorems; applications. Prerequisites: MATH 3110/4110 or MATH 3260/4260.
MATH 4300 Graduate Seminar (1-4 Credits)
Students research a topic of their choosing with the aid of a faculty member, and then prepare and present a formal lecture on the subject. Prerequisite: graduate standing or consent of the instructor.
MATH 4311 Linear Programming (4 Credits)
Linear optimization models, simplex algorithm, sensitivity analysis and duality, network models, dynamic programming, applications to physical, social and management sciences.
MATH 4312 Markov Chains (4 Credits)
Discrete-time and continuous Markov Chains, ergodic theorems, random processes, elementary queueing theory, applications.
MATH 4351 Introduction to Dynamical Systems (4 Credits)
Dynamical systems (one-parameter families such as circle rotations/tent maps, shift spaces); global properties (transitivity/mixing/sensitivity); behavior of trajectories (chaos, long-term averages, periodicity). Prerequisite: MATH 4161.
MATH 4400 Differential Geometry (4 Credits)
Planar and spatial curves, global properties of curves, surfaces in three dimensions, the first fundamental form, curvature of surfaces, Gaussian curvatures, geodesics, Theorema Egregium, hyperbolic geometry. Prerequisites: MATH 3170 and either MATH 3110/4110 or MATH 3260/4260.
MATH 4451 Chaos, Dynamics & Fractals (4 Credits)
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. Prerequisite: MATH 4161.
MATH 4501 Functional Analysis (4 Credits)
Advanced topics in structure of linear spaces; Banach spaces; Hahn-Banach Theorem and Duality; Uniform Boundedness Theorem; Open Mapping and Closed Graph Theorems; Stone-Weierstrass Theorem; Topics in Hilbert Spaces. Prerequisite: MATH 4280.
MATH 4550 Introduction to Theory of Numbers (4 Credits)
Concepts of nonanalytic number theory and its history; prime numbers, divisibility, continued fractions, modular arithmetic, Diophantine equations and unsolved conjectures.
MATH 4600 Numerical Analysis (4 Credits)
An introduction to numerical methods. Topics include: iterative methods, Banach fixed point method, Runge-Kutta and finite element methods for partial differential equations, numerical optimization and gradient descent algorithm, and back propagation in neural networks.
MATH 4605 Mathematics of Complex Networks (4 Credits)
An introduction to the study of complex networks, focusing on the modeling, classification and geometrical properties of complex systems. Topics include stochastic and non-stochastic models of complex networks, measures of centrality and clustering, influence propagation and geometric data (expansion, the small world phenomena and PageRank).
MATH 4610 Machine Learning: Linear Models and Regression (4 Credits)
An introduction to modern regression techniques, with an emphasis in theoretical foundations and applications in artificial intelligence and machine learning. Topics include multilinear regression, polynomial regression, logistic regression, and support vector machines, including kernels methods. Prerequisite: MATH 4151 & MATH 4080.
MATH 4615 Statistics and Stochastic Methods (4 Credits)
Statistical decision theory, estimation, testing, confidence intervals. Bayesian statistics, introduction to Markov chains, and hidden Markov chains with applications to artificial intelligence. Prerequisite: MATH 4080.
MATH 4651 Ordinary Differential Equations (4 Credits)
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.
MATH 4661 Partial Differential Equations (4 Credits)
First and second order linear equations, Fourier series, the wave equation, the Cauchy problem, the heat equation, maximum principles, Laplace's equation, Green's functions.
MATH 4700 Special Topics in Mathematics (1-4 Credits)
MATH 4701 Combinatorial Algorithms (4 Credits)
Basic enumeration techniques; representations of combinatorial objects; algorithms for searching, sorting, generating combinatorial objects, graph algorithms. Prerequisites: MATH 3701 or MATH 3710.
MATH 4705 Special Topics Applied Math (1-5 Credits)
Varying selected advanced topics in mathematics, depending on student demand. Possible alternatives include of variations, partial differential equations, algebraic topology, differential manifolds, special functions.
MATH 4710 Graph Theory (4 Credits)
Paths, cycles, trees, Euler tours and Hamilton cycles, bipartite graphs, matchings, basic connectivity theorems, planar graphs, Kuratowski's theorem, chromatic number, n-color theorems, introduction to Ramsey theory.
MATH 4720 Coding Theory (4 Credits)
Goals of coding theory and information theory, instantaneous and Huffman codes, Shannon theorems, block and linear codes, generating and parity-check matrices, Hamming codes, perfect codes, binary Golay code, Reed-Muller codes, cyclic codes, BCH codes, Reed-Solomon codes, ideas of convolutional and turbo codes. Prerequisite: MATH 4170.
MATH 4851 Functions Complex Variable (4 Credits)
Complex numbers, analytic functions, complex integration, series expansions, residue theory, conformal maps, advanced topics and applications.
MATH 4900 Mathematics Internship (0-1 Credits)
The purpose of this course is to provide graduate students in mathematics with the opportunity to perform mathematically related work and to apply their mathematical skills in a real-world employment setting.
MATH 4991 Independent Study (1-10 Credits)
Cannot be arranged for any course that appears in course schedule for that particular year.
MATH 4995 Independent Research (1-10 Credits)
Research projects undertaken in conjunction with a faculty member.
MATH 5000 Doctoral Seminar (3 Credits)
Techniques, methods used in mathematical, computing research. Includes proofs, bibliographic searching, writing styles, what constitutes an acceptable dissertation.
MATH 5991 Independent Study (1-10 Credits)
Cannot be arranged for any course that appears in the regular course schedule for that particular year.
MATH 5995 Independent Research (1-10 Credits)
Research leading to a dissertation.