MAT 701 Vector and Tensor Analysis (3 Credit Hours)

Suggested topics are: Definition of vectors and transformation equations, general Cartesian co-ordinates; vector and scalar products, geometry of space curves; introduction to differential forms and tensors.

MAT 702N Research in Mathematics Education I (3 Credit Hours)

Prerequisites: Completion of a course in statistics and completion of the Measurement and Evaluation Standard

This course will allow the student to find and study models of accomplished researchers on the teaching of Mathematics at the secondary level. The course will examine necessary concepts in research validity; data gathering; instrumentation selection and construction; validation and reliability determinations; sampling techniques; and research designing. Further, the course will review the application of statistical models salient to designs utilized in conducting research which requires the testing of hypotheses that have been generated from problems in secondary school Mathematics. Open only to M.A.T. and M.Ed. candidates for degree credit.

MAT 703 Research in Mathematics Education II (3 Credit Hours)

Prerequisites: MAT 702N

This course is an extension of MAT 702N and will build upon the competencies and skills obtained in that Research Methods I course. Students will further develop their abilities to find and analyze contemporary research in the teaching of secondary school Mathematics.  Again, both qualitative and quantitative research will be emphasized. In this second course in Research Methods, students will be paying special attention to hypotheses, designs, statistical models and data gathering techniques of published research for the special purpose of designing their own research projects on topics germane to the teaching of secondary school Mathematics. Class instruction will also emphasizes guidelines for such research designing.  Open only to M.A.T. and M.Ed. candidates for degree credit.

MAT 704 Linear Algebra (3 Credit Hours)

Prerequisite: 6 Hours of Calculus and 3 Hours of linear algebra.

Topics include modules, linear dependence, matrix algebra, linear transformations, determinants, eigenvalues, linear systems, inner products, classical groups, diagonalization, symmetric matrices, function spaces, and differential operators.

MAT 705 Modern Plane Geometry (3 Credit Hours)

Prerequisite: Two semesters of Calculus

Suggested topics are: axiomatic approach to plane geometry, parallel postulate, Euclidean and hyperbolic geometries; quadratic extensions and angle trisection; plane measure.

MAT 706 Theory of Numbers (3 Credit Hours)

Prerequisite: Permission of the Department

Suggested topics are: properties of divisibility, linear congruences; quadratic congruences; prime numbers, continued fractions; number-theoretic functions; primitive roots and quadratic residues.

MAT 707 Mathematical Statistics (3 Credit Hours)

Prerequisite: 12 Hours of Calculus

A calculus-based study of probability and statistics. Topics include probability models, discrete and continuous random variables and their distributions, bivariate and multivariate distributions, sampling distributions, limit theorems, point and interval estimation, theory and applications of hypothesis testing, linear regression and correlation.

MAT 708 Introduction to Cryptography (3 Credit Hours)

Prerequisite: Mathematical maturity as demonstrated by any one of the following – at least 12 credits of undergraduate or graduate math courses, or a score of 700 or higher on the math SAT or GRE.

The study of methods of sending messages in disguised form, including some recent applications of number theory and group theory to public key cryptography. Topics include elementary number theory, finite fields, group theory, cryptosystems, and public key cryptography.

MAT 709 Complex Variables (3 Credit Hours)

Prerequisite: Real Analysis I or the equivalent

Complex numbers, analytic functions, derivatives and integrals of complex functions, Cauchy integral theorem and formula, Taylor and Laurent series, residues, maximum principles, conformal mapping, families of analytic functions and analytic continuation.

MAT 710 Foundations of Mathematics (3 Credit Hours)

Suggested topics are: propositional and predicate calculi, consistency and completeness of axiom systems, Godel’s theorem, axiomatic set theory, cardinal and ordinal numbers.

MAT 711 Real Analysis I (3 Credit Hours)

Prerequisite: 12 Hours of Calculus or the equivalent

Completeness, limits, continuity, convergence of sequences and series, derivatives, the Riemann integral, and theorems of Taylor, Bolzano-Weierstrass, and Heine-Borel together with applications.

MAT 712 Topology I (3 Credit Hours)

Prerequisite: Real Analysis I or the equivalent

Topics in analytic, geometric and combinatorial topology, with an emphasis on specific examples. Concepts covered include continuity, separation, compactness, connectedness, matrix spaces and the fundamental group.

MAT 713 Ordinary Differential Equations (3 Credit Hours)

Prerequisites: 9 hours of Calculus

Suggested topics are solutions of linear differential equations and systems of equations. Bessel and Legendre functions, Laplace transforms, series solutions, Sturm-Liouville theory, stability theory and singular points.

MAT 714 Algebraic Structures (3 Credit Hours)

Prerequisites: 3 hours of modern algebra or abstract algebra

The basic properties of groups, rings, integral domains and fields are quickly reviewed. A theoretical treatment of specific structures such as permutation groups, the ring of integers, polynomial rings, modular systems, and algebraic number fields is given.

MAT 715 Topics in Modern Geometry (3 Credit Hours)

Suggested topics are: homogeneous co-ordinates, cross ratio, quadratic involution on a line, fixed points, binary forms, binary quadratic forms, Jacobians, Hessians, Pluckerian line co-ordinates, cross ratio of a line pencil, poles and polars, conic as defined by Steiner, pencils of conics, tensors, measure in the plane, elliptic and hyperbolic geometry.

MAT 716 Analytic Number Theory (3 Credit Hours)

Prerequisites: Theory of Numbers and Complex Variables

Suggested topics are: Riemann zeta function, prime number theorem, L-functions, Dirichlet’s theorem, Waring’s problem, partitions, Goldbach’s conjecture.

MAT 721 Real Analysis II (3 Credit Hours)

Prerequisite: Real Analysis I

Suggested topics are functions of several real variables, Jacobians, implicit and inverse function theorems, vector analysis, theorems of Green, Gauss, and Stokes, with applications and additional topics as time permits.

MAT 723 Numerical Analysis (3 Credit Hours)

Prerequisite: Ordinary Differential Equations

Suggested topics are least-square polynomial approximation, numerical integration, root finding, numerical solution of differential equations, direct and iterative methods in matrix theory, analysis of numerical stability.

MAT 724 Abstract Algebra (3 Credit Hours)

Prerequisite: Linear Algebra

Suggested topics are: Sylow theorems, Jordan-Holder theorem, algebraic and transcendental field extensions, Galois theory, solvability of polynomial equations, ideal theory, modules.

MAT 725 Fractal Geometry (3 Credit Hours)

Prerequisite: Acceptance into either the Master of Science in Mathematics, Master of Arts in Teaching Mathematics or the Master of Science in Geo-Information Science program or permission of the Mathematics Graduate Program Coordinator.

A study of the geometry of fractal sets, self-similarity and fractal dimension. Suggested topics are: Iteration using the computer, graphical analysis, the Julia and Mandelbrot sets, chaos and applications to image compression, to dynamical systems and to computing the limiting perimeter and area enclosed by fractal sets.

MAT 731 Measure and Integration (3 Credit Hours)

Prerequisite: Real Analysis I or the equivalent

Suggested topics are: metric spaces, topological spaces, abstract measure; outer measure, absolute continuity, measure spaces, measurable functions, Lebesgue-Stieltjes integration, product measure, Caratheodory outer measure, L-spaces, the Radon-Nikodym theorem.

MAT 734 Linear and Multilinear Algebra (3 Credit Hours)

Prerequisite: Linear Algebra

Suggested topics are: canonical forms for matrices and linear transformations, quadratic forms, principal axis theorem, tensor products, exterior and symmetric algebras.

MAT 737 Operations Research (3 Credit Hours)

Prerequisite: 6 Hours of Calculus

The objective of this course is to teach students to design, solve, and apply operations research models to the analysis of systems problems in industry, business, or government. Suggested topics are linear programming, network analysis, dynamic programming, integer programming, nonlinear programming, queueing theory and inventory.

MAT/CSC 740 Computer Applications in Mathematics I (3 Credit Hours)

The FORTRAN language is introduced and used to illustrate computer methods in Calculus, Number Theory, Algebra, Statistics and Economics. Attention is paid to machine accuracy, error estimation and multiple-precision arithmetic. Assignments include the coding and running of programs in the Computer Laboratory. No previous computer experience required.

MAT/CSC 741 Computer Applications in Mathematics II (3 Credit Hours)

Prerequisite: MAT/CSC 740

Continuation of MAT/CSC 740. Further techniques of FORTRAN programming are discussed, with applications to transcendental equations, interpolation, optimization, modeling, simulation, and Physical Science.

MAT 747 Applied Statistical Inference (3 Credit Hours)

Prerequisite: Acceptance into either the Master of Science in Mathematics, Master of Arts in Teaching Mathematics or the Master of Science in Geo- Information Science program or permission of the Mathematics Graduate Program Coordinator.

A study of probability and statistical inference. Suggested topics are: Probability, discrete and continuous probability distributions, sampling distribution theory, confidence intervals, tests of statistical hypotheses, linear regression, and a nonparametric method: the Kolmogorov-Smirnov Goodness-of-Fit Test; applications to spatial statistics. The emphasis of the course is on applications and conceptual understanding, rather than on mathematical derivations.

MAT 750 History of Mathematics (3 Credit Hours)

Prerequisite: 9 Hours of Calculus

A survey course designed to deepen the student’s knowledge of the vast literature of mathematics. Historically influential concepts will be examined for their effects on mathematics and the culture in which they evolved. Philosophical and psychological comparisons will be made between the mathematical and scientific developments in Ancient Greek times, in the Renaissance and Newtonian times, and in the 19th and 20th centuries.

MAT 801 Differential Geometry (3 Credit Hours)

Prerequisite: Vector and Tensor Analysis

Suggested topics are: curves, vectors, curvature, hypersurfaces in R3, the sphere map and the Weingarten map, lines of curvature, tensors and forms, Gaussian curvatures, theorems on surfaces in the large, intrinsic geometry, connections, geodesics, Gauss-Bonnet formula.

MAT 804 Advanced Topics in Algebra (3 Credit Hours)

Prerequisite: Abstract Algebra

Suggested topics are: Module and ideal theory, Noetherian rings, local rings, structure of rings, introduction to categorical algebra.

MAT 807 Statistical Inference (3 Credit Hours)

Prerequisite: Mathematical Statistics

A continuation of MAT 707. Suggested topics are multiple regression, analysis of variance, decision functions, Bayes solutions, and nonparametric methods.

MAT 809 Theory of Functions of a Complex Variable (3 Credit Hours)

Prerequisite: Complex Variables

Suggested topics are: conformal mapping, Riemann mapping theorem, harmonic functions, Riemann surfaces, theorems of Weierstrass and Mittag Leffler, infinite products, entire functions.

MAT 812 Topology II (3 Credit Hours)

Prerequisite: Topology I or the equivalent

Suggested topics are: product topologies, Tychonoff’s theorem, paracompactness, metrization theorems, uniform spaces, topological groups.

MAT 813 Partial Differential Equations and Fourier Series (3 Credit Hours)

Prerequisite: Ordinary Differential Equations

Suggested topics are: wave equations, elliptic and parabolic equations; Fourier series; Sturm-Liouville theory and general Fourier expansions; eigenvalue expansions and Bessel functions.

MAT 816 Algebraic Number Theory (3 Credit Hours)

Prerequisite: Abstract Algebra

Suggested topics are: algebraic number fields, ideal theory in rings of algebraic integers, finiteness of class number, Dirichlet unit theorem, zeta functions.

MAT 821 Functional Analysis (3 Credit Hours)

Prerequisite: Measure and Integration

Suggested topics are: metric spaces, topological linear spaces, general theory of linear operators, spectral analysis of linear operators, spectral analysis in Hilbert space, the Stone-Weierstrass Theorem, introduction to Banach spaces, Hahn-Banach Theorem.

MAT 822 Introduction to Algebraic Topology (3 Credit Hours)

Prerequisites: Topology I, Abstract Algebra

Suggested topics are: homotopy theory (fundamental group, covering spaces), simplicial complexes, singular homology theory, products and Kunneth theorems.

MAT 831 Manifolds and Differential Forms (3 Credit Hours)

Suggested topics are: differential manifolds, differential forms, connections; Riemannian manifolds; operators on forms and integrations; Gauss-Bonnet formula and theory of rigidity; Pfaffian forms; Lie groups; DeRham’s Theorem.

MAT 900 Seminar: Independent Study (3 Credit Hours)

Open only for graduate students seeking the degree of M. Ed. in Elementary Education with a specialization in mathematics.

MAT 910 Seminar in Mathematics (3 Credit Hours)

Intended primarily for graduate students seeking the degree of Master of Arts in Teaching (Mathematics). The seminar will explore various topics in mathematics on an individual and group basis.

MAT 920 Seminar and Workshop in Teaching Mathematics (3 Credit Hours)

Intended primarily for graduate students seeking the degree of Master of Arts in Teaching (Mathematics). The purpose of this course is to explore applications of mathematical topics to the teaching of mathematics on the junior high and high school levels.

MAT 930 Seminar: Independent Study (3 Credit Hours)

Open only to students seeking the degree of Master of Science.