# Graduate Mathematics Courses

**MATH 5086. Advanced Special Problems in Mathematics. 1-3 Credit Hours (Lecture: 0 Hours, Lab: 1-3 Hours). **

Special problems in mathematics. Work may be either theory or laboratory. May be repeated with approval of the department head for additional credit. Prerequisite: Approval of department head.

**MATH 5088. Thesis. 1-6 Credit Hours (Lecture: 1-6 Hours, Lab: 0 Hours). **

Scheduled when the student's committee chair determines the student is ready to begin the thesis. No credit is earned until the student has enrolled in at least 6 credit hours of thesis and the thesis is certified as completed by the student's committee, at which time the student will be awarded 6 credit hours of thesis. Prerequisites: 18 hours of approved graduate credit toward the degree and consent of the student's committee.

**MATH 5301. Nonparametric Statistics. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Introduction to nonparametric statistics. Topics will include hypothesis testing, contingency tables, rank tests, and goodness-of-fit tests. Prerequisite: Junior or senior level statistics course.

**MATH 5302. Mathematical Foundations for Data Science. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

An overview of calculus, probability theory, linear algebra, and proof writing at an accelerated pace. Mathematical software will be used throughout the course.

**MATH 5303. Programming Skills for Data Science. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Introduction to core technology and programming skills for data science such as SQL, Python, and R. Additional topics may include parallelized algorithms, no-code workbenches, model/environment storage and deployment, GIS tools, AutoML, and user interfaces.

**MATH 5304. Scientific Computing. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Introduction to scientific computing, emphasizing C/C++, Cuda, symbolic computing, and other topics selected by instructor such as Matlab, Mathematica, Fortran, Linux scripting, JavaScript, Python, R, OpenGL, and ArcGIS. Prerequisites: Graduate standing.

**MATH 5305. Statistical Models. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Basics of experimental design, mathematical theory for linear and logistic regression models in the multivariate case, and diagnostics and remedial measures for these models. Other topics will be selected from ridge/lasso regression, principle components, canonical correlations, factor analysis, and discriminant analysis. Students may not receive credit for both MATH 5305 and STAT 5305. Prerequisite: The equivalent of an undergraduate course in probability and statistics or STAT 5304.

**MATH 5306. Dynamical Systems. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Advanced study of dynamical systems. Topics will be selected from discrete and continuous dynamical systems, sensitivity analysis, models of the physical, life, and social sciences, and bifurcation analysis. Prerequisites: Differential Equations and Linear Algebra.

**MATH 5308. Abstract Algebra. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

A course in abstract algebra, starting with group theory, quotient groups, homomorphisms, permutation representations, and the Sylow theorems. Additional topics will be selected from direct and semidirect products; the fundamental theorem of finitely generated abelian groups; ring theory; module theory; vector spaces; field theory; Galois theory; and algebraic geometry. Credit will not be awarded for both MATH 5308 and MATH 6308. Prerequisites: An undergraduate course in abstract algebra.

**MATH 5309. Complex Analysis I. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

An introduction to complex analysis. Topics will be selected from elementary operations and analytic functions, curves and integrals, power series, Cauchy¿s theorem, zeroes and singularities of analytic functions, Laurent series, maximum principle, analytic continuation, harmonic functions, conformal mapping and transformations. Prerequisite: A two semester sequence in calculus.

**MATH 5310. Complex Analysis II. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Conformal mapping, harmonic functions, infinite products, Weierstrass factorization theorem, Mittag-Leffler's theorem, normal families, Riemann mapping theorem, analytic continuation, Picard's theorems and selected topics. Prerequisite: MATH 5309.

**MATH 5312. Design of Experiments. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Students will learn about planning and conducting an experiment. Data analysis using appropriate software is covered. Prerequisite: MATH 5305 or approval of department head.

**MATH 5320. Real Analysis I. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

An introduction to measure theory and integration, beginning with outer measures, sigma-algebras, Borel sets, measurable functions, and Lebesgue measure. Further topics include convergence of measurable functions, Luzin’s theorem, the monotone convergence theorem, the dominated convergence theorem, differentiation, the Hardy-Littlewood maximal inequality, and the Lebesgue differentiation theorem. Prerequisite: An undergraduate course in real analysis.

**MATH 5321. Real Analysis II. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

A second course in real analysis, with topics selected from product measures, the Fubini-Tonelli theorem, Lebesgue integration in n-dimensional Euclidean space, metric spaces, normed vector spaces, L-p spaces, Holder’s inequality, Hilbert spaces, and Fourier analysis. Prerequisite: MATH 5320.

**MATH 5330. Mathematical Modeling. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

An advanced course in mathematical modeling. Topics will be selected from scaling, dimensional analysis, regular and singular perturbation theory, stability theory, and asymptotic analysis. Prerequisites: Differential Equations and Linear Algebra.

**MATH 5340. Topology. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Axioms of a topological space; open and closed sets; compactness; connectedness; basis; product topology; subspaces; metric spaces; and quotient topologies. Additional topics will be selected from completeness, continua, separation axioms, metrization theorems, Baire spaces, and algebraic topology. Credit will not be awarded for both MATH 5340 and MATH 6340. Prerequisites: An undergraduate course in topology or analysis.

**MATH 5350. Linear Algebra. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

An advanced course in linear algebra. Topics to be selected from linear spaces and operators, canonical forms, quadratic forms and optimization, computation and condition, and compatible systems. Prerequisite: The equivalent of an undergraduate course in linear algebra.

**MATH 5351. Applied Numerical Linear Algebra. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Topics include methods for solving linear systems; Gram-Schmidt process; least squares; inverse and pseudoinverse operators; LU, QR, SVD and other decompositions with applications of linear algebra selected from: Markov Chains, Hilbert spaces, spectral theory, Fourier and associated transforms, difference equations, curve fitting, Green’s functions, extremal problems, graph Laplacian, PageRank, operator representation and interpolation, Jordan form, and LAPACK. Prerequisite: A course in linear algebra or instructor approval.

**MATH 5360. Numerical Analysis. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

An advanced study of numerical analysis. Topics will be selected from linear systems, approximation theory, numerical differential and integral equations, integration theory. Prerequisite: MATH 3360.

**MATH 5361. Iterative Methods. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Matrix and vector norms, conditioning, iterative methods for the solution of larger linear systems and eigenvalue problems. Krylov subspace methods and methods for stiff systems of differential equations. Other topics to be chosen by the instructor. Prerequisites: MATH 5360 and a course in computer programming.

**MATH 5362. Data Warehousing. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Use SQL for manipulation and exploration of large data sets by creating tables, transforming data, using joins, and performing simple queries. Prerequisite: COSC 1310 or equivalent.

**MATH 5364. Data Science I. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

This course centers on the identification, exploration, and description of new patterns contained within data sets using appropriate software. Selected topics will be chosen from data exploration, classification, cluster analysis, and model evaluation and comparison. Credit will not be awarded for both MATH 5364 and MATH 6364. Prerequisites: Probability and Statistics.

**MATH 5365. Applications of Data Science. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Data science topics that are not among the core components covered in MATH 5364 and MATH 5366 but are widely used in the field. Topics will be selected to align with student interest and the current state of data science. Prerequisite: MATH 5364.

**MATH 5366. Data Science II. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

This course centers on the identification, exploration, and extraction of new patterns from natural language text documents using appropriate software. Selected topics will be chosen from association analysis, anomaly detection, text mining, dimensionality reduction, and model evaluation and comparison. Credit will not be awarded for both MATH 5366 and MATH 6366. Prerequisite: MATH 5364.

**MATH 5370. History of Mathematics. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

A historical and philosophical development of mathematics from antiquity to the present. Mathematical topics are presented in a historical and philosophical setting not only to provide a unifying theme, but also to illustrate how the evolution of mathematical ideas finally led to modern concepts in the field. Credit will not be awarded for both MATH 5370 and MATH 6370. Prerequisite: 6 advanced hours in mathematics.

**MATH 5371. Euclidean and Non-Euclildean Geometries. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

This course focuses on important geometric concepts of Euclidean and non-Euclidean geometries from an axiomatic perspective. Technology will be included where appropriate. Prerequisite: 3 hours of undergraduate geometry.

**MATH 5373. Theory of Functions. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

This course is designed to emphasize the role of function as the key unifying concept of mathematics and to extend the understanding of the structural foundations of mathematics. The properties of various families of functions will also be studied. Prerequisite: 24 hours of MATH, including MATH 2413.

**MATH 5375. Statistical Reasoning and Probability. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

This course focuses on statistical reasoning and decision making by extending the elements of probability and statistics introduced in an undergraduate course. Topics may include probability theory, distribution functions, statistical inference, sampling methods, regressional analysis, and ANOVA. Technology will be incorporated where appropriate. Prerequisite: 3 hours of undergraduate statistics.

**MATH 5376. Algebraic Structures. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

This course examines algebraic structures in secondary and post-secondary mathematics from an advanced perspective. Analysis of algebraic concepts and underlying theory, along with the appropriate integration of manipulatives and technology in accordance with the standards of the National Council of Teachers of Mathematics, will be emphasized. Prerequisite: 24 hours of MATH at the undergraduate level, including Calculus.

**MATH 5377. In-Depth Mathematical Reasoning. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

The study of mathematics from an advanced perspective, taking into account not only the interconnections among topics but their relationship to higher mathematics. Important new mathematical insights and understandings will be revealed in its structure and its applicability. The focus will be on concept analysis, problem analysis, and mathematical connections as well as mathematical habits of mind. Prerequisite: 24 hours from MATH, including MATH 2413.

**MATH 5378. Technology-Aided Mathematics-. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Students will engage in mathematical problem-solving using technological tools. Technologies may include graphing handhelds, data collection devices, computer software packages, and internet resources. This course may be repeated for credit as the topic changes. Prerequisite: 24 hours of MATH, including MATH 2413.

**MATH 5379. Trends and Issues in Research. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

In this seminar-style course, students have a forum for discussion and presentation of inquiries into the history, current trends, and issues pertaining to analysis of research trends in mathematics education and its effect on policy, curriculum, and the teaching and learning of mathematics. Prerequisite: 24 hours of MATH, including MATH 120.

**MATH 5380. Selected Topics in Mathematical Theory. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

An examination of topics in mathematical theory appropriate for secondary mathematics educators. Topics will be selected from geometry and topology, number theory, modern algebra, and library research in mathematics. This course may be repeated for credit as the topic changes. Prerequisite: Approval of department head.

**MATH 5386. Advanced Special Problems in Mathematics. 1-3 Credit Hours (Lecture: 0 Hours, Lab: 1-3 Hours). **

Special problems in mathematics. Work may be either theory or laboratory. May be repeated with approval of the department head for additional credit. Prerequisite: Approval of department head.

**MATH 5390. Selected Topics in Mathematics. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

An examination of topics in applied mathematics. Topics for study will be selected from advanced mathematical modeling, advanced numerical techniques, practical optimizations, calculus of variations, dynamic programming, integral equations, optimal control, perturbation methods, and library research in applied mathematics. This course may be repeated for credit as the topic changes. Prerequisite: Approval of department head.

**MATH 5398. Research Analysis. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

An overview of the components of research in various areas of mathematics. These areas will include pure mathematics, applied mathematics, mathematics education, and statistics. The course will include a study of reviewing contemporary and classical literature, presenting research, and how to submit an article for publication. Prerequisites: Graduate standing in the mathematics department or approval of the department head.

**MATH 5699. Internship. 1-6 Credit Hours (Lecture: 1-6 Hours, Lab: 0 Hours). **

The student will complete a supervised and comprehensive work experience in a mathematics-related position with a public or private business organization for career preparation in a mathematics-related enterprise. Credit in this course does not count towards the 24 hour requirement for the M.S. in Mathematics. Prerequisite: Mathematics graduate student with department head approval. Field assignment fee $75.

**MATH 6086. Advanced Special Problems in Mathematics.. 1-3 Credit Hours (Lecture: 0 Hours, Lab: 0 Hours). **

Special problems in mathematics. Work may be either theory or laboratory. May be repeated with approval of the department head for additional credit. Prerequisite: Approval of department head.

**MATH 6098. Research. 1-6 Credit Hours (Lecture: 0 Hours, Lab: 0 Hours). **

Doctoral students conduct original research on a variety of topics in applied mathematics toward a doctoral dissertation. Course will be graded as satisfactory or unsatisfactory. Prerequisites: Doctoral candidacy in applied mathematics.

**MATH 6185. Seminar. 1 Credit Hour (Lecture: 0 Hours, Lab: 0 Hours). **

A weekly colloquium consisting of research presentations by faculty and students, including speakers from Tarleton and other institutions. Prerequisites: Graduate standing.

**MATH 6303. Programming Skills for Data Science. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Introduction to core technology and programming skills for data science such as SQL, Python, and R. Additional topics may include parallelized algorithms, no-code workbenches, model/environment storage and deployment, GIS tools, AutoML, and user interfaces. Credit will not be awarded for both MATH 5303 and MATH 6303.

**MATH 6308. Abstract Algebra. 3 Credit Hours (Lecture: 0 Hours, Lab: 0 Hours). **

A course in abstract algebra, starting with group theory, quotient groups, homomorphisms, permutation representations, and the Sylow theorems. Additional topics will be selected from direct and semidirect products; the fundamental theorem of finitely generated abelian groups; ring theory; module theory; vector spaces; field theory; Galois theory; and algebraic geometry. Credit will not be awarded for both MATH 5308 and MATH 6308. Prerequisite: A course in abstract algebra.

**MATH 6309. Complex Analysis I. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

A course in complex analysis, starting with axioms of the complex numbers, analytic functions, the Cauchy-Riemann equations, harmonic functions, and conformal mappings. Additional topics will be selected from line integrals, power series, Laurent series, the residue calculus, and hyperbolic geometry. Credit will not be awarded for both MATH 5309 and MATH 6309. Prerequisite: An undergraduate course in analysis.

**MATH 6310. Complex Analysis II. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Conformal mapping, harmonic functions, infinite products, Weierstrass factorization theorem, Mittag-Leffler's theorem, normal families, Riemann mapping theorem, analytic continuation, Picard's theorems and selected topics. Credit will not be awarded for both MATH 5310 and MATH 6310 Prerequisite: MATH 5309, MATH 6309, or a graduate course in complex analysis.

**MATH 6313. Probability Theory I. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Develops probability axioms in a measure theoretic setting, starting with sigma-fields, Lebesgue measure, random variables, and extensions using the pi-lambda theorem. Additional topics include Borel’s normal number theorem; the weak and strong laws of large numbers; the Borel-Cantelli lemmas; and Markov chains. Prerequisites: MATH 5320 or approved graduate course work in real analysis that includes measure theory and integration.

**MATH 6314. Probability Theory II. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

A second course in probability theory, with topics selected from the Poisson process, the ergodic theorem, convergence of distributions, characteristic functions, the central limit theorem, the Radon-Nikodym theorem, conditional distributions, martingales, and stochastic processes. Prerequisite: MATH 6313.

**MATH 6322. Ordinary Differential Equations. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Study and applications of linear and nonlinear ordinary differential equations and their systems. Existence and uniqueness of solutions, stability theory and applications, singularities, periodic and oscillatory solutions, and other topics as time allows. Prerequisites: Graduate level real analysis and graduate level linear algebra or permission of department head.

**MATH 6323. Partial Differential Equations. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Study and applications of partial differential equations. Existence and uniqueness for boundary value problems. Wave equation, heat equation, and Laplace equation will be studied. Theory for elliptic, hyperbolic, and parabolic partial differential equations. Other topics as time allows. Prerequisite: MATH 6322 or permission of department head.

**MATH 6324. Dynamical Systems I. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Study and applications of nonlinear difference equations (maps) and systems of difference equations, stability of solutions, phase plane, periodic doubling, bifurcations, oscillations, and chaos. Stochastic systems and other topics as time allows. Prerequisite: Graduate Linear Algebra or graduate Applied Linear Algebra, or permission by department head.

**MATH 6325. Dynamical Systems II. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Study and applications of nonlinear differential equations and systems of differential equations, stability of solutions, phase plane, bifurcations, periodic coefficients, and Poincare maps. Stochastic systems other topics as time allows. Prerequisite: MATH 6324 or permission of department head.

**MATH 6328. Functional Analysis I. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

A study of vector spaces in the infinite-dimensional setting, starting with Hilbert spaces, the Riesz representation theorem, diagonalization of operators, Banach spaces, and the Hahn-Banach theorem. Further topics include dual spaces, the principle of uniform boundedness, locally convex spaces, and weak topologies. Prerequisites: MATH 5321 or approved graduate coursework in real analysis that includes topics such as product measures, metric spaces, and Fourier analysis.

**MATH 6329. Functional Analysis II. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

A second course in functional analysis, with topics selected from linear operators on Banach space, the Banach-Stone theorem, Banach algebras, the Riesz functional calculus, spectral theory, C*-algebras, normal operators on Hilbert space, unbounded operators, and Fredholm theory. Prerequisite: MATH 6328.

**MATH 6340. Topology. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Axioms of a topological space; open and closed sets; compactness; basis; product topology; subspaces; metric spaces; and quotient topologies. Additional topics will be selected from connectedness, completeness, continua, separation axioms, metrization theorems, Baire spaces, and algebraic topology. Credit will not be awarded for both MATH 5340 and MATH 6340. Prerequisite: A course in topology or analysis.

**MATH 6362. Computational Optimization Methods.. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Fundamentals of mathematical analysis underlying theory of constrained optimizations for a finite number of variables, necessary and sufficient conditions for constrained extrema of equality constraint problems, sufficient conditions for fulfillment of constraint qualification, computational methods for concave programming problems and applications. Prerequisites: MATH 5360 and MATH 5320 or approved graduate coursework in numerical analysis and real analysis.

**MATH 6363. Numerical Solutions to Partial Differential Equations. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Covers finite difference methods for elliptic, parabolic, and hyperbolic problems in partial differential equations. Also, stability, consistency, and convergence results. Attention is given to computer implementations. Prerequisites: MATH 5360 or approved graduate coursework in numerical analysis; and MATH 6323.

**MATH 6364. Data Science I. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

This course centers on the identification, exploration, and description of new patterns contained within data sets using appropriate software. Selected topics will be chosen from data exploration, classification, cluster analysis, and model evaluation and comparison. Credit will not be awarded for both MATH 5364 and MATH 6364. Prerequisites: The equivalent of an undergraduate course in Probability and Statistics.

**MATH 6365. Applications of Data Science. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

Data science topics that are not among the core components covered in MATH 6364 and MATH 6366 but are widely used in the field, such as anomaly detection, reinforcement learning, recommender systems, geospatial analysis, natural language processing, image processing, and generative models. Due to the rapid evolution of data science and its widespread use in diverse fields, topics will be selected to align with student interest and the current state of data science. Credit will not be awarded for both MATH 5365 and MATH 6365. Prerequisite: MATH 5364 or MATH 6364.

**MATH 6366. Data Science II. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

This course centers on the identification, exploration, and extraction of new patterns from natural language text documents using appropriate software. Selected topics will be chosen from association analysis, anomaly detection, text mining, dimensionality reduction, and model evaluation and comparison. Credit will not be awarded for both MATH 5366 and MATH 6366. Prerequisite: MATH 5364 or MATH 6364.

**MATH 6370. History of Mathematics. 3 Credit Hours (Lecture: 3 Hours, Lab: 0 Hours). **

A historical and philosophical development of mathematics from antiquity to the present. Mathematical topics are presented in a historical and philosophical setting not only to provide a unifying theme, but also to illustrate how the evolution of mathematical ideas finally led to modern concepts in the field. Credit will not be awarded for both MATH 5370 and MATH 6370. Prerequisite: 6 advanced hours in mathematics.

**MATH 7088. Dissertation. 1-6 Credit Hours (Lecture: 0 Hours, Lab: 0 Hours). **

Scheduled when the student is prepared to begin the scholarly investigation of a topic acceptable to the dissertation committee. The dissertation must provide evidence that the candidate has pursued a coherent program of research related to the student’s areas of academic specialization, the results of which make a significant, original contribution to the discipline. Prerequisites: Doctoral candidacy in applied mathematics.