2026-2027 Academic Catalog

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Mathematics (MATH)

MATH 1004 - Discovering Mathematics I (1 credit) 
Introduction to the scope and applicability of mathematics and its many sub-disciplines. Introduction to the process of thinking, learning, and writing as a mathematician through topics such as logic systems, recreational mathematics, LaTeX programming, history, ethics, open problems, and research in mathematics. Also includes advising topics such as planning a Virginia Tech course of study. P/F only. Math majors.
Instructional Contact Hours: (1 Lec, 1 Crd) 
MATH 1014 - Precalculus with Transcendental Functions (3 credits) 
Precalculus college algebra, basic functions (algebraic, exponential, logarithmic, and trigonometric), conic sections, graphing techniques, basic probability. Usage of mathematical models, analytical calculations, and graphical or numerical representations of data to analyze problems from multiple disciplines that address intercultural and global challenges in areas such as chemistry, environmental science, the life sciences, finance, and statistics. Use of spreadsheet software. Two units of high school algebra and one of plane geometry are required.
Pathway Concept Area(s): 5F Quant & Comp Thnk Found., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 1025 - Elementary Calculus (3 credits) 
Quantitative and computational thinking to address relevant global issues. Unified calculus course covering techniques and applications of differential and integral calculus for functions of one variable. Constitutes the standard first-year mathematics courses for the life sciences. 1025: Differential calculus, graphing, applications for the life sciences, use of spreadsheet software. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit of trigonometry and precalculus. 1026: Integral calculus, numerical techniques, elementary differential equations, applications for the life sciences, use of spreadsheet and scientific software. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1026 and 1226.
Pathway Concept Area(s): 5F Quant & Comp Thnk Found., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 1026 - Elementary Calculus (3 credits) 
Quantitative and computational thinking to address relevant global issues. Unified calculus course covering techniques and applications of differential and integral calculus for functions of one variable. Constitutes the standard first-year mathematics courses for the life sciences. 1025: Differential calculus, graphing, applications for the life sciences, use of spreadsheet software. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit of trigonometry and precalculus. 1026: Integral calculus, numerical techniques, elementary differential equations, applications for the life sciences, use of spreadsheet and scientific software. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1026 and 1226.
Prerequisite(s): MATH 1025 or MATH 1225 
Pathway Concept Area(s): 5F Quant & Comp Thnk Found., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 1044 - Discovering Mathematics II (2 credits) 
Introduction to the scope and applicability of mathematics and its many sub-disciplines. Introduction to the process of thinking, learning, and writing as a mathematician through topics in pure and applied mathematics and a brief experience with mathematical research. Also includes advising topics such as planning a Virginia Tech course of study. Math majors.
Instructional Contact Hours: (2 Lec, 2 Crd) 
MATH 1214 - Preparation for Calculus (3 credits) 
Linear equations, polynomials, relations and functions, rational functions, quadratic equations, radicals and functions with rational exponents, exponentials, logarithms, trigonometric functions, trigonometric identities. Designed as preparation for MATH 1225: Calculus of a Single Variable.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 1225 - Calculus of a Single Variable (4 credits) 
1225-1226: CALCULUS OF A SINGLE VARIABLE Quantitative and computational thinking to address relevant intercultural and global issues. Unified calculus course covering techniques of differential and integral calculus for functions of one variable. Constitutes the standard first-year mathematics courses for science and engineering. 1225: limits, continuity, differentiation, transcendental functions, applications of differentiation, introduction to integration. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit each of trigonometry and precalculus, and placement by Math Dept. 1226: techniques and applications of integration, trapezoidal and Simpson’s rules, improper integrals, sequences and series, power series, parametric curves and polar coordinates, software-based techniques. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1525 and 1225. A student can earn credit for at most one of 1026 and 1226. Pre: Grade of at least C- in 1225 for 1226.
Prerequisite(s): MATH 1214 
Pathway Concept Area(s): 5F Quant & Comp Thnk Found., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (4 Lec, 4 Crd) 
MATH 1226 - Calculus of a Single Variable (4 credits) 
1225-1226: CALCULUS OF A SINGLE VARIABLE Quantitative and computational thinking to address relevant intercultural and global issues. Unified calculus course covering techniques of differential and integral calculus for functions of one variable. Constitutes the standard first-year mathematics courses for science and engineering. 1225: limits, continuity, differentiation, transcendental functions, applications of differentiation, introduction to integration. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit each of trigonometry and precalculus, and placement by Math Dept. 1226: techniques and applications of integration, trapezoidal and Simpson’s rules, improper integrals, sequences and series, power series, parametric curves and polar coordinates, software-based techniques. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1525 and 1225. A student can earn credit for at most one of 1026 and 1226. Pre: Grade of at least C- in 1225 for 1226.
Prerequisite(s): MATH 1225 
Pathway Concept Area(s): 5F Quant & Comp Thnk Found., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (4 Lec, 4 Crd) 
MATH 1454 - Introduction to Programming for Mathematical Problem-Solving (3 credits) 
Introduction to programming for mathematical problem solving. Programming software interfaces. Logic and conditional computations. Iterative computations and recursion. Data arrays. Compartmentalized computations using functions. Data visualization. Data input/output. Programming applications such as Monte Carlo simulation, random walks, computational geometry, and graph theory.
Corequisite(s): MATH 1225 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 1524 - Business Calculus (4 credits) 
Differential calculus techniques for functions of one and two variables. Emphasis on graphs, rates of change, and optimization of linear, quadratic, exponential, and logistic functions. Terminology and applications for business, including spreadsheet software. Mathematical models of real-world business problems, including discrete and continuous models, that address intercultural and global challenges in such areas as finance, marketing, and accounting. Assumes 2 units of high school algebra and 1 unit of geometry.
Pathway Concept Area(s): 5F Quant & Comp Thnk Found., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (4 Lec, 4 Crd) 
MATH 1535 - Geometry and Mathematics of Design (3 credits) 
A standard first-year mathematics sequence for architecture majors. Mathematical models of real-world problems, including discrete and continuous models, that address relevant global challenges in such areas as urban planning, building construction, and home design. 1535: Euclidean geometry, trigonometry, sequences and the golden ratio, graph theory, tilings, polygons and polyhedra, applications for 2- and 3-dimensional design and construction, use of geometric software. 1536: vectors in the plane and space, descriptive and projective geometry, differential and integral calculus, applications for 2- and 3-dimensional design and construction, including areas, volumes, centroids, and optimization. Assumes 2 unites of high school algebra and 1 unit of high school geometry.
Pathway Concept Area(s): 5F Quant & Comp Thnk Found., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 1536 - Geometry and Mathematics of Design (3 credits) 
A standard first-year mathematics sequence for architecture majors. Mathematical models of real-world problems, including discrete and continuous models, that address relevant global challenges in such areas as urban planning, building construction, and home design. 1535: Euclidean geometry, trigonometry, sequences and the golden ratio, graph theory, tilings, polygons and polyhedra, applications for 2- and 3-dimensional design and construction, use of geometric software. 1536: vectors in the plane and space, descriptive and projective geometry, differential and integral calculus, applications for 2- and 3-dimensional design and construction, including areas, volumes, centroids, and optimization. Assumes 2 unites of high school algebra and 1 unit of high school geometry.
Pathway Concept Area(s): 5F Quant & Comp Thnk Found., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 1614 - Numbers and Operations for Teachers (3 credits) 
Study of the nature and structure of numbers for prospective elementary and middle school teachers; number theory, number systems, operations and algebraic thinking, problem solving, and mathematical modeling. 1614 may not be taken by math majors for credit.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 1624 - Geometry for Teachers (3 credits) 
Study of key geometry concepts for prospective elementary and middle school teachers; multiple perspectives including transformational, coordinate, Euclidean and analytical geometry; geometric modeling; geometric and spatial reasoning. 1624 may not be taken by math majors for credit.
Prerequisite(s): MATH 1614 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 1984 - Special Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 1984E - Special Study (3 credits) 
Pathway Concept Area(s): 5F Quant & Comp Thnk Found. 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 2114 - Introduction to Linear Algebra (3 credits) 
Vector and matrix algebra systems of linear equations, linear equations, linear independence, bases, orthonormal bases, rank, linear transformations, diagonalization, implementation with contemporary software. Math 1226 or a grade of at least B in VT MATH 1225. A student can earn credit for at most one of 2114 and 2405H.
Prerequisite(s): MATH 1225 or MATH 1226  
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 2114H - Introduction to Linear Algebra (3 credits) 
Vector and matrix algebra systems of linear equations, linear equations, linear independence, bases, orthonormal bases, rank, linear transformations, diagonalization, implementation with contemporary software. Math 1226 or a grade of at least B in VT MATH 1225. A student can earn credit for at most one of 2114H and 2405H.
Prerequisite(s): MATH 1225 or MATH 1226  
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 2204 - Introduction to Multivariable Calculus (3 credits) 
Calculus for functions for several variables. Planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, software-based techniques. A student can earn credit for at most one of 2204 and 2406H. A student can earn credit for at most one of 2024 and 2204. A student can earn credit for at most one of 2204 and CMDA 2005.
Prerequisite(s): MATH 1226 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 2204H - Introduction to Multivariable Calculus (3 credits) 
Calculus for functions of several variables. Planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, software-based techniques. A student can earn credit for at most one of 2204H and 2406H. A student can earn credit for at most one of 2024 and 2204H. A student can earn credit for at most one of 2204H and CMDA 2005.
Prerequisite(s): MATH 1226 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 2214 - Introduction to Differential Equations (3 credits) 
Unified course in ordinary differential equations. First-order equations, second-and-higher-order constant coefficient linear equations, systems of first-order linear equations, and numerical methods. Mathematical models describing motion and cooling, predator-prey population models, SIR-models, mechanical vibrations, electric circuits, rates of chemical reactions, radioactive decay. Quantitative and computational thinking to address relevant intercultural and global issues. A student can earn credit for at most one of 2214 and 2406H. A student can earn credit for at most one of 2214 and CMDA 2006.
Prerequisite(s): (MATH 2114 or MATH 2114H or MATH 2405H) and MATH 1226 
Pathway Concept Area(s): 5A Quant & Comp Thnk Adv., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 2214H - Introduction to Differential Equations (3 credits) 
Unified course in ordinary differential equations. First-order equations, second-and-higher-order constant coefficient linear equations, systems of first-order linear equations, and numerical methods. Mathematical models describing motion and cooling, predator-prey population models, SIR-models, mechanical vibrations, electric circuits, rates of chemical reactions, radioactive decay. Quantitative and computational thinking to address relevant intercultural and global issues. A student can earn credit for at most one of 2214H and 2406H. A student can earn credit for at most one of 2214H and CMDA 2006
Prerequisite(s): (MATH 1114 or MATH 2114 or MATH 2114H or MATH 2405H or ISC 2105) and MATH 1226 
Pathway Concept Area(s): 5A Quant & Comp Thnk Adv., 11 Intercultural&Global Aware. 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 2405H - Mathematics in a Computational Context (5 credits) 
Unified course covering topics from linear algebra, differential equations, and calculus for functions of several variables. Comprises the standard second year mathematics courses for science and engineering. 2405H: Vector and matrix algebra, systems of linear equations, linear independence, bases, orthonormal bases, rank, linear transformations and diagonalization. Ordinary linear homogeneous differential equations, implementation with contemporary software. 2406H: Ordinary nonhomogeneous differential equations, calculus for functions of several variables, planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, with software-based techniques. A student can earn credit for at most one of 2114, 2114H, and 2405H. A student can earn credit for at most one of 2204, 2204H, and 2406H. A student can earn credit for at most one of 2214, 2214H, and 2406H.
Prerequisite(s): MATH 1226 
Instructional Contact Hours: (5 Lec, 5 Crd) 
MATH 2406H - Mathematics in a Computational Context (5 credits) 
Unified course covering topics from linear algebra, differential equations, and calculus for functions of several variables. Comprises the standard second year mathematics courses for science and engineering. 2405H: Vector and matrix algebra, systems of linear equations, linear independence, bases, orthonormal bases, rank, linear transformations and diagonalization. Ordinary linear homogeneous differential equations, implementation with contemporary software. 2406H: Ordinary nonhomogeneous differential equations, calculus for functions of several variables, planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, with software-based techniques. A student can earn credit for at most one of 2114, 2114H, and 2405H. A student can earn credit for at most one of 2204, 2204H, and 2406H. A student can earn credit for at most one of 2214, 2214H, and 2406H.
Prerequisite(s): MATH 2405H 
Instructional Contact Hours: (5 Lec, 5 Crd) 
MATH 2534 - Introduction to Discrete Mathematics (3 credits) 
Emphasis on topics relevant to computer science. Topics include logic, propositional calculus, set theory, relations, functions, mathematical induction, elementary number theory and Boolean algebra. Does not carry credit for mathematics majors, but may be used as though it were a 3000-level elective course for the mathematics minor. Two units of high school algebra, one unit of geometry, one-half unit each of trigonometry and precalculus mathematics required. 2534 may not be taken by math majors for credit without special permission. A student can earn credit for at most one of 2534 and 3034.
Prerequisite(s): CS 1114 or ECE 1574 or ECE 1004 or CS 2064 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 2644 - Mathematics Tutoring (1 credit) 
Introduction to professional, culturally respectful mathematics tutoring. Development of listening and questioning skills, assessment of students’ mathematical difficulties. Exploration of teaching and learning processes, effectively utilizing technology, and adjusting instruction to diversity in students’ mathematical reasoning. Concurrent mathematics tutoring experience required. May be repeated twice with different leadership expectations for a maximum of 3 credits.
Prerequisite(s): MATH 1226 
Instructional Contact Hours: (1 Lec, 1 Crd) 
Repeatability: up to 3 credit hours 
MATH 2964 - Field Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 2974 - Independent Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 2974H - Independent Study (1-19 credits) 
Honors section.
Instructional Contact Hours: Variable credit course 
MATH 2984 - Special Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 2984H - Special Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 2994 - Undergraduate Research (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 2994H - Undergraduate Research (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 3034 - Introduction to Proofs (3 credits) 
Practice in writing mathematical proofs. Exercises from set theory, number theory, and functions. Propositional logic, set operations, equivalence relations, methods of proof, mathematical induction, the division algorithm and images and pre-images of sets. A student can earn credit for at most one of 2534 and 3034.
Prerequisite(s): MATH 2114 or MATH 2114H or MATH 2405H 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 3124 - Modern Algebra (3 credits) 
Introduction to abstract algebraic structures (groups, rings, and fields) and structure-preserving maps (homomorphisms) for these structures. Proof-intensive course illustrating the rigorous development of a mathematical theory from initial axioms.
Prerequisite(s): MATH 3034 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 3134 - Applied Combinatorics and Graph Theory (3 credits) 
Emphasis on concepts related to computational theory and formal languages. Includes topics in graph theory such as paths, circuits, and trees. Topics from combinatorics such as permutations, generating functions, and recurrence relations.
Prerequisite(s): MATH 1226 and (MATH 2534 or MATH 3034) 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 3144 - Linear Algebra I (3 credits) 
Introductory course in linear algebra. Abstract vector spaces, linear transformations, algorithms for solving systems of linear equations, matrix analysis. This course involves mathematical proofs.
Prerequisite(s): (MATH 3034 or MATH 2534) and (MATH 2114 or MATH 2114H or MATH 2405H) 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 3214 - Vector Calculus (3 credits) 
Calculus concepts for scalar fields and vector fields. Derivative operators including gradient, curl, divergence, Jacobians, total derivatives. Parametrized curves and surfaces. Change-of-variable formulas for multiple integrals. Line integrals and surface integrals and their physical meaning. Fundamental Theorem of Line Integrals, Green's Theorem, Gauss's Theorem, Stokes's Theorem, and applications.
Prerequisite(s): MATH 2204 or MATH 2204H or MATH 2406H or CMDA 2005 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 3224 - Advanced Calculus (3 credits) 
Theory of limits, continuity, differentiation, integration, series. 3224 duplicates 4525.
Prerequisite(s): (MATH 2204 or MATH 2204H or MATH 2406H or CMDA 2005) and MATH 3034 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 3414 - Numerical Methods (3 credits) 
Computational methods for numerical solution of non-linear equations, differential equations, approximations, iterations, methods of least squares, and other topics. A grade of C or better required in CS prerequisite 1044 or 1705. A student can earn credit for at most one of 3414 and 4404.
Prerequisite(s): (CS 1044 or CS 1705 or CS 1114 or CS 1124) and MATH 2406H or (CMDA 2005 and CMDA 2006) or (MATH 2214 or MATH 2214H) and (MATH 2204 or MATH 2204H) 
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: CS 3414 
MATH 4044 - History of Mathematics (3 credits) 
Historical development of mathematics from antiquity to modern times. Senior standing in mathematics or mathematics education required.
Prerequisite(s): MATH 3034 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4124 - Introduction to Abstract Algebra (3 credits) 
An introduction to the theory of groups and rings. Topics include normal subgroups, permutation groups, Sylows Theorem, Abelian groups, Integral Domains, Ideals, and Polynomial Rings.
Prerequisite(s): MATH 3124 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4134 - Number Theory (3 credits) 
Introduction to elementary number theory. Topics covered may include divisibility, greatest common divisors, unique prime factorization, congruences, Fermat's Little Theorem, Chinese Remainder Theorem, multiplicative number-theoretic functions, Diophantine equations, primitive roots, and the Quadratic Reciprocity Law.
Prerequisite(s): MATH 3034 or MATH 3134 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4144 - Linear Algebra II (3 credits) 
Second course in linear algebra. Similarity invariants, Jordan canonical form, inner product spaces, self-adjoint operators, selected applications.
Prerequisite(s): MATH 3144 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4175 - Cryptography (3 credits) 
4175: Introduction to classical and modern symmetric-key cryptography; alphabetic ciphers, block ciphers and stream ciphers; background in modular arithmetic and probability; perfect secrecy; linear and differential cryptanalysis; Advanced Encryption Standard; hashing. 4176: Introduction to modern public-key cryptography and cryptanalysis; RSA algorithm, ElGamal algorithm, Diffie-Hellman algorithm; digital signatures; background in group theory and number theory; algorithms for primality testing, factoring, and discrete logarithms; elliptic curves.
Prerequisite(s): MATH 3034 or CMDA 3605 or MATH 3134 or MATH 3144 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4176 - Cryptography (3 credits) 
4175: Introduction to classical and modern symmetric-key cryptography; alphabetic ciphers, block ciphers and stream ciphers; background in modular arithmetic and probability; perfect secrecy; linear and differential cryptanalysis; Advanced Encryption Standard; hashing. 4176: Introduction to modern public-key cryptography and cryptanalysis; RSA algorithm, ElGamal algorithm, Diffie-Hellman algorithm; digital signatures; background in group theory and number theory; algorithms for primality testing, factoring, and discrete logarithms; elliptic curves.
Prerequisite(s): MATH 4175 or CMDA 3606 or MATH 3034 or MATH 3134 or MATH 3144 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4225 - Elementary Real Analysis (3 credits) 
Real number system, point set theory, limits, continuity, differentiation, integration, infinite series, sequences and series of functions.
Prerequisite(s): MATH 3224 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4226 - Elementary Real Analysis (3 credits) 
Real number system, point set theory, limits, continuity, differentiation, integration, infinite series, sequences and series of functions.
Prerequisite(s): MATH 4225 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4234 - Elementary Complex Analysis (3 credits) 
Analytic functions, complex integration, series representation of analytic functions, residues, conformal mapping, applications
Prerequisite(s): MATH 3224 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4245 - Intermediate Differential Equations (3 credits) 
Solution techniques, linear systems, the matrix exponential, existence theorems, stability, non-linear systems, eigenvalue problems.
Prerequisite(s): MATH 3224 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4254 - Chaos and Dynamical Systems (3 credits) 
Survey of basic concepts in chaotic dynamical systems. Includes material on bifurcation theory, conjugacy, stability, and symbolic dynamics.
Prerequisite(s): (MATH 3034 or MATH 2534) and MATH 3214 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4324 - Elementary Topology (3 credits) 
Basic concepts of topological spaces, continuous functions, connected spaces, compact spaces, and metric spaces.
Prerequisite(s): MATH 3224 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4334 - College Geometry (3 credits) 
Transformational approach to Euclidean geometry including an in-depth study of isometries and their application to symmetry, geometric constructions, congruence, coordinate geometry, and non-Euclidean geometries.
Prerequisite(s): MATH 3034 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4404 - Applied Numerical Methods (3 credits) 
Interpolation and approximation, numerical integration, solution of equations, matrices and eigenvalues, systems of equations, approximate solution of ordinary and partial differential equations. Applications to physical problems. A student can earn credit for at most one of 3414 and 4404.
Prerequisite(s): MATH 4564 and (ESM 2074 or AOE 2074) 
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: AOE 4404 
MATH 4414 - Issues in Scientific Computing (3 credits) 
Theory and techniques of modern computational mathematics, computing environments, computational linear algebra, optimization, approximation, parameter identification, finite difference and finite element methods and symbolic computation. Project-oriented course; modeling and analysis of physical systems using state-of-the-art software and packaged subroutines.
Prerequisite(s): (MATH 2214 or MATH 2214H or MATH 2406H or CMDA 2006) and MATH 3214 and (CS 1114 or CS 1064 or CS 1054 or CS 1044 or MATH 1454) 
Instructional Contact Hours: (2 Lec, 3 Lab, 3 Crd) 
Course Crosslist: CS 4414 
MATH 4425 - Fourier Series and Partial Differential Equations (3 credits) 
Separation of variables for heat, wave, and potential equations. Fourier expressions. Application to boundary value problems. Bessel functions. Integral transforms and problems on unbounded domains.
Prerequisite(s): MATH 2406H or CMDA 2006 or MATH 2214 or MATH 2214H and MATH 3224 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4426 - Fourier Series and Partial Differential Equations (3 credits) 
Separation of variables for heat, wave, and potential equations. Fourier expressions. Application to boundary value problems. Bessel functions. Integral transforms and problems on unbounded domains.
Prerequisite(s): MATH 4425 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4445 - Introduction to Numerical Analysis (3 credits) 
4445: Vector spaces and review of linear algebra, direct and iterative solutions of linear systems of equations, numerical solutions to the algebraic eigenvalue problem, solutions of general non-linear equations and systems of equations. 4446: Interpolation and approximation, numerical integration and differentiation, numerical solutions of ordinary differential equations. Computer programming skills required.
Prerequisite(s): MATH 2406H or (CMDA 2005 and CMDA 2006) or (MATH 2214 or MATH 2214H) and (MATH 2204 or MATH 2204H) 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4446 - Introduction to Numerical Analysis (3 credits) 
4445: Vector spaces and review of linear algebra, direct and iterative solutions of linear systems of equations, numerical solutions to the algebraic eigenvalue problem, solutions of general non-linear equations and systems of equations. 4446: Interpolation and approximation, numerical integration and differentiation, numerical solutions of ordinary differential equations. Computer programming skills required.
Prerequisite(s): MATH 2406H or (CMDA 2005 and CMDA 2006) or (MATH 2214 or MATH 2214H) and (MATH 2204 or MATH 2204H) 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4454 - Applied Mathematical Modeling (3 credits) 
Analysis of classical and modern applications of mathematics in the physical, biological and social sciences. Emphasis on problem formulating, modeling, solving, simulating, and analyzing results. Programming language required.
Prerequisite(s): MATH 3214 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4564 - Operational Methods for Engineers (3 credits) 
Laplace transformations, Fourier series, partial differential equations and separation of variables, boundary value problems, and Sturm-Liouville theory.
Prerequisite(s): (MATH 2214 or MATH 2214H) or MATH 2406H or CMDA 2006 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4574 - Vector and Complex Analysis for Engineers (3 credits) 
Vector Analysis: Greens theorem, potential theory, divergence, and Stokes theorem. Complex Analysis: Analyticity, complex integration, Taylor series, residues, conformal mapping, applications. 4574 may not be taken by math majors for credit.
Prerequisite(s): MATH 2204 or MATH 2204H or MATH 2406H or CMDA 2005 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4625 - Mathematics for Secondary Teachers (3 credits) 
Course activities will emphasize the curricular themes of problem solving, reasoning and proof, communication, connections, and representation. 4625: Topics in discrete mathematics and algebra from a secondary teaching perspective. 4626: Topics in trigonometry, geometry, measurement, statistics, and probability from a secondary teaching perspective.
Prerequisite(s): MATH 3034 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4626 - Mathematics for Secondary Teachers (3 credits) 
Course activities will emphasize the curricular themes of problem solving, reasoning and proof, communication, connections, and representation. 4625: Topics in discrete methematics and algebra from a secondary teaching perspective. 4626: Topics in trigonometry, geometry, measurement, statistics, and probability from a secondary teaching perspective.
Prerequisite(s): MATH 3034 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4644 - Secondary School Mathematics With Technology (3 credits) 
Use and impact of technology in secondary mathematics curriculum. Various technologies including graphing calculators, calculator based laboratory and probes (CBLs), computer algebra systems, spreadsheets, dynamic geometry software and the Internet will be used to explore secondary mathematical concepts from an advanced viewpoint.
Prerequisite(s): MATH 3034 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 4754 - Internship (1-19 credits) 
May be repeated for a maximum of 12 credits.
Instructional Contact Hours: Variable credit course 
Repeatability: up to 12 credit hours 
MATH 4964 - Field Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 4974 - Independent Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 4974H - Independent Study (1-19 credits) 
Honors section.
Instructional Contact Hours: Variable credit course 
MATH 4984 - Special Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 4994 - Undergraduate Research (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 4994H - Undergraduate Research (1-19 credits) 
Honors section.
Instructional Contact Hours: Variable credit course 
MATH 5114 - Specialized Topics in Algebra (3 credits) 
Various graduate level topics in algebra such as field theory, non-commutative ring theory, group representation theory, Lie algebras. May be taken for credit more than once with departmental permission.
Prerequisite(s): MATH 4124 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5125 - Abstract Algebra (3 credits) 
Groups, rings and ideals, modules, brief introduction to homological algebra.
Prerequisite(s): MATH 4124 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5126 - Abstract Algebra (3 credits) 
Groups, rings and ideals, modules, brief introduction to homological algebra.
Prerequisite(s): MATH 5125 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5134 - Coding Theory and Applications (3 credits) 
Introduction to error-correcting codes and coding theory. Linear codes, parameters, bounds, classical code families. Decoding algorithms, modern code design, applications. Pre: Graduate Standing
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5144 - Inverse Theory and Geoscience Applications (3 credits) 
Overview of inverse theory, utilizing geophysical examples to illustrate the concepts of model construction, parameter estimation, resolution, and non-uniqueness. Emphasis is on the linear problem, concluding with an overview of nonlinear inversion.
Prerequisite(s): MATH 2214 and MATH 2224 
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: GEOS 5144 
MATH 5174 - Mathematics of Public-Key Cryptography (3 credits) 
A study of the fundamentals of modern public-key cryptographic protocols and their underlying mathematics. Number theoretic and algebraic problems including primality testing, integer factorization, discrete logarithms, and quadratic residuosity. Algorithms for classical number theoretic problems. Public-key cryptosystems. Zero-knowledge proofs and the Fiat-Shamir transform. Digital signatures. Provable security. Selected advanced protocols. Pre: Graduate Standing. Background knowledge in abstract algebra or number theory.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5214 - Topics in Analysis (3 credits) 
Topics in mathematical analysis. Algebraic structures in analysis. Function spaces. Approximation schemes. This course may be repeated for credit, without limit on the number of credits earned, if the topic subtitle is not repeated. Pre: Graduate standing.
Instructional Contact Hours: (3 Lec, 3 Crd) 
Repeatability: up to 99 credit hours 
MATH 5225 - Real Analysis (3 credits) 
Basic set theoretic and topological notions, fundamental theorems of measure and integration, differentiation, applications to linear analysis.
Prerequisite(s): MATH 4226 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5226 - Real Analysis (3 credits) 
Basic set theoretic and topological notions, fundamental theorems of measure and integration, differentiation, applications to linear analysis.
Prerequisite(s): MATH 4226 and MATH 5225 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5235 - Complex Analysis (3 credits) 
Holomorphic functions, complex integration and residues, series expansions, normal families and the Riemann mapping theorem, entire functions, power series in the unit disk.
Prerequisite(s): MATH 3224 or MATH 4225 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5245 - Ordinary Differential Equations (3 credits) 
Existence theorems, linear theory, stability theory, periodic solutions, Poincare-Bendixon theory, boundary-value problems, functional differential equations.
Prerequisite(s): MATH 4225 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5254 - Nonlinear Systems (3 credits) 
Dynamics of conservative and nonconservative systems; phase planes; local and global stability; damping mechanisms; self-excited oscillators. Forced oscillations of one-degree-of-freedom systems; primary, secondary, and multiple resonances; period-multiplying bifurcations; strange attractors; chaos. Parametric excitations; Floquet theory; influence of damping and nonlinearity. Multi-degree-of-freedom systems; concepts of internal and external resonances; Hopf bifurcation. Applications to continuous systems; strings, beams, plates, and shells.
Prerequisite(s): ESM 5754 and ESM 5304 
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: ESM 5414, ME 5414 
MATH 5344 - Specialized Topics in Topology and Geometry (3 credits) 
Various graduate level topics in topology and geometry. May be taken for credit more than once with department consent.
Prerequisite(s): MATH 4324 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5414 - Topics in Applied Mathematics (3 credits) 
Various graduate level topics in applied mathematics such as discontinuous Galerkin methods for partial differential equations, stochastic differential equations and advanced partial differential equations. May be repeated with different topics and department consent. Pre: Graduate Standing.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5424 - Numerical Linear Algebra (3 credits) 
Matrix factorizations and iterative algorithms. Stability analysis, accuracy, and problem conditioning. Solving large systems of linear equations, linear least-squares, and eigenvalue problems. Pre: Graduate standing.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5425 - Applied Partial Differential Equations (3 credits) 
Partial differential equations of first and second order, hyperbolic equations, elliptic equations and Greens functions, parabolic equations, canonic forms, application to physics and engineering.
Prerequisite(s): MATH 4426 or MATH 4564 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5444 - Numerical Methods for Ordinary Differential Equations (3 credits) 
Computational procedures for ordinary differential equations including Runge-Kutta methods, variable-step Runge-Kutta methods, predictor-corrector methods, applications to two-point boundary-value problems and parameter estimation. Error control, relative and absolute stability, methods for stiff equations; with computer assignments. Senior standing in engineering, science, or mathematics, and some programming ability required.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5454 - Graph Theory (3 credits) 
Graphs, trees, connectivity, Euler tours, matching, independent sets and cliques, planar graphs, directed graphs. Consent required.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5464 - Combinatorics (3 credits) 
Counting problems, generating functions, recurrence relations, principle of inclusion and exclusion, experimental designs. Consent required.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5474 - Finite Difference Methods for Partial Differential Equations (3 credits) 
Finite difference methods for initial and boundary value problems for partial differential equations. Consistency, stability, convergence, dispersion, and dissipation. Methods for linear and nonlinear elliptic and parabolic equations, first- and second-order hyperbolic equations, and nonlinear conservation laws. High-level programming language required.
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: CS 5474 
MATH 5484 - Finite Element Methods for Partial Differential Equations (3 credits) 
Weak formulations of boundary-value problems for elliptic partial differential equations. Finite element spaces. Approximation theory for finite element spaces. Error estimates. Effects of numerical integration and curved boundaries. Nonconforming methods. Concrete examples of the application of the finite element method. Efficient implementation strategies. Time dependent problems. High-level programming language required.
Prerequisite(s): MATH 3414 and MATH 4525 
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: CS 5484 
MATH 5485 - Numerical Analysis and Software (3 credits) 
Presentation and analysis of numerical methods for solving common mathematical and physical problems. Methods of solving large sparse linear systems of equations, algebraic eigenvalue problems, and linear least squares problems. Numerical algorithms for solving constrained and unconstrained optimization problems. Numerical solutions of nonlinear algebraic systems. Convergence, error analysis. Hardware and software influences. Efficiency, accuracy, and reliability of software. Robust computer codes.
Prerequisite(s): MATH 4445 and MATH 4446 
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: CS 5485 
MATH 5486 - Numerical Analysis and Software (3 credits) 
Presentation and analysis of numerical methods for solving common mathematical and physical problems. Methods of solving large sparse linear systems of equations, algebraic eigenvalue problems, and linear least squares problems. Numerical algorithms for solving constrained and unconstrained optimization problems. Numerical solutions of nonlinear algebraic systems. Convergence, error analysis. Hardware and software influences. Efficiency, accuracy, and reliability of software. Robust computer codes.
Prerequisite(s): MATH 4445 and MATH 4446 
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: CS 5486 
MATH 5514 - Topics in Mathematical Biology (3 credits) 
Various graduate level topics in mathematics as they apply to biological problems. Representation of biological phenomena as mathematical models, and derivation of model features with analytical techniques and numerical simulation. Depending on the semester, may cover the following topics: applied stochastic modeling and Markov chains; multi-scale dynamics of infectious disease; or network systems. May be repeated 2 times with different content for a maximum of 9 credits. Pre: Graduate standing.
Instructional Contact Hours: (3 Lec, 3 Crd) 
Repeatability: up to 9 credit hours 
MATH 5515 - Mathematical Methods for Modeling and Simulation of Biological Systems (3 credits) 
Introduction to mathematical techniques for modeling and simulation, parameter identification and analysis of biological systems. Emphasis on both theoretical and practical issues and methods of computation, with concrete applications. Suitable for students from the mathematical and life sciences who have a basic foundation in multivariate calculus and ordinary differential equations. 5515: Continuous models and methods. 5516: Discrete models and methods.
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: GBCB 5515 
MATH 5516 - Mathematical Methods for Modeling and Simulation of Biological Systems (3 credits) 
Introduction to mathematical techniques for modeling and simulation, parameter identification and analysis of biological systems. Emphasis on both theoretical and practical issues and methods of computation, with concrete applications. Suitable for students from the mathematical and life sciences who have a basic foundation in multivariate calculus and ordinary differential equations. 5515: Continuous models and methods. 5516: Discrete models and methods.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5524 - Matrix Theory (3 credits) 
Determinants, rank, linear systems, eigenvalues, diagonalization, Gram-Schmidt process, Hermitian and unitary matrices, Jordan canonical form, variational principles, perturbation theory, Courant minimax theorem, Weyls inequality, numerical methods for solving linear systems and for determining eigenvalues. science or engineering.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5544 - Mathematical Optimization for Machine Learning (3 credits) 
Formulation and analyses of mathematical problems to minimize or maximize functions. Convex, nonlinear, constrained, largescale, and stochastic optimization problems. Gradient based and higher order optimization methods. Convergence analysis and implementation of optimization strategies. Accuracy and performance trade-offs. Introduction to neural networks, backpropagation, supervised learning, and machine and deep learning strategies, including stochastic approximation and sample average approximation. Regression and classification applications. Pre: Graduate standing.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5545 - Calculus of Variations and Optimal Control Theory (3 credits) 
Unified course in the calculus of variations and control theory including multiple integral problems and distributed parameter control systems. Necessary conditions, sufficient conditions, nonclassical problems, optimal control, distributed parameter control, computational methods.
Prerequisite(s): MATH 4525 and MATH 4425 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5546 - Calculus of Variations and Optimal Control Theory (3 credits) 
Unified course in the calculus of variations and control theory including multiple integral problems and distributed parameter control systems. Necessary conditions, sufficient conditions, nonclassical problems, optimal control, distributed parameter control, computational methods.
Prerequisite(s): MATH 4525 and MATH 4425 and MATH 5545 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5554 - Approximation Theory (3 credits) 
Algorithms for calculating the best approximations in the uniform norm and in inner-product spaces. Weierstrass and Jackson theorems. Lagrange and Hermite interpolation. Function approximation by piecewise and rational polynomial approximation. Approximation of linear functionals including numerical differentiation and numerical integration rules. Applications include numerical methods for approximating solutions to differential equations and linear systems theory. Pre: Graduate standing.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5564 - Model Reduction: System-Theoretic Methods (3 credits) 
Transfer functions, reachability and observability, Gramians. Balanced truncation. Interpolatory model reduction and optimal interpolation. Data-driven modeling. Parametric Systems. Balanced truncation for nonlinear systems. Interpolatory model reduction for nonlinear systems. Pre: Graduate standing.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5624 - Research on Mathematical Knowing and Learning (3 credits) 
Methods for conducting scientific research on how K-12 students learn mathematics. Students will develop skills for designing and conducting such research across various domains of mathematical learning, such as whole numbers, fractions, algebra, and calculus. Pre: Graduate Standing.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5634 - Research in Undergraduate Mathematics Education (3 credits) 
Research on undergraduate mathematics education. Student understanding of particular concepts from undergraduate math courses, from calculus to abstract algebra. Student engagement in mathematical practices, such as proving and defining, that transcend particular course concepts. Pre: Graduate Standing.
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 5754 - Introduction to Perturbation Methods (3 credits) 
Asymptotic expansions and series, approximate solutions of algebraic equations, straightforward expansions and their regions of nonuniformities, the Lindstedt-Poincare technique, the method of renormalization, the method of averaging, the method of matched asymptotic expansions.
Prerequisite(s): MATH 2214 or MATH 2514 or MATH 4544 
Instructional Contact Hours: (3 Lec, 3 Crd) 
Course Crosslist: ESM 5754 
MATH 5974 - Independent Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 5984 - Special Study (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 5994 - Research and Thesis (1-19 credits) 
Instructional Contact Hours: Variable credit course 
MATH 6255 - Functional Analysis (3 credits) 
Banach spaces, Hilbert spaces, linear operators on Banach and Hilbert spaces, Riesz Representation Theorems, spectral theory, topological vector spaces, other topics in functional analysis.
Prerequisite(s): MATH 5226 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 6256 - Functional Analysis (3 credits) 
Banach spaces, Hilbert spaces, linear operators on Banach and Hilbert spaces, Riesz Representation Theorems, spectral theory, topological vector spaces, other topics in functional analysis.
Prerequisite(s): MATH 5226 and MATH 6255 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 6324 - Topics in Topology and Geometry (3 credits) 
Advanced topics in topology and geometry. May be taken for credit more than once with department consent, with different content.
Prerequisite(s): MATH 5344 
Instructional Contact Hours: (3 Lec, 3 Crd) 
MATH 7994 - Research and Dissertation (1-19 credits) 
Instructional Contact Hours: Variable credit course