Graduate Courses

MATH 6308 TEACHING OF ALGEBRAIC CONCEPTS

This course examines issues, trends, and research related to the teaching/learning of secondary school algebra. Topics include historical items with major influence in algebra and a study of relationships between abstract and school algebra. Prerequisite: Math 4351 or equivalent.

MATH 6310 TOPICS IN MATHEMATICS TEACHING

This course examines issues, trends, and research related to the teaching/learning of secondary school mathematics. Specific topics will vary, but could include: technology in the classroom, mathematical problem solving and the use of applications in the teaching of mathematics.
Prerequisite: Graduate standing in mathematics.

MATH 6312 TEACHING OF GEOMETRIC CONCEPTS

This course examines issues, trends, and research related to the teaching/learning of secondary school geometry. Topics include the historical significance of geometry, the relationship between modern geometry and the geometry taught in schools, and the van Hiele model of geometric understanding.
Prerequisite: Math 3304 or consent of instructor.

MATH 6328 SPECIAL PROBLEMS IN TEACHING MATHEMATICS

A critical analysis of issues, trends and historical developments in elementary and/or secondary mathematics teaching with emphasis on the areas of curriculum and methodology. This course may be repeated for credit when topic changes.

MATH 6330 LINEAR ALGEBRA

Topics include the proof-based theory of matrices, determinants, vector spaces, linear spaces, linear transformations and their matrix representations, linear systems, linear operators, eigenvalues and eigenvectors, invariant subspaces of operators, spectral decompositions, functions of operators, and applications to science, industry, and business. Prerequisite: Math 2345 Elementary Linear Algebra
Student Learning Outcomes: After completing this course students will

Develop analytical skills to prove propositions, theorems of linear algebra, i.e., understand what constitutes a valid proof of results in linear algebra and learn how to create such proofs;
Understand the definitions, properties, and examples of vector spaces, subspaces, and their sums and direct sums;
Know the theory of linear maps, eigenvalues and eigenvectors, characteristic polynomials;
Understand inner-product spaces, norms, orthonormal bases, operators on inner-product spaces;
Learn spectral theory, singular value decomposition and applications of linear algebra;
Develop skills to use linear algebra in other related mathematical fields.

MATH 6331 ALGEBRA I

This course is an extension of the undergraduate course in abstract algebra. Topics include polynomial rings over a field and finite field extensions.
Prerequisite: Math 4351 or Math 6401.

Student Learning Outcomes: Students will, by the conclusion of this course:

Demonstrate a knowledge of the definitions of groups, rings, fields, and associated structures.
Demonstrate knowledge, understanding, and a proficiency in utilizing the core theorems of abstract algebra.
Demonstrate a sound conceptual understanding of abstract algebra through the construction of mathematically rigorous and logically correct proofs.
Be able to give examples of the structures of abstract algebra from various subjects in mathematics and the real world.
Demonstrate how mathematical material taught in Elementary, Middle, and High School mathematics (and more generally, science and art classes) reflects the structures of groups, rings, and fields.

MATH 6332 ALGEBRA II

The purpose of this course is to provide essential background in groups, rings, and fields, train the student to recognize algebraic structures in various settings, and apply the tools and techniques made available by algebraic structures. Topics include: groups, structure of groups, rings, modules, Galois theory, structure of fields, commutative rings and modules. Prerequisite: Math 6331.

Student Learning Outcomes: Students will, by the conclusion of this course:

Demonstrate a knowledge of the definitions of groups, rings, fields, and associated structures.
Demonstrate knowledge, understanding, and a proficiency in utilizing the core theorems of abstract algebra.
Demonstrate a sound conceptual understanding of abstract algebra through the construction of mathematically rigorous and logically correct proofs.
Be able to give examples of the structures of abstract algebra from various subjects in mathematics and the real world.
Demonstrate how mathematical material taught in Elementary, Middle, and High School mathematics (and more generally, science and art classes) reflects the structures of groups, rings, and fields.

MATH 6337 ADVANCED NUMBER THEORY

Topics include the Mobius Inversion Formula, primitive roots, quadratic reciprocity, continued fractions, nonlinear Diophantine equations, sums of squares, and primality testing.
Prerequisite: MATH 4351 or MATH 6401 with a grade of "C" or better.
Student Learning Outcomes: After completing this course students will be able to

Prove theorems involving number-theoretic concepts including divisibility, congruences, multiplicative functions, Pythagorean triples, primitive roots, quadratic residues, and continued fractions;
Use summatory functions and the Mobius Inversion formula to create complex multiplicative functions from simpler ones;
Use number-theoretic concepts to solve certain types of linear and nonlinear Diophantine equations;
Use quadratic reciprocity to determine whether a given quadratic equation in modular arithmetic has solutions;
Apply number-theoretic concepts to solve problems involving cryptology, check digits, scheduling round-robin tournaments, primality testing, and electronic coin flipping.

MATH 6352 ANALYSIS I

The purpose of this course is to provide the necessary elementary background for all branches of modern mathematics involving analysis and to train the student in the use of axiomatic methods. Topics include: metric spaces, sequences, limits, continuity, function spaces, series, differentiation, the Riemann integral.
Prerequisite: Math 4357 or Math 6402.
Student Learning Outcomes: After completing this course students will be able to

Carefully state and be able to apply the major definitions and theorems of real analysis.
Understand the definitions, axioms, and major theorems underlying the terms metric space, open and closed sets, connected sets, convergent and Cauchy sequences, complete metric space, limit, continuity, derivative, and Riemann integral as these concepts relate to metric spaces.
Be able to apply the concepts of metric spaces, continuity, derivative, and Riemann integral to solve problems in which their use is fundamental to obtaining and understanding the solution.
Understand what constitutes a valid proof of results in analysis and create such proofs.
Be able to write mathematics in a precise, effective, and understandable way.

MATH 6353 ANALYSIS II

The purpose of this course is to present advanced topics in analysis. Topics may be chosen from (but not restricted to) normed linear spaces, Hilbert spaces, elementary spectral theory, complex analysis, measure and integration theory. Prerequisite: Math 6352.
Student Learning Outcomes: After completing this course students will be able to

Carefully state and be able to apply the major definitions and theorems of modern analysis.
Understand the definitions, axioms, and major theorems underlying the terms infinite series of functions, measurable sets, Lebesque measure, measurable functions, the Lebesgue integral, normed linear spaces, inner product spaces, and Lp function spaces.
Be able to apply the concepts of infinite series of functions, Lebesque integral, and Lp spaces to solve problems in which their use is fundamental to obtaining and understanding the solution.
Understand what constitutes a valid proof of results in analysis and create such proofs.
Be able to write mathematics in a precise, effective, and understandable way.

MATH 6359 APPLIED ANALYSIS

This course provides an introduction to methods and applications of mathematical analysis. Topics include: function spaces, linear spaces; linear product spaces, Banach and Hilbert spaces; linear operators on Hubert spaces, eigenvalues and eigenvectors of operators, and orthogonal systems; Green's functions as inverse operators; relations between integral and ordinary differential equations, and methods of solving integral equations. Some special functions important fur applications arc shown.
Prerequisite: Math 2345, Math 3349, and Math 4318, or equivalent. Math 6352 is recommended.
Student Learning Outcomes: After completing this course students will be able to

Know the basic function spaces.
Understand the properties of linear operators on these spaces.
Know the Banach fixed point theorem and its applications.
Understand relations between differential and integral equations.
Know the basic types and methods of solving integral equations.
Construct Green functions for ordinary differential equations.
Know some basic special functions.

MATH 6360 ORDINARY DIFFERENTIAL EQUATIONS

This course examines existence and uniqueness of and methods for calculating solutions to systems of ordinary differential equations, the study of algebraic and qualitative properties of solutions, iterative methods for numerical solutions of ordinary differential equations, and an introduction to the finite element method.
Prerequisite: Math 3349 or consent of instructor.
Student Learning Outcomes: After completing this course students will

Master the standard theorems on the existence, uniqueness, continuation, and continuity properties of solutions that apply to a wide class of ordinary differential equations.
Understand the theory of linear systems of first order ordinary differential equations, the methods of their solution, and the qualitative properties of their solutions.
Know the stability properties of linear, almost linear, and nonlinear systems and how to identify these properties.
Understand the notation and language of ordinary differential equations and be able to apply the theory discussed to applied problems.

MATH 6361 PARTIAL DIFFERENTIAL EQUATIONS

This course considers the existence, uniqueness, and approximation of solutions to linear and non-linear ordinary, partial, and functional differential equations. It also considers the relationships of differential equations with functional analysis. Computer-related methods of approximation are also discussed.
Prerequisite: Math 3349 or consent of instructor.

MATH 6362 FOURIER ANALYSIS

This course includes trigonometric series and Fourier series, Dirichlet Integral, convergence and summability of Fourier series, uniform convergence and Gibbs Phenomena, L2 space, properties of Fourier coefficients, Fourier Transform and applications, Laplace Transform and applications, distributions, Fourier series of distributions, Fourier Transforms of generalized functions, orthogonal systems.
Prerequisite: Math 6353 or consent of instructor
Student Learning Outcomes: After completing this course students will

Understand the terminology, scope, main results, and applications of mathematical signal and image processing and Fourier analysis.
Be able to compute and apply Fourier series and transforms, and use them to solve problems in mathematics, science, and engineering.
Know the basic terminology and results of inner product spaces, Hilbert spaces, and normed linear spaces, such as the L^p spaces, and how they relate to signal and image processing.
Understand wavelet analysis and multiresolution analysis and their applications.
Know how to apply computer and graphing calculator technology to gain insight into the topics discussed in class and to aid in performing computations.

MATH 6364 STATISTICAL METHODS

This is a course in the concepts, methods, and usage of statistical data analysis. Topics include test of hypotheses and confidence intervals; linear and multiple regression analysis; concepts of experimental design, randomized blocks and factorial analysis; a brief introduction to non-parametric methods; and the use of statistical software.
Prerequisite: Consent of instructor.

MATH 6365 PROBABILITY AND STATISTICS

Topics in this course include set theory and concepts of probability, random variables, discrete and continuous probability distributions, distribution and expectations of random variables, moment generating functions, transformation of random variables, order statistics, central limit theorem and limiting distributions. [sample syllabus]
Prerequisite: Math 3337 or 4339 or consent of instructor.
Student Learning Outcomes: After completing this course students will

Have the basic set theoretic concepts to calculate probabilities of various combinations of events.
Understand the basics of combinatorial theory to calculate probabilities.
Understand the difference among discrete, continuous and mixture probability distributions and calculate probabilities involving them.
Understand the concepts of moments and moment generating functions.
Understand the special discrete and continuous probability distributions and their main properties.
Have thorough ideas on joint and conditional probability distributions.
Calculate joint and conditional moments using conditional probability distributions.
Emphasize on bivariate normal distributions and understand the concept of joint moment generating functions.
Understand the concepts of transformation of variables, concepts of jacobians for one-dimensional and two-dimensional set up.
Understand the concepts of order statistics and their probability distributions.
Understand limit theorems in probabilities like central limit theorems and understand various types of convergence.
Have some ideas on asymptotic distributions.

MATH 6366 MATHEMATICAL STATISTICS

This course is about the theory of estimation and hypothesis testing. Topics include point estimation and its properties, interval estimation, sufficient statistics, decision theory, most powerful tests, likelihood ratio tests, linear models and estimation by least squares.
Prerequisite: Math 6365 or consent of instructor.

MATH 6370 TOPOLOGY

This course is a foundation for the study of analysis, geometry, and algebraic topology. Topics include set theory and logic, topological spaces and continuous functions, connectedness, compactness, countability, and separation axioms. Prerequisite: Math 4360 or consent of instructor.

MATH 6375 NUMERICAL ANALYSIS

This course provides a fundamental introduction to numerical techniques used in mathematics, computer science, physical sciences and engineering. The course covers basic theory on classical fundamental topics in numerical analysis such as: computer arithmetic, approximation theory, numerical differentiation and integration, solution of linear and nonlinear algebraic system, numerical solution of ordinary differential equations, and error analysis of the mentioned topics. Connections are made to contemporary research in mathematics and it applications to the real world.
Prerequisites: MATH 2345, 2401 with a grade of C or better and computer programming or consent of instructor.

MATH 6376 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

This course focuses on examples of applications in model problems from engineering and science. Model equations include parabolic equations in one, two and three dimensions, hyperbolic equations in one dimension, elliptic equations in two dimensions. The course covers basic theory of scheme consistency, convergence and stability; various methods including finite difference methods (explicit and implicit), finite volume methods and finite element methods (if time allows).
Prerequisites: MATH 1470, MATH 2345, MATH 3349, MATH 3368 or MATH 4318 or equivalent.
Student Learning Outcomes: After completing this course students will be able to

Derive an explicit scheme for a parabolic equation in one space variable.
Implement this scheme to obtain the approximated solutions.
Derive an implicit scheme for a parabolic equation in one space variable.
Implement this scheme to obtain the numerical solutions.
For parabolic equations in one space variable, analyze the convergence of the explicit and implicit schemes.
Derive the ADI and LOD schemes for a specific parabolic equation in two or three dimensions.
Implement the schemes to obtain the numerical solutions.
Analyze the upwind/downwind scheme for hyperbolic equations in one space dimension.
Derive the Lax Wendroff, box and leap-frog schemes for a specific hyperbolic equation in one space dimension.
Implement the schemes to obtain the numerical solutions.
For schemes, discuss the consistency, convergence and stability, and do error analysis.
Derive a finite element scheme for a simple elliptic equation in two dimensions and implement the scheme to obtain the numerical solutions. (optional)

MATH 6385 CRYPTOLOGY AND CODES

Topics in this applied mathematics course include: elementary ciphers, error-control codes, public key ciphers, random number generator- error codes, and Data Encryption Standard. Supporting topics from number theory, linear algebra, group theory, and ring theory will also be Studied.
Prerequisite: MATH 4351 or MATH 6401
Student Learning Outcomes: After completing this course students will be able to

use concepts from number theory, group theory, and ring theory to encrypt and decrypt messages using elementary ciphers such as affine ciphers, polyalphabetic ciphers, substitution ciphers, permutations ciphers, block ciphers, Playfair ciphers, and the Enigma machine;
encrypt and decrypt messages using public-key encryption systems such as the knapsack encryption system, the RSA encryption system, and the ElGamal encryption system;
create pseudorandom numbers using concepts from number theory.
construct and validate electronic signatures produced by the RSA and ElGamal cryptosystems;
construct and decode codes including Hamming codes and BCH codes that detect and correct transmission errors using concepts from linear algebra, group theory, and finite field theory.

MATH 6387 MATHEMATICAL MODELING

This course presents the theory and application of mathematical modeling. Topics will be selected from dynamic models, stable and unstable motion, stability of linear and nonlinear systems, Liapunov functions, feedback, growth and decay, the logistic model, population models, cycles, bifurcation, catastrophe, biological and biomedical models, chaos, strange attractors, deterministic and random behavior.
Prerequisite: Consent of instructor.
Student Learning Outcomes: After completing this course students will be able to

Solve problems involving optimization models, dynamic models, growth and decay, the logistic model and population models.
Understand concepts of stable and unstable motion, stability of linear and nonlinear systems, Liapunov theory.
Identify bifurcation and catastrophe phenomena.
Recognize dynamics and bifurcation in biological and biomedical models.
Distinguish "deterministic" and "random" behavior.
Implement a variety of problem-solving strategies, and evaluate and transform mathematical expressions.
Find and utilize mathematical models which reasonably approximate the behavior of real world problems.

MATH 6388 DISCRETE MATHEMATICS

This course is an introduction to modern finite mathematics. Topics include methods of enumeration, graphs, partially ordered sets, and an introduction to Polya's theory of enumeration.
Prerequisite: Math 4351 or consent of instructor.

MATH 6390 MATHEMATICS SEMINAR

An introduction is given to the methods and tools of mathematical research. Independent work on assign topics is expected of the student, with presentations on the results in both oral and written form.
Prerequisite: Consent of the instructor.

MATH 6391 MASTER'S PROJECT

Individual work or research on advanced mathematical problems conducted under the direct supervision of a faculty member. The course, including a written report, could be taken twice.

MATH 6399 SPECIAL TOPICS IN MATHEMATICS

This course covers special topics in graduate level mathematics that are not taught elsewhere in the department. May be repeated for credit when topic is different.
Prerequisite: Consent of the instructor.

MATH 6401 SURVEY OF ABSTRACT ALGEBRAIC STRUCTURES

This course provides an extensive survey of abstract algebraic structures from the areas of modern algebra, linear algebra, and number theory. Topics include logic, set theory, groups, rings, fields, relations, matrices, vector spaces, mathematical induction, congruences, and number-theoretic functions. Emphasis is placed on the development and presentation of rigorous proofs of elementary results in these areas.
Prerequisite: Graduate standing and consent of instructor. Students seeking a M.S. in Mathematics with prior credit for Math 6331 or Math 6332 with a grade of "B" or higher may not receive credit for this course as an approved elective.

MATH 6402 SURVEY OF ANALYSIS

Topics are chosen from point set topology in the plane (open sets, compactness, connected set, continuity), analysis (sequence, series, continuity and differentiability of functions in two-dimensional Euclidean space) and modern geometry (metric postulates for the Euclidean plane, postulates for the non-Euclidean plane). Emphasis is placed the development and presentation of rigorous proofs of elementary results in these areas.
Prerequisite: Graduate standing and consent of instructor. Students seeking an M.S. in Mathematics with prior credit for MATH 6352 or MATH 6353 with a grade of "B" or higher may not receive credit for this course as an approved elective.
Student Learning Outcomes: After completing this course students will

Understand the intellectual structure of differential calculus and its major theorems, definitions, axioms, and problems;
Understand the definitions, axioms, and major theorems underlying the terms sequence, series, limit, continuity, derivative, and convergence as these concepts relate to real numbers and real-valued functions of a real variable;
Be able to apply the concepts of sequence, series, limit, continuity, derivative, and convergence to solve problems in which their use is fundamental to obtaining and understanding the solution;
Understand what constitutes a valid proof of results in real analysis and learn how to create such proofs;
Be able to write mathematics in a precise, effective, and understandable way.

MATH 7300 Master Thesis I.

First part of two course sequence.
Prerequisites: Graduate standing and consent of thesis advisor.

MATH 7301 Master Thesis II.

Second part of two course sequence.
Prerequisites: Graduate standing and consent of thesis advisor.