# Mathematics

## Program Description

An **Associate of Science in Mathematics **provides students with a reasonable exposure to foundational college-level mathematics. This degree program includes the Colorado Statewide General Education Core and meets the lower division general education requirements at most public institutions in Colorado. By completing this degree, students should be able to matriculate into a baccalaureate degree in mathematics with only 60 additional hours of coursework.

Pursuing a **Baccalaureate degree in Mathematics**, students develop powerful problem-solving, logical and critical thinking skills. By completing the required coursework, students gain an understanding of the nature of proof, a broad general understanding of mathematics and a deep understanding of at least one area of mathematics. Math majors also develop independent learning skills and oral and written mathematical communication skills. Mathematics majors get jobs in a wide variety of areas. CMU graduates have worked for local businesses, have run their own businesses and have worked for scientific companies. Other graduates have continued their educations by attending graduate school (in mathematics, computer science and engineering), law school, medical school and veterinary school.

Students who pursue a** Baccalaureate degree in Mathematics with a concentration in Applied Mathematics **will develop powerful problem-solving, logical, and critical thinking skills. Students will use methods of applied mathematics from areas including modeling, linear algebra, ordinary and partial differential equations, numerical methods, computer programming, and statistics to model and solve applied problems. Students will also gain an understanding of the nature of proof and will complete a senior capstone project that includes developing research skills, writing skills, and presentation skills. Applied mathematics graduates can choose to find work in a variety of areas within business, industry, and government or may choose to continue their educations by attending graduate school in areas such as applied mathematics, computer science, and engineering.

The** Baccalaureate degree in Mathematics with a concentration in Secondary Education **will prepare students to teach in both middle schools and in high schools. While completing this degree, students develop problem-solving and critical thinking skills and are introduced to the logical and historical development of mathematical ideas. Students also learn the professional skills in teaching methods and content necessary for secondary mathematics teachers. Nationally recommended curriculum guidelines are followed in order to ensure that graduates have the mathematical content and conceptual understanding necessary for all high school mathematics courses. Graduates from this program are in demand both locally and statewide with the scarcity of mathematics teachers.

With a** Baccalaureate degree in Mathematics with a concentration in Statistics**, students develop problem-solving, logical and critical thinking skills. While completing the required coursework, students gain an understanding of the nature of proof, a general understanding of mathematics and an understanding of statistical reasoning, necessary assumptions and the correct use of statistical analysis procedures. Students also develop statistical software skills and oral and written mathematical communication skills. The statistics concentration in mathematics prepares students for graduate work in statistics or to enter the job force. With some additional job-specific training, students entering the job market could function as applied statisticians working in areas such as actuarial science, wildlife management, marketing, quality control and epidemiology to name a few.

With a **Baccalaureate degree in Mathematics with a concentration in Actuarial Science**, students develop problem-solving, logical and critical thinking skills. While completing this degree, students develop a general understanding of mathematics and an understanding of statistical reasoning including using statistical software to aid in problem-solving and investigation, applying appropriate statistical procedures and drawing valid statistical conclusions. Coursework in economics and finance also helps prepare students for graduate work in actuarial science or to enter the job force. With some additional job-specific training, students entering the job market could function as actuaries in the insurance field or as applied statisticians working in areas such as risk management and marketing.

A** Minor in Mathematics **is a natural enhancement to many majors outside mathematics where an understanding of mathematics is needed (e.g. physics, computer science, chemistry, biology, geology). A minor in mathematics enables non-mathematics majors to complete a focused course of study in mathematics on a smaller scale.

A M**inor in Statistics **is a natural enhancement to many majors outside mathematics where an understanding of statistical analysis of data is needed (e.g. biology, business, psychology, sociology, history, human performance and wellness, political science). A minor in statistics enables non-mathematics majors to complete a focused course of study in statistics on a smaller scale.

The **Graduate Certificate in Applied Mathematics** is intended to provide licensed secondary mathematics teachers the credentials required by the Higher Learning Commission to teach concurrent enrollment classes and to enable other professionals to enhance their knowledge of applied mathematics. For more complete program information: Applied Mathematics (Graduate Certificate).

## Special Requirements

Additional expenses: A graphing calculator is recommended for several mathematics and statistics courses. See department for recommended models.

## Contact Information

Department of Mathematics and Statistics

Wubben Science 132

970.248.1407

## Math - Foundations (MATC)

**MATC 090 Introductory Algebra4 Credits**

Introduction to algebra with a review of basic arithmetic. Includes decimals, fractions, percentage, ratio, proportion, signed numbers, algebraic expressions, factoring, exponents and radicals, linear equations, functions and graphs.

**MATC 091 Intermediate Algebra4 Credits**

Further study in topics of algebra. Includes properties of real and complex numbers; laws of exponents and radicals; factoring polynomials; solving linear and quadratic equations and inequalities; rational expressions and complex fractions; introduction to functions and relations; applications.

**Prerequisites:** MATC 090 or equivalent, or appropriate Accuplacer score.

**MATC 096 Topics:1-4 Credits**

Course may be taken multiple times up to maximum of 15 credit hours.

## Mathematics (MATH)

**MATH 101 Review in Mathematics1 Credit**

Review of mathematical concepts and computations. Content will vary and topics will be chosen to prepare students for a specific subsequent course.

**Prerequisites:** Permission of instructor.

**MATH 105 Elements of Mathematics I3 Credits**

Mathematics for the prospective elementary teacher with an emphasis on understanding mathematical reasoning and processes. Topics include problem solving, set theory, number theory, numeration systems, the integers and rational numbers.

**Prerequisites:** Appropriate mathematics placement test score and interview, and permission of instructor.

**MATH 107 Career Math3 Credits**

Covers material designed for career technical or general studies students who need to study particular mathematical topics. Topics include measurement, algebra, geometry, trigonometry, graphs, and/or finance. These are presented on an introductory level and the emphasis is on applications.

**MATH 108 Technical Mathematics4 Credits**

Covers material designed for career technical or general studies students who need to study particular mathematical topics. Topics may include measurement, algebra, geometry, trigonometry, graphs, and/or finance. These are presented on an introductory level and the emphasis is on applications.

**MATH 110 College Mathematics-GTMA13 Credits**

Essential mathematical concepts for B.A. students. Topics include logic, set theory, solving equations, basic inequalities, combinatorics, probability, descriptive statistics, geometry, consumer mathematics and the appropriate use of calculators

**Prerequisites:** MATC 091 or equivalent or appropriate mathematics placement test score.

**Essential Learning Categories:** Mathematics See the program requirements list to determine the minimum level math needed**Colorado Guaranteed Transfer (GT) Pathways General Education Curriculum**

**MATH 113 College Algebra-GTMA14 Credits**

A college-level treatment of algebra. Topics include algebraic properties of the integers, rationals, real and complex numbers; techniques for manipulation of expressions; techniques for solving linear, non-linear, absolute value equations, and inequalities; techniques for solving systems of equations; the Cartesian plane, relations and functions; properties and graphs of polynomial, rational, exponential, logarithmic and inverse functions; conic sections.

**Prerequisites:** MATC 091 or equivalent, or appropriate mathematics placement test score.

**Essential Learning Categories:** Mathematics See the program requirements list to determine the minimum level math needed**Colorado Guaranteed Transfer (GT) Pathways General Education Curriculum**

**MATH 119 Precalculus Mathematics-GTMA15 Credits**

An in-depth treatment of the mathematics essential to Calculus. Topics include the Cartesian plane, functions; polynomial, rational, exponential, logarithmic, inverse, circular and trigonometric functions; solving inequalities and systems of equations Additional topics may include matrices, determinants and vectors.

**Prerequisites:** MATH 113 or equivalent, or appropriate mathematics placement test score.

**Essential Learning Categories:** Mathematics See the program requirements list to determine the minimum level math needed**Colorado Guaranteed Transfer (GT) Pathways General Education Curriculum**

**MATH 121 Calculus for Business3 Credits**

An introduction to calculus with an emphasis on applications to business and economics. Topics include linear and quadratic functions, limits, continuity, differentiation, integration, the logarithmic and exponential functions, and applications. Computer algebra systems will be used where applicable. Current college algebra skills and graphic calculator are required.

**Prerequisites:** MATH 113 or equivalent, or appropriate mathematics placement test score.

**MATH 127 Mathematics of Finance3 Credits**

Simple interest, simple discount, compound interest, continuously compounded interest, annuities, perpetuities, capitalization, determining payment size, determining outstanding principal, and constructing amortization schedules, including the derivation of mathematical formulae and the methods for solving many financial problems.

**Prerequisites:** MATH 113 or permission of instructor.

**MATH 130 Trigonometry3 Credits**

A college-level treatment of trigonometry. Topics include the Cartesian plane, functions, inverse functions, the circular function, trigonometric functions, graphs of trigonometric functions, trigonometric identities, solving trigonometric equations, inverse trigonometric functions, triangle solution techniques and vectors.

**Prerequisites:** MATH 113 or equivalent, or appropriate mathematics placement test score.

**MATH 135 Engineering Calculus I4 Credits**

Introduction to differentiation and integration of functions of a single variable. Emphasis on computational aspects. Includes functions, limits, continuity, differentiation, related rates, optimization problems, graphing, integration and applications.

**Prerequisites:** MATH 119, or appropriate mathematics placement score.

**MATH 136 Engineering Calculus II4 Credits**

Continuation of MATH 135 Engineering Calculus I. Includes techniques of integration, trigonometric and hyperbolic functions, inverse, logarithmic and exponential functions, sequences, series, conic sections, polar coordinates and parametric equations.

**Prerequisites:** MATH 135 or MATH 151.

**MATH 141 Analytical Geometry3 Credits**

A college-level treatment of analytic geometry. Topics include Cartesian coordinate systems, distance, parallel and perpendicular lines and planes, the locus of a condition, generalizations of lines, planes and parabolas, polar coordinates and vectors in two and three dimensions.

**Prerequisites:** MATH 130 or permission of instructor.

**MATH 146 Calculus for Biological Sciences5 Credits**

An introduction to calculus with an emphasis on applications to biology. Topics include functions, properties and graphs of polynomials, rational functions, the trigonometric, inverse, exponential and logarithmic functions, limits, continuity, differentiation, related rates, min-max problems,integration and applications of biology.

**Prerequisites:** MATH 113 or permission of instructor.

**MATH 147 Introduction to Computer Algebra Systems1 Credit**

Introduction to computer algebra using an appropriate computer algebra system (CAS) such as Maple, Mathematica, Derive, etc. Topics will include the syntax and simple programming of the CAS used. Assignments and projects will emphasize applications in Calculus.

**Prerequisites:** MATH 119.

**Corequisites:** MATH 151.

**MATH 149 Honors Mathematics-GTMA13 Credits**

An in-depth exploration of mathematical concepts, with an emphasis on the process of mathematical discovery. Topics are left to the discretion of the instructor, and typically include an introduction to more advanced topics such as group theory or graph theory. This course fulfills the essential learning requirement for students in the Honors Program.

**Prerequisites:** Permission of instructor.

**MATH 150 Topics and Careers in Mathematics1 Credit**

Introduction to the nature of mathematical thinking. Advanced topics and applications of mathematics and statistics will be presented at an introductory level. Career options will be investigated.

**Prerequisites:** MATH 151 or MATH 135 or MATH 146 (any of these courses may be taken concurrently with MATH 150).

**MATH 151 Calculus I-GT-MA15 Credits**

An introduction to differentiation and integration of functions of a single variable. Topics include functions, limits, continuity, differentiation, related rates, min-max problems, graphing, integration and applications.

**Prerequisites:** MATH 119, or appropriate mathematics placement test score.

**Essential Learning Categories:** Mathematics See the program requirements list to determine the minimum level math needed**Colorado Guaranteed Transfer (GT) Pathways General Education Curriculum**

**MATH 152 Calculus II5 Credits**

A continuation of MATH 151 Calculus I. Topics include techniques of integration, trigonometric and hyperbolic functions, inverse, logarithmic and exponential functions, sequences, series, conic sections, polar coordinates and parametric equations.

**Prerequisites:** MATH 151.

**MATH 196 Topics1-4 Credits**

Course may be taken multiple times up to maximum of 15 credit hours.

**MATH 205 Elements of Mathematics II-GTMA13 Credits**

Decimal numbers, probability, statistics, geometry, and the metric system. A continuation of MATH 105 designed for the prospective elementary teacher.

**Prerequisites:** MATH 105 or permission of instructor.

**Essential Learning Categories:** Mathematics See the program requirements list to determine the minimum level math needed**Colorado Guaranteed Transfer (GT) Pathways General Education Curriculum**

**MATH 225 Computational Linear Algebra3 Credits**

Computational approach to systems of equations, vector spaces, matrices, matrix transformations, eigenvalues, as well as their applications. Software for linear algebra computations is introduced and utilized.

**Prerequisites:** MATH 151 or MATH 135 or MATH 146.

**MATH 236 Differential Equations and Linear Algebra4 Credits**

Introduction to ordinary differential equations and linear algebra. Topics covered include ordinary differential equations, systems of linear equations, matrices, determinants, vector spaces, linear transformations, and systems of linear differential equations.

**Prerequisites:** MATH 152 or MATH 136.

**MATH 240 Introduction to Advanced Mathematics4 Credits**

An introduction to writing mathematical proofs. This course is designed to provide students with a transition from computationally-based lower level classes to proof-based upper level classes. The primary goal of the course is to train students to construct and analyze rigorous mathematical proofs. Topics include introductory logic, set theory, relations, functions, induction, equivalence relations, partitions and combinatorics.

**Prerequisites:** MATH 152.

**MATH 253 Calculus III4 Credits**

Vectors in three-dimensional space, vector functions, partial derivatives, directional derivative and multiple integrals.

**Prerequisites:** MATH 136 or MATH 152.

**MATH 260 Differential Equations3 Credits**

Techniques of solving differential equations of order one, linear differential equations, linear equations with constant coefficients, non-homogeneous equations, variation of parameter techniques, and Laplace transform methods.

**Prerequisites:** MATH 152 or MATH 136.

**MATH 296 Topics1-3 Credits**

Course may be taken multiple times up to maximum of 15 credit hours.

**MATH 301 Mathematics for Elementary Teachers3 Credits**

A selection of mathematics topics addressing content and standards for elementary education. Strong emphasis on written and oral communication.

**Prerequisites:** MATH 205 and formal acceptance into the Teacher Education Program, or permission of instructor.

**MATH 305 Euclidean Geometry3 Credits**

Development of Euclidean Geometry. Topics include basic concepts of logic, axiomatic proofs, inductive reasoning, analytic geometry, applications of technology, and van Hiele levels of learning. Intended for students seeking elementary teacher licensure.

**Prerequisites:** MATH 301, and MATH 151 or MATH 146.

**MATH 310 Number Theory3 Credits**

Classical number theory including the fundamental theorem of arithmetic, congruences, and linear diophantine equations.

**Prerequisites:** MATH 240.

**MATH 325 Linear Algebra3 Credits**

Proof-based treatment of linear algebra. Topics include vector spaces, linear transformations, eigenvalues, and orthogonality.

**Prerequisites:** MATH 225 and MATH 240.

**MATH 340 Ethnomathematics3 Credits**

Study of mathematics within cultures, especially small-scale indigenous cultures. Through the lens of culture, students can compare/contrast mathematics systems, their logical structures, and their modes of expression.

**Prerequisites:** MATH 240 or MATH 301 or permission of instructor.

**MATH 352 Advanced Calculus3 Credits**

Proof-based treatment of calculus of one real variable with focus on axiomatic development. Topics include completeness of the real numbers, limits, continuity, derivatives, integrals, and the Fundamental Theorem of Calculus.

**Prerequisites:** MATH 240.

**MATH 360 Methods of Applied Mathematics3 Credits**

Selection of techniques in applied mathematics of particular use to scientists and engineers. Topics include vector analysis, partial differential equations and transform techniques. Applications are stressed.

**Prerequisites:** MATH 253, and MATH 236 or MATH 260.

**MATH 361 Numerical Analysis4 Credits**

Elementary numerical analysis using the hand-held programmable calculator including Taylor's theorem, truncating errors, iteration processes, least squares methods, numerical solution of algebraic and transcendental equations, systems of equations, ordinary and partial differential equations, integral equations, interpolation, finite differences, eigenvalue problems, relaxation techniques, approximations, and error analysis.

**Prerequisites:** MATH 152 or MATH 136.

**MATH 362 Fourier Analysis3 Credits**

Introduction to continuous and discrete Fourier analysis. Topics include signals as vectors, matrices, and functions; orthogonality and correlation; expansions and transforms; Fourier series and frequency analysis; filtering, thresholding and compression; analysis of accuracy, including aliasing and convergence; Fourier and inverse Fourier transforms; discrete and inverse discrete Fourier transforms.

**Prerequisites:** MATH 152 or MATH 136.

**MATH 365 Mathematical Modeling3 Credits**

A bridge between calculus and the application of mathematics. Investigation of meaningful and practical problems chosen from experiences, encompassing the disciplines of mathematical sciences, operations research, engineering, management sciences and life sciences.

**Prerequisites:** STAT 200, MATH 152 or MATH 136, and one of the following: MATH 236, MATH 240, MATH 253, MATH 260, MATH 325, or permission of instructor.

**MATH 366 Methods of Applied Mathematics II3 Credits**

Treatment of numerical methods used to solve problems in applied mathematics. Topics include iteration, interpolation, numerical integration and differentiation, numerical linear algebra, numerical solutions of matrix eigenvalue problems, and numerical solutions of ordinary and partial differential equations.

**Prerequisites:** MATH 360; and CSCI 110/CSCI 110L or CSCI 111 or CSCI 130 or CSCI 310.

**MATH 369 Discrete Structures I3 Credits**

Elementary logic, induction, recursion, recurrence relations, sets, combinatorics, relations, functions, graphs, trees, and elementary abstract structures.

**Prerequisites:** MATH 152 or MATH 136; and CSCI 110/CSCI 110L or CSCI 111 or CSCI 130.

**MATH 370 Discrete Structures II3 Credits**

Applications of logic, Boolean algebra and computer logic, abstract structures, coding theory, finite-state machines, and computability.

**Prerequisites:** MATH 369 or both MATH 240 and CSCI 111.

**MATH 380 History of Mathematics3 Credits**

History of mathematics from antiquity to the present with emphasis upon the development of mathematics concepts and the people involved.

**Prerequisites:** MATH 152.

**MATH 386 Geometries4 Credits**

A study of Euclidean and non-Euclidean geometries. This course examines the differences in their axiom systems and their models, and how notions in Euclidean geometry are interpreted in non-Euclidean systems.

**Prerequisites:** MATH 240.

**MATH 389 Explorations in Mathematics for Elementary Educators1 Credit**

Broadening of future elementary educators' exposure to, and understanding of, diverse fields of mathematics through directed readings, analysis, and discussion.

**Prerequisites:** MATH 301 and permission of instructor.

**MATH 394 Mathematics Colloquium1 Credit**

A weekly series of talks on a wide range of contemporary mathematics will be given by local faculty and others. Students must provide written commentary on these talks.

**Prerequisites:** Permission of instructor.

**MATH 395 Independent Study1-3 Credits**

Course may be taken multiple times up to maximum of 6 credit hours.

**MATH 396 Topics1-3 Credits**

Course may be taken multiple times up to maximum of 15 credit hours.

**MATH 397 Structured Research1-4 Credits**

Mathematical research under the direct guidance of a faculty member. Designed for junior and senior level students.

**Prerequisites:** Permission of instructor.

Course may be taken multiple times up to maximum of 12 credit hours.

**MATH 420 Introduction to Topology3 Credits**

Introduction to point set topology. Topics include topological spaces, metric spaces, connectedness, compactness, the separation axioms, and the Tychonoff theorem.

**Prerequisites:** MATH 310 or MATH 325 or MATH 352.

**MATH 430 Mathematical Logic3 Credits**

Introduction to the classical areas of mathematical logic (model theory, proof theory, the theory of computation, complexity theory and set theory), the relationships these sub-disciplines have with each other and their relationships to the foundations of mathematics, computational science, computer science and the philosophy of mathematics.

**Prerequisites:** MATH 240 or MATH 369.

**MATH 450 Complex Variables3 Credits**

Algebra of complex numbers, analyticity, differentiation and integration of complex functions, Cauchy's integral formulae, and series.

**Prerequisites:** MATH 240.

**MATH 452 Intro to Real Analysis I3 Credits**

Introduction to real analysis from a general metric space perspective. Topics may include point set topology, completeness, compactness and connected sets, sequences, series, continuity, integration and sequences and series of functions.

**Prerequisites:** MATH 253 and MATH 352.

**MATH 453 Intro to Real Analysis II3 Credits**

Selected topics in advanced real analysis chosen by instructor.

**Prerequisites:** MATH 452.

**MATH 460 Advanced Linear Algebra3 Credits**

Characteristics and minimal polynomial, Cayley-Hamilton Theorem, invariant subspaces, bilinear forms, primary decomposition theorem, dual vector spaces.

**Prerequisites:** MATH 325.

**MATH 466 Methods of Applied Mathematics III3 Credits**

Exploration of advanced methods of applied mathematics with an emphasis on extending basic methods and concepts. Specific content may vary but will typically include contemporary techniques in applied mathematics, modeling and data analysis.

**Prerequisites:** MATH 366.

**MATH 484 Senior Seminar I2 Credits**

An introduction to conducting mathematical research with discussion of various research topics, including how to read and analyze articles in mathematics. Presentations and papers will be required.

**Prerequisites:** MATH 452 or MATH 490 or MATH 366 or STAT 350.

**MATH 490 Abstract Algebra I3 Credits**

Introduction to the theory of algebraic structures. Topics include groups, subgroups, cyclic groups, groups of permutations, homomorphisms, isomorphisms, the order of group elements, cosets, quotient structures, isomorphism theorems and an introduction to rings and fields.

**Prerequisites:** MATH 310.

**MATH 491 Abstract Algebra II3 Credits**

A continuation of MATH 490 Abstract Algebra I. Topics include properties of rings, subrings, ideals, quotient structures; ring homomorphisms and isomorphisms, integral domains, polynomial rings, properties of fields, subfields, field extensions, finite fields and Galois Theory.

**Prerequisites:** MATH 490.

**MATH 492 Senior Capstone3 Credits**

Exploration and communication of mathematical ideas and problems relevant to individual mathematics concentrations by integrating and extending material covered in mathematics coursework. Investigations may also include placing mathematics in historical, applied, professional and social contexts.

**Prerequisites:** Senior standing.

**Terms Typically Offered:** Fall.

**MATH 494 Senior Seminar II2 Credits**

Capstone course, with discussion of specialized topics and analysis of mathematical results, requiring students to interpret and present research. Subject matter will vary. Presentations and/or written research papers will be required.

**Prerequisites:** Permission of instructor.

**MATH 495 Independent Study1-3 Credits**

Course may be taken multiple times up to maximum of 6 credit hours.

**MATH 496 Topics1-3 Credits**

Course may be taken multiple times up to maximum of 15 credit hours.

**MATH 500 Introduction to Graduate Studies in Applied Mathematics3 Credits**

Introduction to methods and concepts of applied mathematics, including differentiation and integration of single and multivariate functions, vector calculus methods, matrix-vector computations, vector space concepts, and mathematical proofs.

**Prerequisites:** Acceptance into the Graduate Certificate in Applied Mathematics program.

**MATH 510 Applied Probability and Statistics3 Credits**

Develop a comprehension of, and an ability to perform, statistical methods that are most common in educational research. Emphases on statistical concepts that will further prepare teachers to teach introductory-level college statistics and critically examine and comprehend the data analysis in educational literature. Graphing calculators and computer software may be used to analyze and display data.

**Prerequisites:** Acceptance into the Graduate Certificate in Applied Mathematics program.

**MATH 520 Applied Numerical Methods3 Credits**

Exploration of fundamental algorithms and analysis of numerical methods commonly used by scientists, engineers, and mathematicians to approximately solve mathematical problems that are analytically impossible or intractable.

**Prerequisites:** MATH 500.

**MATH 530 Applied Mathematical Modeling3 Credits**

Investigation of applications of mathematics in the natural and social sciences, involving continuous, discrete, and probabilistic models. Survey of historical applications of mathematics in fields including chemistry, engineering, finance, ecology, and management; and creation of new models to address current questions in these fields. Involves model creation and model selection, analytical and computational methods of solving a model, and presentation of original work in a seminar setting.

**Prerequisites:** MATH 500.

**MATH 540 Applied Audio and Image Processing3 Credits**

Investigation of the mathematics behind the processing of sound waves and digital images. Both theory and computer-based applications will be explored, using methods of calculus, matrix-vector algebra, and inner product spaces.

**Prerequisites:** MATH 500.

**MATH 550 Mathematical Logic and Foundations in Mathematics3 Credits**

Study of logical systems, formal languages, satisfaction, deduction, correctness, completeness, applications to algebraic structures and orderings, construction of ordinal and cardinal numbers within axiomatic set theory, models of computation, undecidability, computational complexity, intractability, and introduction to themes within the philosophy of mathematics.

**Prerequisites:** Acceptance into the Graduate Certificate in Applied Mathematics program.

**MATH 560 Applied Number Theory3 Credits**

Applied treatment of number theory including prime numbers, congruences, quadratic residues and primitive roots.

**Prerequisites:** Acceptance into the Graduate Certificate in Applied Mathematics program.

**MATH 570 Applied Cryptography3 Credits**

Exploration of cryptography. Topics include number theory, classical ciphers, integer factorization, primality testing, public-key ciphers, digital signatures schemes, commitment schemes, elliptic curve methods, and applications to e-commerce. Additional topics upon student interest.

**Prerequisites:** MATH 560.

**MATH 596 Topics1-3 Credits**

Course may be taken multiple times up to maximum of 15 credit hours.