Mathematics
Office: C.M. Knudson Hall, Room 300
Mailing Address: 2390 S. York St, Denver, Colorado 80208
Phone: 3038712911
Fax: 3038713173
Email: mathinfo@math.du.edu
Web Site: http://www.math.du.edu
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.
Doctor of Philosophy in Mathematics, Master of Arts in Mathematics, Master of Science in Mathematics
Application Deadlines
 Fall 2018 Priority Deadline: February 16, 2018
 Fall 2018 Final Submission Deadline: September 14, 2018
 Winter 2019 Final Submission Deadline: January 11, 2019
 Spring 2019 Final Submission Deadline: April 5, 2019
Admission Requirements
 Online admission application
 $65.00 Application Fee
 University Minimum Degree and GPA Requirements
 Transcripts: One official transcript from each postsecondary institution.
 GRE: The Graduate Record Examination (GRE) is required. Scores must be received directly from the appropriate testing agency by the deadline. The institution code for the University of Denver is 4842.
 Letters of Recommendation: Three (3) letters of recommendation are required. Letters should be submitted by recommenders through the online application.
 Personal Statement: A personal statement of at least 300 words is required. The statement should include information concerning your life, education, practical experience, special interests and specific purpose in applying to the University of Denver’s Department of Mathematics.
Additional Standards for NonNative English Speakers
Official scores from the Test of English as a Foreign Language (TOEFL), International English Language Testing System (IELTS) or Cambridge English: Advanced (CAE) are required of all graduate applicants, regardless of citizenship status, whose native language is not English or who have been educated in countries where English is not the native language. The minimum TOEFL/IELTS/CAE test score requirements for the degree program are:
 Minimum TOEFL Score (paperbased test): 550
 Minimum TOEFL Score (internetbased test): 80
 Minimum IELTS Score: 6.5
 Minimum CAE Score: 169
 English Conditional Admission Offered: In cases where minimum TOEFL/IELTS/CAE scores were not achieved or no English proficiency test was taken, the Mathematics program may offer English Conditional Admission (ECA) to academically qualified nonnative English speakers.
Read the English Language Proficiency policy for more details.
Read the English Conditional Admission (ECA) policy for more details.
Read the Required Tests for GTA Eligibility policy for more details.
Additional Standards for International Applicants
Per Student & Exchange Visitor Program (SEVP) regulation, international applicants must meet all standards for admission before an I20 or DS2019 is issued, [per U.S. Federal Register: 8 CFR § 214.3(k)] or is academically eligible for admission and is admitted [per 22 C.F.R. §62]. Read the Additional Standards For International Applicants policy for more details.
Financial Aid
There are many different options available to finance your education. Most University of Denver graduate students are granted some type of financial support. Our Office of Financial Aid is committed to helping you explore your options.
Doctor of Philosophy in Mathematics
This degree requires completion of at least 135 graduatelevel 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.
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  135 
Minimum credits required for degree: 135
Noncoursework Requirements

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.

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.

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 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 graduatelevel MATH courses  
Total Credits  45 
Minimum credits required for degree: 45
This degree requires completion of 45 credits of graduatelevel 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.
Noncoursework 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 graduatelevel MATH courses  
Total Credits  45 
Minimum credits required for degree: 45
This degree requires completion of 45 credits of graduatelevel 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.
Noncoursework 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.
Courses
MATH 3000 The Real World Seminar (1 Credit)
Lectures by alumni and others on surviving culture shock when leaving the University and entering the job world. Open to all students regardless of major. Cross listed with COMP 3000.
MATH 3040 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. Prerequisite: MATH 2200.
MATH 3050 Set Theory (4 Credits)
ZermeloFraenkel axioms, axiom of choice, Zorn's Lemma, ordinals, cardinals, cardinal arithmetic. Prerequisite: MATH 2200.
MATH 3060 Mathematical Logic (4 Credits)
Classical propositional calculus (deductive systems and truthtable semantics), firstorder logic (axiomatization and completeness), elements of recursion theory, introduction to nonclassical logics. Prerequisite: MATH 2200.
MATH 3090 Mathematical Probability (4 Credits)
Limit theorems for independent random variables, multivariate distributions, generating functions. Prerequisites: MATH 2080 and MATH 3080.
MATH 3151 Advanced Linear Algebra (4 Credits)
Vector spaces, linear mappings, matrices, inner product spaces, eigenvalues and eigenvectors. Prerequisite: MATH 2060 and MATH 2200.
MATH 3161 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. Prerequisites: MATH 2080 and MATH 2200.
MATH 3162 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. Prerequisite: MATH 3161.
MATH 3166 Group Theory (4 Credits)
Groups and homomorphisms, isomorphism theorems, symmetric groups and Gsets, the Sylow theorems, normal series, fundamental theorem of finitely generated abelian groups. Cross listed with MATH 4166. Prerequisite: MATH 3170.
MATH 3170 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. Prerequisite: MATH 2060 and MATH 2200.
MATH 3260 Metric Spaces (4 Credits)
Metric spaces and continuous functions; completeness and compactness; examples including norm spaces; pointwise and uniform convergence; Baire Category Theorem. Cross listed with MATH 4260. Prerequisite: MATH 3161 or equivalent.
MATH 3311 Linear Programming (4 Credits)
Linear optimization models, simplex algorithm, sensitivity analysis and duality, network models, dynamic programming, applications to physical, social and management sciences. Prerequisite: MATH 2060.
MATH 3312 Markov Chains (4 Credits)
Discretetime and continuous Markov Chains, ergodic theorems, random processes, elementary queueing theory, applications. Prerequisite: MATH 2060 and MATH 3080.
MATH 3400 Introduction to Geometry (4 Credits)
Specific geometrical systems including finite, Euclidean, nonEuclidean and projective geometries. Prerequisite: MATH 2200.
MATH 3451 Chaos, Dynamics & Fractals (4 Credits)
Introduction to onedimensional dynamical systems, fractals; fixed and periodic points; sources and sinks; period doubling and tangent node bifurcations; chaotic dynamical systems; Sarkovskii's Theorem. Prerequisite: MATH 3161.
MATH 3550 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. Prerequisites: MATH 2200.
MATH 3651 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. Prerequisites: MATH 2060 and MATH 2070.
MATH 3661 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. Prerequisites: MATH 2070 and MATH 2080.
MATH 3701 Combinatorics (4 Credits)
The principle of inclusion and exclusion, elementary counting techniques, systems of distinct representatives, partitions, recursion and generating functions, Latin squares, designs and projective planes. Prerequisite: MATH 2200.
MATH 3705 Topics in Mathematics (4 Credits)
Varying selected advanced topics in mathematics, depending on student demand and instructor interest.
MATH 3710 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, ncolor theorems, introduction to Ramsey theory. Prerequisite: MATH 2200.
MATH 3720 Coding Theory (4 Credits)
Goals of coding theory and information theory, instantaneous and Huffman codes, Shannon theorems, block and linear codes, generating and paritycheck matrices, Hamming codes, perfect codes, binary Golay code, ReedMuller codes, cyclic codes, BCH codes, ReedSolomon codes, ideas of convolutional and turbo codes. Prerequisite: MATH 3170.
MATH 3851 Functions Complex Variable (4 Credits)
Complex numbers, analytic functions, complex integration, series expansions, residue theory, conformal maps, advanced topics and applications. Prerequisites: MATH 2060 and MATH 2080 and MATH 2200.
MATH 3900 Mathematics Internship (01 Credits)
Graduate students in mathematics may receive elective credit for mathematically related work performed for employers with the approval of the department. At the end of the term, a student report on the work is required, and a recommendation will be required from the employer before a grade is assigned.
MATH 3991 Independent Study (110 Credits)
Cannot be arranged for any course that appears in regular course schedule for that particular year.
MATH 3992 Directed Study (110 Credits)
MATH 4050 Combinatorial Set Theory (4 Credits)
Beginning with a quick review of ZFC, the standard axioms of set theory, the course covers advanced ordinal and cardinal arithmetic and infinitary combinatorics, including Ramsey theory. Additional axioms such as the Continuum Hypothesis, Martin's Axiom, and combinatorial principles such as Diamond and their consequences for mathematics are studied. Prerequisite: MATH 3050.
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, Bairemeasurability, extensions of continuous functions, separation theorems, and more. Prerequisite: MATH 3050.
MATH 4070 Proof Theory (4 Credits)
Hilbertstyle systems, Natural deduction, (simply typed) lambda calculus, combinatory logic, the CurryHoward 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 firstorder 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 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 4162 Rings and Modules (4 Credits)
Ideals, left and right Rmodules, simple modules, totally decomposable modules, WedderburnArtin 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 Gsets, 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 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, 3nets, isotopy, multiplication groups, inner mapping groups, nuclei, commutant, center, associator subloop, inverse properties, powerassociative loops, Bruck loops, Bol loops, Moufang loops, octonions. Prerequisites: MATH 3166 or MATH 4166.
MATH 4260 Metric Spaces (4 Credits)
Metric spaces and continuous functions; completeness and compactness; examples including norm spaces; pointwise and uniform convergence; Baire Category Theorem. Cross listed with MATH 3260. Prerequisite: MATH 3161 or equivalent.
MATH 4270 Hilbert Spaces (4 Credits)
Schwarz and triangle inequalities, Reisz lemma, subspaces and othogonal projections, orthonormal bases, spectrum of bounded linear operators, compact, selfadjoint, 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; RaydonNikodym 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 (14 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 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 4501 Functional Analysis (4 Credits)
Advanced topics in structure of linear spaces; Banach spaces; HahnBanach Theorem and Duality; Uniform Boundedness Theorem; Open Mapping and Closed Graph Theorems; StoneWeierstrass Theorem; Topics in Hilbert Spaces. Prerequisite: MATH 4280.
MATH 4700 Special Topics in Mathematics (14 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 (15 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 4991 Independent Study (110 Credits)
Cannot be arranged for any course that appears in course schedule for that particular year.
MATH 4992 Directed Study (110 Credits)
MATH 4995 Independent Research (110 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 (110 Credits)
Cannot be arranged for any course that appears in the regular course schedule for that particular year.
MATH 5995 Independent Research (110 Credits)
Research leading to a dissertation.
Faculty
Petr Vojtechovsky, Professor and Department Chair, PhD, Iowa State University
John Antonioli, Teaching Assistant Professor, PhD, University of Victoria
Alvaro Arias, Professor, PhD, Texas A&M University
Richard Ball, Professor, Emeritus, PhD, University of Wisconsin
Joel Cohen, Associate Professor, Emeritus, PhD, University of Maryland
Natasha Dobrinen, Professor, PhD, University of Minnesota
Kelly Flaherty, Teaching Associate Professor, PhD, University of Kansas
Thomas French, Teaching Assistant Professor, PhD, University of Denver
Nick Galatos, Professor, PhD, Vanderbilt University
Stanley Gudder, Professor, Emeritus, PhD, University of Illinois
James Hagler, Professor, Emeritus, PhD, University of California, Berkeley
Paul Horn, Associate Professor , PhD, University of California, San Diego
Shashank Kanade, Assistant Professor, PhD, Rutgers University
Michael Kinyon, Professor, PhD, University of Utah
Frederic Latremoliere, Professor, PhD, University of California, Berkeley
Andrew Linshaw, Associate Professor, PhD, Brandeis University
Nicholas Ormes, Associate Professor, PhD, University of Maryland
L. Alayne Parson, Professor, Emerita, PhD, University of Illinois at Chicago
Ronnie Pavlov Jr, Associate Professor, PhD, The Ohio State University
Katherine Perry, Visiting Assistant Professor, PhD, Auburn University
Allegra Reiber, Teaching Instructor, MS, University of Notre Dame
Franklin Schroeck, Research Professor, PhD, University of Rochester
Sara Ugolini, Visiting Assistant Professor, PhD, University of Pisa
Mei Yin, Assistant Professor, PhD, University of Arizona