# Mathematics (MATH)

Courses numbered 500 to 799 = *undergraduate/graduate*. (Individual courses may be limited to undergraduate students only.); 800 to 999 = *graduate*.

**MATH 501. Elementary Mathematics 5 credit hours**

A study of topics necessary to an understanding of the elementary school curriculum, such as set theory, real numbers and geometry. Not for major or minor credit. Prerequisites: elementary education major and MATH 111 or equivalent with a grade point of 2.000 or better, or departmental consent.

**MATH 502. Mathematics for Middle School Teachers 5 credit hours**

A study of the mathematical knowledge which forms the theoretical foundations of, the applications of, and extensions of middle school mathematics. This capstone course serves to reinforce mathematics skills learned in prerequisite courses and assists students in recognizing the unifying principles within their mathematical experiences. Prerequisites: MATH 111, 121, 123, 144, 501, and STAT 370 or equivalent with a grade point of 2.000 or better in each.

**MATH 511. Linear Algebra 3 credit hours**

An elementary study of linear algebra, including an examination of linear transformations and matrices over finite dimensional spaces. Prerequisite: MATH 243 with a grade point of 2.000 or better.

**MATH 513. Fundamental Concepts of Algebra 3 credit hours**

Defines group, ring and field, and studies their properties. Prerequisites: MATH 415 and 511 with a grade point of 2.000 or better, or departmental consent.

**MATH 525. Elementary Topology 3 credit hours**

Studies topological spaces, open and closed sets, bases for topology, continuous mappings, homeomorphisms, connectedness and compactness, Hausdorff and other spaces, with special emphasis on metric spaces. Prerequisite: MATH 415 with a grade point of 2.000 or better.

**MATH 530. Applied Combinatorics 3 credit hours**

Basic counting principles, occupancy problems, generating functions, recurrence relations, principles of inclusion and exclusion, the pigeonhole principle, Fibonacci sequences and elements of graph theory. Prerequisite: MATH 344 with a grade point of 2.000 or better.

**MATH 531. Introduction to the History of Mathematics 3 credit hours**

*General education advanced issues and perspectives course*. Studies the development of mathematics from antiquity to modern times. Solves problems using the methods of the historical period in which they arose. Requires mathematical skills. Prerequisites: MATH 511 and two additional courses at the 500 level or above, with a grade point of 2.000 or better in each.

**MATH 531H. Introduction to the History of Mathematics - Honors 3 credit hours**

*General education advanced issues and perspectives course*. Studies the development of mathematics from antiquity to modern times. Solves problems using the methods of the historical period in which they arose. Requires mathematical skills. Honors section. Prerequisites: MATH 511 and two additional courses at the 500 level or above, with a grade point of 2.000 or better in each.

**MATH 545. Integration Techniques and Applications 3 credit hours**

Studies the basic integration techniques used in applied mathematics. Includes the standard vector calculus treatment of line and surface integrals, Green's Theorem, Stokes's Theorem, and the Divergence Theorem. Also includes the study of improper integrals with application to special functions. Prerequisite: MATH 344 with a grade point of 2.000 or better.

**MATH 547. Advanced Calculus I 3 credit hours**

Covers the calculus of Euclidean space including the standard results concerning functions, sequences and limits. Prerequisites: MATH 344 and 415 with a grade point of 2.000 or better in each.

**MATH 548. Introduction to Complex Variables 3 credit hours**

Study of complex numbers, analytic functions, differentiation and integration of complex functions, line integrals, power series, residues and poles, and conformal mapping with applications. Prerequisites: MATH 344 with a grade point of 2.000 or better.

**MATH 551. Numerical Methods 3 credit hours**

Approximating roots of equations, interpolation and approximation, numerical differentiation and integration, and the numerical solution of first order ordinary differential equations. Some computer use. Prerequisites: MATH 344 and 451 with a grade point of 2.000 or better, or departmental consent.

**MATH 553. Mathematical Models 3 credit hours**

Covers case studies from the fields of engineering technology and the natural and social sciences. Emphasizes the mathematics involved. Each student completes a term project which is the solution of a particular problem approved by the instructor. Prerequisite: Math 344 with a grade point of 2.000 or better, or departmental consent.

**MATH 555. Differential Equations I 3 credit hours**

A study of first order equations including separation of variables and exact equations, second order equations including the general theory of initial value problems, constant coefficients, undetermined coefficients, variation of parameters and special methods of solution using power series and the Laplace transform methods. A standard course in differential equation for students in the sciences and engineering. Prerequisite: MATH 243 with a grade point of 2.000 or better, or departmental consent.

**MATH 555H. Differential Equations I - Honors 3 credit hours**

A study of first order equations including separation of variables and exact equations, second order equations including the general theory of initial value problems, constant coefficients, undetermined coefficients, variation of parameters and special methods of solution using power series and the Laplace transform methods. A standard course in differential equation for students in the sciences and engineering. Honors section. Prerequisite: MATH 243 with a grade point of 2.000 or better, or departmental consent.

**MATH 580. Selected Topics In Math 1-3 credit hours**

Topic chosen from topics not otherwise represented in the curriculum. May be repeated up to a maximum of 6 hours credit with departmental consent. Prerequisite: departmental consent.

**MATH 615. Elementary Number Theory 3 credit hours**

Studies properties of the integers by elementary means. Prerequisite: MATH 344 with a grade point of 2.000 or better, or departmental consent.

**MATH 621. Elementary Geometry 3 credit hours**

Studies Euclidean geometry from an advanced point of view. Prerequisite: MATH 344 with a grade point of 2.000 or better, or departmental consent.

**MATH 640. Advanced Calculus II 3 credit hours**

A continuation of MATH 547. Prerequisites: MATH 511 and 547 with a grade point of 2.000 or better in each.

**MATH 655. Differential Equations II 3 credit hours**

A continuation of MATH 555 (but with more emphasis on theoretical issues) that covers higher order differential equations, systems of first order equations (including the basics of linear algebra), some numerical methods, and stability and behavior of solutions for large times. Prerequisite: MATH 555 with a grade point of 2.000 or better, or departmental consent.

**MATH 657. Optimization Theory 3 credit hours**

Introduces selected topics in linear and nonlinear optimization. Develops the revised simplex method along with a careful treatment of duality. Then extends the theory to solve parametric, integer and mixed integer linear programs. Prerequisite: MATH 511 with a grade point of 2.000 or better.

**MATH 713. Abstract Algebra I 3 credit hours**

Treats the standard basic topics of abstract algebra. Prerequisite: MATH 513 with a grade point of 2.000 or better, or departmental consent.

**MATH 720. Modern Geometry 3 credit hours**

Examines the fundamental concepts of geometry. Prerequisite: MATH 513 with a grade point of 2.000 or better, or departmental consent.

**MATH 725. Topology I 3 credit hours**

Studies the results of point set and algebraic topology. Prerequisite: MATH 547 with a grade point of 2.000 or better, or departmental consent.

**MATH 743. Real Analysis I 3 credit hours**

Includes a study of the foundations of analysis and the fundamental results of the subject. Prerequisite: MATH 640 with a grade point of 2.000 or better, or departmental consent.

**MATH 745. Complex Analysis I 3 credit hours**

Studies the theory of analytic functions. Prerequisite: MATH 640 with a grade point of 2.000 or better, or departmental consent.

**MATH 751. Numerical Linear Algebra 3 credit hours**

Includes analysis of direct and iterative methods for the solution of linear systems, linear least squares problems, Eigenvalue problems, error analysis, and reduction by orthogonal transformations. Prerequisites: MATH 511, 547, 551 with a grade point of 2.000 or better in each, or departmental consent.

**MATH 753. Ordinary Differential Equations 3 credit hours**

Covers existence, uniqueness, stability and other qualitative theories of ordinary differential equations. Prerequisite: MATH 545 or 547 with a grade point of 2.000 or better, or departmental consent.

**MATH 755. Partial Differential Equations I 3 credit hours**

Studies the existence and uniqueness theory for boundary value problems of partial differential equations of all types. Prerequisite: MATH 547 with a grade point of 2.000 or better, or departmental consent.

**MATH 757. Partial Differential Equations for Engineers 3 credit hours**

Includes Fourier series, the Fourier integral, boundary value problems for the partial differential equations of mathematical physics, Bessel and Legendre functions, and linear systems of ordinary differential equations. Prerequisite: MATH 555 with a grade point of 2.000 or better.

**MATH 758. Complex and Vector Analysis for Engineers 3 credit hours**

A survey of some of the mathematical techniques needed in engineering including an introduction to vector analysis, line and surface integrals, and complex analysis, contour integrals and the method of residues. Not applicable toward a graduate degree in mathematics. Prerequisite: MATH 555 with a grade point of 2.000 or better.

**MATH 813. Abstract Algebra II 3 credit hours**

A continuation of MATH 713. Prerequisite: MATH 713 or equivalent.

**MATH 825. Topology II 3 credit hours**

A continuation of MATH 725. Prerequisite: MATH 725 or equivalent.

**MATH 828. Selected Topics Topology 2-3 credit hours**

Repeatable with departmental consent. Prerequisite: departmental consent.

**MATH 828G. Smooth Geometry 3 credit hours**

Begins with basic smooth manifold theory including topics such as vector bundles, tensor fields, flows, Lie derivatives and the theorems of Green, Gauss and Stokes. Introduces some elementary topics from Riemannian geometry.

**MATH 828I. Topics in Geometric Topology 3 credit hours**

Studies a variety of topics in low-dimensional topology including knot theory, the topological classification of surfaces and the classification of three-dimensional manifolds. With regard to the latter, students also briefly discuss Thurston's Geometrisation Conjecture.

**MATH 828J. Smooth Geometry II 3 credit hours**

Survey of topics in the areas of smooth manifold theory and Riemannian geometry. Emphasizes the Riemann Curvature Tensor and the relationship between topology and positive curvature.

**MATH 828K. Partial Differential Operator Theory 3 credit hours**

Begins with an introduction to the theory of smooth manifolds, fiber bundles, their section spaces, and operators on sections spaces from a categorical point of view. Introduces jet theory with the goal of proving Peetre's theorem. Studies sheaves, ringed spaces, and morphisms of each with the ultimate goal of proving Slovak's generalization of Peetre's theorem. Finally, investigates applications of the two main theorems.

**MATH 829. Selected Topics in Geometry 2-3 credit hours**

Repeatable with departmental consent. Prerequisite: departmental consent.

**MATH 843. Real Analysis II 3 credit hours**

A continuation of MATH 743. Prerequisite: MATH 743 or equivalent.

**MATH 845. Complex Analysis II 3 credit hours**

A continuation of MATH 745. Prerequisite: MATH 745 or equivalent.

**MATH 848. Calculus of Variations 3 credit hours**

Includes Euler-Lagrange equations, variational methods and applications to extremal problems in continuum mechanics. Prerequisite: MATH 547 or 757.

**MATH 849. Selected Topics in Analysis 2-3 credit hours**

Repeatable with departmental consent. Prerequisite: departmental consent.

**MATH 849G. Broad Topics in Analysis 3 credit hours**

Seminal publications in mathematical analysis will be introduced, examined and discussed and incorporated into the research of participating students.

**MATH 851. Numerical Analysis of Ordinary Differential Equations 3 credit hours**

Includes single-step and multi-step methods of ordinary differential equations, stability, consistency and convergence, error estimation, step size selection, stiff systems, and boundary value problems. Prerequisites: MATH 555, 751.

**MATH 852. Numerical Analysis of Partial Differential Equations 3 credit hours**

Includes analysis of algorithms for the solution of initial value problems and boundary value problems for systems of PDEs with applications to fluid flow, structural mechanics, electromagnetic theory and control theory. Prerequisite: MATH 751.

**MATH 854. Tensor Analysis with Applications 3 credit hours**

After introducing tensor analysis, considers applications to continuum mechanics, structural analysis and numerical grid generation. Prerequisite: MATH 545 or 757.

**MATH 856. Partial Differential Equations II 3 credit hours**

**MATH 857. Selected Topics in Engineering Mathematics 3 credit hours**

Advanced topics in mathematics of interest to engineering students, including tensor analysis, calculus of variations and partial differential equations. Not applicable toward the MS in mathematics.

**MATH 858. Selected Topics in Engineering Mathematics II 3 credit hours**

Advanced topics in mathematics of interest to engineering students, including tensor analysis, calculus of variations and partial differential equations. Not applicable toward the MS in mathematics.

**MATH 859. Selected Topics in Applied Mathematics 2-3 credit hours**

Repeatable with departmental consent.

**MATH 859K. Mathematical Physics 3 credit hours**

Advanced topics of mathematics as applied to problems in quantum computing and quantum information. Includes mappings, simulations and diffeomorphisms of quantum measures.

**MATH 859L. Mathematical Physics II 3 credit hours**

Advanced topics of mathematics as applied to problems in quantum computing and quantum information. Includes mappings, simulations and diffeomorphisms of quantum measures.

**MATH 859M. Computational Fluid Dynamics 3 credit hours**

A study of the fundamental equations in fluid mechanics. In particular, discusses numerical solvers of the Euler equations and, in general, conservation laws. The numerical methods are applied to fluid dynamical problems in plasma physics and nuclear fusion.

**MATH 880. Proseminar 1 credit hour**

Oral presentation of research in areas of interest to the students. Prerequisite: major standing.

**MATH 881. Individual Reading 1-5 credit hours**

Repeatable up to a maximum of 6 hours with departmental consent. Prerequisite: departmental consent.

**MATH 881Y. Inverse Problems 3 credit hours**

Based on the book: "Introduction to the Theory of Inverse Problems" A.L. Bukhgeim, VSP BV 2000, ISBN 90-6764-319-X. Covers chapters 2–5. Chapter 2. A priori estimates and the uniqueness of solutions of integro differential equations with operator coefficients; Chapter 2.1. Estimates of the Carleman type and their connection with the uniqueness of solutions of inverse problems; Chapter 2.2. Estimates for the Schrodinger equation with operator coefficients. Chapter 3. Inverse problems for differential equations; Chapter 3.1. One-dimensional inverse problem for the wave equation in linearized setting; Chapter 3.2. The method of transformation operators; Chapter 3.3. Uniqueness in multidimensional inverse problems in nonstationary and spectral statements. Chapter 4. Volterra operator equations and their applications; Chapter 4.1. Volterra operator equations in scales of Banach spaces; Chapter 4.2. Nonhyperbolic Cauchy problem for the wave equation; Chapter 4.3. The problem of integral geometry in a strip; Chapter 4.4. The inverse problem of variational calculus. Chapter 5. Foundations of the theory of conditionally well-posed problems; Chapter 5.1. Conditional well-posedness; Chapter 5.2. l-well-posedness of difference schemes; Chapter 5.3. Variational methods of solution of l-stable difference schemes.

**MATH 881Z. Topics in Geometry 3 credit hours**

Introductory course in geometry. Students learn about various classical geometries in two- and three- dimensions: Euclidean, projective, elliptic and hyperbolic.

**MATH 885. Thesis 1-4 credit hours**

May be repeated to a maximum of 6 hours credit. Graded S/U. Prerequisite: departmental consent.

**MATH 941. Applied Functional Analysis I 3 credit hours**

Introduces functional analysis and its applications. Prerequisites: MATH 843, 755 (MATH 755 may be a corequisite).

**MATH 942. Applied Functional Analysis II 3 credit hours**

Introduces functional analysis and its applications. Prerequisites: MATH 843, 755 (MATH 755 may be a corequisite).

**MATH 947. Theory Fluid Dynamics I 3 credit hours**

Mechanics of fluid flow, momentum and energy principles, Navier-Stokes and Euler equations, potential flows, vortex dynamics, stability analysis and numerical methods applied to fluid dynamics. Prerequisite: MATH 745.

**MATH 948. Theory Fluid Dynamics II 3 credit hours**

Mechanics of fluid flow, momentum and energy principles, Navier-Stokes and Euler equations, potential flows, vortex dynamics, stability analysis and numerical methods applied to fluid dynamics. Prerequisite: MATH 745.

**MATH 952. Advanced Topics in Numerical Analysis 3 credit hours**

Advanced topics of current research interest in numerical analysis. Topics chosen at instructor's discretion. Possible areas of concentration are numerical methods in ordinary differential equations, partial differential equations and linear algebra. Repeatable with departmental consent. Prerequisites: MATH 751, 851, and instructor's consent.

**MATH 958. Selected Advanced Topics in Applied Mathematics 3 credit hours**

Topics of current research interest in applied mathematics. Repeatable for credit with departmental consent. Prerequisite: instructor's consent.

**MATH 959. Selected Advanced Topics in Applied Mathematics 3 credit hours**

Topics of current research interest in applied mathematics. Repeatable for credit with departmental consent. Prerequisite: instructor's consent.

**MATH 981. Advanced Independent Study in Applied Mathematics 1-3 credit hours**

Arranged individual directed study in an area of applied mathematics. Repeatable to a maximum of 6 hours. Prerequisites: must have passed the PhD qualifying exam and instructor's consent.

**MATH 985. PhD Dissertation 1-9 credit hours**

Repeatable to a maximum of 24 hours. Graded S/U. Prerequisite: must have passed the PhD preliminary exam.