2019-2020 Undergraduate Bulletin

Mathematics

Office: C.M. Knudson Hall, Room 300
Mailing Address:  2390 S. York St, Denver, CO 80208
Phone: 303-871-2911
Fax: 303-871-3173
Email: math-info@math.du.edu
Web Site: http://www.math.du.edu

The Department of Mathematics offers a bachelor of arts in mathematics, bachelor of arts in mathematics with concentration in finance and bachelor of science in mathematics. These programs provide a strong foundation in theoretical and applied mathematics with particular emphasis on the development of logical and analytical problem-solving skills. This major is often combined with a major or minor in natural sciences, computer science, business and related fields. It is an excellent preparation for graduate school in quantitative subjects. Math majors find jobs in academia, high-tech industry, financial industry and government agencies, with positions including research mathematician, applied mathematician, engineer, computer programmer, financial analyst, economist, actuary and teacher.

Mathematics

Bachelor of Arts Major Requirements

(183 credits required for the degree)

This degree requires completion of 48 credits of MATH courses numbered 1951 or higher, including at least 20 credits at the 3000-level or higher. The following courses are required:

MATH 1951Calculus I4
MATH 1952Calculus II4
or MATH 1962 Honors Calculus II
MATH 1953Calculus III4
or MATH 1963 Honors Calculus III
MATH 2060Elements of Linear Algebra4
MATH 2070Introduction to Differential Equations4
MATH 2080Calculus of Several Variables4
MATH 2200Mathematical Reasoning & Proof4
MATH 3161Introduction to Real Analysis4
MATH 3170Introduction to Abstract Algebra4
Additional Courses12
Total Credits48

Note

  • In addition, students must complete the ETS Major Field Test in Mathematics as instructed by the department, and satisfy all requirements for the Bachelor of Arts degree as outlined in the University of Denver Undergraduate Bulletin.
  • It is recommended that students take MATH 3151 Advanced Linear Algebra after MATH 2200 Mathematical Reasoning & Proof but before MATH 3161 Introduction to Real Analysis and MATH 3170 Introduction to Abstract Algebra.

Bachelor of Science Major Requirements 

(183 credits required for the degree)

This degree requires completion of 52 credits of MATH courses numbered 1951 or higher, including at least 24 credits at the 3000-level or higher. The following courses are required:

MATH 1951Calculus I4
MATH 1952Calculus II4
or MATH 1962 Honors Calculus II
MATH 1953Calculus III4
or MATH 1963 Honors Calculus III
MATH 2060Elements of Linear Algebra4
MATH 2070Introduction to Differential Equations4
MATH 2080Calculus of Several Variables4
MATH 2200Mathematical Reasoning & Proof4
MATH 3161Introduction to Real Analysis4
MATH 3170Introduction to Abstract Algebra4
A MATH course at the 3000-level or higher that (a) is different from MATH 3161 and MATH 3170 and (b) has either MATH 2200 or another 3000-level MATH course as a prerequisite.4
Additional courses12
Total Credits52

Notes

  • In addition, students must complete the ETS Major Field Test in Mathematics as instructed by the department, and satisfy all requirements for the Bachelor of Science degree as outlined in the University of Denver Undergraduate Bulletin.
  • It is recommended that students take MATH 3151 Advanced Linear Algebra after MATH 2200 Mathematical Reasoning & Proof but before MATH 3161 Introduction to Real Analysis and MATH 3170 Introduction to Abstract Algebra.

  • Students are encouraged to complete the analysis sequence MATH 3161 Introduction to Real Analysis, MATH 3162 Introduction to Real Analysis II, MATH 3110 Topology, or the algebra sequence MATH 3170 Introduction to Abstract Algebra, MATH 3166 Group Theory, MATH 3176 Rings and Fields.

Bachelor of Arts with a Concentration in Finance Major Requirements

(183 credits required for the degree)

This degree requires completion of 48 credits of MATH courses numbered 1951 or higher, including at least 20 credits at the 3000-level or higher. In addition, ACTG 2200, FIN 2800 and at least 12 credits in FIN courses at the 3000-level or higher are required. Students should work with Daniels advisors in order to make appropriate choices. The following courses are required:

Mathematics 1
MATH 1951Calculus I4
MATH 1952Calculus II4
or MATH 1962 Honors Calculus II
MATH 1953Calculus III4
or MATH 1963 Honors Calculus III
MATH 2060Elements of Linear Algebra4
MATH 2070Introduction to Differential Equations4
MATH 2080Calculus of Several Variables4
MATH 2200Mathematical Reasoning & Proof4
MATH 3080Introduction to Probability4
MATH 3161Introduction to Real Analysis4
Select one of the following:4
Introduction to Abstract Algebra
Advanced Linear Algebra
Math Electives8
Business 2
ACTG 2200Introduction to Financial Reporting4
FIN 2800Financial Decision Making4
FIN Electives12
Total Credits68

Notes

  • In addition, students must complete the ETS Major Field Test in Mathematics as instructed by the department, and satisfy all requirements for the Bachelor of Arts degree with concentration in Finance as outlined in the University of Denver Undergraduate Bulletin.
  • Students may enroll in ACTG 2200 Introduction to Financial Reporting only if they are formally enrolled in this degree, have completed MATH 2200 Mathematical Reasoning & Proof and have Microsoft Excel certification.
  • Students in this program should work with Daniels advisors in order to make appropriate choices for 3000-level FIN electives, and ensure that prerequisites are followed.
  • Students in this degree are not eligible for a Business Administration minor or a Finance minor. Good choices for complementary minors include economics, statistics, or computer science.

SECONDARY MAJOR

48 credits. Same requirements as for BA degree.

Minor Requirements

This minor requires completion of at least 20 credits in MATH courses numbered 1951 or higher. COMP 2300 Discrete Structures in Computer Science may be counted toward the math minor. Courses not covered by these requirements must be approved in writing by a mathematics faculty advisor.

Requirements for Distinction in the Major in Mathematics

  • Minimum 3.5 major GPA 
  • Completion of a thesis

The course plans below are intended to give students an example of how they might complete their degree requirements in four years. All of the required courses are included, but the quarters in which certain courses are taken can vary significantly from student to student.

For the first quarter of the first year, math majors are typically advised to take an FSEM, at least one math course (MATH 1951, or something else depending on credit earned through examinations) and enough other credits so that the student is enrolled in 16 or more credits total. Those additional other credits may include common curriculum courses, courses in a second major or minor, or electives.

Bachelor of Arts in Mathematics

First Year
FallCreditsWinterCreditsSpringCredits
FSEM 11114WRIT 11224WRIT 11334
MATH 19514MATH 19524MATH 19534
SI: Natural4SI: Natural4SI: Natural4
AI: Society4SI: Society4AI: Society4
 16 16 16
Second Year
FallCreditsWinterCreditsSpringCredits
MATH 20604MATH 20704MATH 20804
MATH 22004SI: Society4MATH 315114
Foreign Language4Foreign Language4Foreign Language4
Minor4Minor4Minor4
 INTZ 250131-2 
 16 17-18 16
Third Year
FallCreditsWinterCreditsSpringCredits
Study Abroad16MATH 317024MATH 316124
 Minor4Minor4
 Elective4Elective4
 Elective4Elective4
 16 16 16
Fourth Year
FallCreditsWinterCreditsSpringCredits
ASEM4MATH Elective4MATH Elective4
Elective4Elective4Elective4
Elective4Elective4Elective4
Elective4Elective4Elective4
 16 16 16
Total Credits: 193-194

Bachelor of Science in Mathematics

First Year
FallCreditsWinterCreditsSpringCredits
FSEM 11114WRIT 11224WRIT 11334
MATH 19514MATH 19524MATH 19534
SI: Natural4SI: Natural4SI: Natural4
AI: Society4SI: Society4AI: Society4
 16 16 16
Second Year
FallCreditsWinterCreditsSpringCredits
MATH 20604MATH 20704MATH 20804
MATH 22004SI: Society4MATH 315134
Foreign Language4Foreign Language4Foreign Language4
Minor 114Minor 14Minor 14
 INTZ 250121-2 
 16 17-18 16
Third Year
FallCreditsWinterCreditsSpringCredits
Study Abroad16MATH 317044MATH 316144
 Minor 14Minor 14
 Minor 24Minor 24
 Elective4Elective4
 16 16 16
Fourth Year
FallCreditsWinterCreditsSpringCredits
ASEM4MATH Elective4MATH Elective4
MATH Elective4Minor 24Minor 24
Minor 24Elective4Elective4
Elective4Elective4Elective4
 16 16 16
Total Credits: 193-194

Students are encouraged to complete the analysis sequence MATH 3161 Introduction to Real AnalysisMATH 3162 Introduction to Real Analysis IIMATH 3110 Topology, or the algebra sequence MATH 3170 Introduction to Abstract AlgebraMATH 3166 Group TheoryMATH 3176 Rings and Fields.

Bachelor of Arts with Concentration in Finance

First Year
FallCreditsWinterCreditsSpringCredits
FSEM 11114WRIT 11224WRIT 11334
MATH 19514MATH 19524MATH 19534
SI: Natural4SI: Natural4SI: Natural4
AI: Society4SI: Society4AI: Society4
 16 16 16
Second Year
FallCreditsWinterCreditsSpringCredits
MATH 20604MATH 20704MATH 20804
MATH 22004SI: Society4MATH 315124
Foreign Language4Foreign Language4Foreign Language4
Minor4Minor4Minor4
 INTZ 250111-2 
 16 17-18 16
Third Year
FallCreditsWinterCreditsSpringCredits
Study Abroad16MATH 30804MATH 316124
 ACTG 22004FIN 28004
 Minor4Minor4
 Elective4Elective4
 16 16 16
Fourth Year
FallCreditsWinterCreditsSpringCredits
MATH Elective4MATH Elective4FIN Elective4
FIN Elective4FIN Elective4Elective4
Elective4Elective4Elective4
Elective4Elective4Elective4
 16 16 16
Total Credits: 193-194

Courses

MATH 1010 Elements of College Algebra (2,4 Credits)

This course is designed to review the required algebra skills to be successful in Business Calculus. The following topics are covered: review of basic algebra, solving equations and inequalities, rectangular coordinate systems and graphing, polynomial and rational functions, exponential and logarithmic functions, and solving exponential and logarithmic equations. Students who completed a MATH course numbered 1200 or higher may not take this course.

MATH 1070 College Algebra and Trigonometry (4 Credits)

Selected topics in algebra and analytic trigonometry intended to prepare students for the calculus sequence. Cannot be used to satisfy the Analytical Inquiry: The Natural and Physical World requirement. Students who completed a MATH course numbered 1951 or higher may not take this course.

MATH 1150 Foundations Seminar (4 Credits)

The seminars offer challenging and interesting mathematical topics that require only high school mathematics. Examples of seminars are Introduction to Crytography, Patterns and Symmetry, Mathematical Art and Patterns of Voting. This course counts toward the Analytical Inquiry: The Natural and Physical World requirement.

MATH 1200 Calculus for Business and Social Sciences (4 Credits)

This is a one-quarter course for students in business, social sciences, and liberal arts. It covers elementary differential calculus with emphasis on applications to business and the social sciences. Topics include functions, graphs, limits, continuity, differentiation, and mathematical models. Students are required to attend weekly labs. This course counts toward the Analytical Inquiry: The Natural and Physical World requirement.

MATH 1951 Calculus I (4 Credits)

Limits, continuity, differentiation of functions of one variable, applications of the derivative. This course counts toward the Analytical Inquiry: The Natural and Physical World requirement. Prerequisite: MATH 1070 or equivalent.

MATH 1952 Calculus II (4 Credits)

Differentiation and integration of functions of one variable especially focusing on the theory, techniques and applications of integration. Prerequisite: MATH 1951.

MATH 1953 Calculus III (4 Credits)

Integration of functions of one variable, infinite sequences and series, polar coordinates, parametric equations. Prerequisite: MATH 1952 OR MATH 1962.

MATH 1962 Honors Calculus II (4 Credits)

Same topics as MATH 1952 treated rigorously and conceptually. Topics include differentiation and integration of functions of one variable especially focusing on the theory, techniques and applications of integration. Prerequisites: MATH 1951 and permission of instructor.

MATH 1963 Honors Calculus III (4 Credits)

Same topics as MATH 1953 treated rigorously and conceptually. Topics include integration of functions of one variable, infinite sequences and series, polar coordinates, parametric equations. Prerequisites: MATH 1952 or MATH 1962 and permission of instructor.

MATH 1992 Directed Study (1-10 Credits)

MATH 2050 Symbolic Logic (4 Credits)

Modern propositional logic; symbolization and calculus of predicates, especially predicates of relation. This course counts toward the Analytical Inquiry: The Natural and Physical World requirement. Cross listed with PHIL 2160.

MATH 2060 Elements of Linear Algebra (4 Credits)

Matrices, systems of linear equations, vectors, eigenvalues and eigenvectors; idea of a vector space; applications in the physical, social, engineering and life sciences. Prerequisite: MATH 1200 or MATH 1951.

MATH 2070 Introduction to Differential Equations (4 Credits)

Solution of linear differential equations; special techniques for nonlinear problems; mathematical modeling of problems from physical and biological sciences. Prerequisite: MATH 1953 or MATH 1963.

MATH 2080 Calculus of Several Variables (4 Credits)

Multivariable processes encountered in all sciences; multiple integration, partial differentiation and applications; algebra of vectors in Euclidean three-space; differentiation of scalar and vector functions. Prerequisite: MATH 1953 or MATH 1963.

MATH 2200 Mathematical Reasoning & Proof (4 Credits)

Introduction to theory of sets; relations and functions; logic, truth tables and propositional calculus; proof techniques; introduction to combinatorial techniques.

MATH 2992 Directed Study (1-10 Credits)

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 3010 History of Mathematics (4 Credits)

This course surveys major mathematical developments beginning with ancient Egyptians and Greeks and tracing the development through Hindu-Indian mathematics, Arabic mathematics, and European mathematics up to the 18th century. Prerequisite: MATH 1953 or MATH 1963.

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)

Zermelo-Fraenkel 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 truth-table semantics), first-order logic (axiomatization and completeness), elements of recursion theory, introduction to nonclassical logics. Prerequisite: MATH 2200.

MATH 3080 Introduction to Probability (4 Credits)

Basic probability models, combinatorial methods, random variables, independence, conditional probability, probability laws, applications to classical problems. Prerequisite: MATH 1953 or MATH 1963.

MATH 3090 Mathematical Probability (4 Credits)

Limit theorems for independent random variables, multivariate distributions, generating functions. Prerequisites: MATH 2080 and MATH 3080.

MATH 3110 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 4110.

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 G-sets, 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 3176 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 4176. Prerequisite: MATH 3170 or equivalent.

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)

Discrete-time 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, non-Euclidean and projective geometries. Prerequisite: MATH 2200.

MATH 3451 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 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, n-color 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 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 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 3991 Independent Study (1-10 Credits)

Cannot be arranged for any course that appears in regular course schedule for that particular year.

MATH 3992 Directed Study (1-10 Credits)

MATH 3995 Independent Research (1-10 Credits)

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

Jeremy Haefner, Professor, PhD, University of Wisconsin-Madison

James Hagler, Professor, Emeritus, PhD, University of California, Berkeley

Paul Horn, Associate Professor , PhD, University of California, San Diego

Chris Jennings-Shaffer, Visiting Assistant Professor, PhD, University of Florida

Shashank Kanade, Assistant Professor, PhD, Rutgers University

Michael Kinyon, Professor, PhD, University of Utah

Wojciech Kossek, Visiting Teaching Assistant Professor, PhD, North Dakota State University

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

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, Associate Professor, PhD, University of Arizona

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