MATH 5111. Graduate Assistant Training.

This course is concerned with techniques used in the teaching of mathematics and prepares graduate teaching and instructional assistants for responsibilities in collegiate mathematics instruction. Topics include classroom management, communication with students, grading practices, academic integrity, equitable and professional teaching practices, and strategies for supporting student learning in undergraduate mathematics courses. Emphasis is placed on effective instructional preparation, professional expectations, and reflective development as a teaching assistant. This course is required as a condition of employment for graduate teaching and instructional assistants, does not earn graduate degree credit, and may be repeated with different emphasis.

MATH 5199B. Thesis.

This course provides students with the opportunity to conduct an independent, original research project under the supervision of a faculty advisor. Students formulate a research question, engage with relevant scholarly literature, and apply appropriate methods to investigate a focused topic within mathematics, mathematics education, or a closely related field. Emphasis is placed on rigorous reasoning, clear exposition, and adherence to disciplinary standards for scholarship. The course culminates in the completion of a written thesis that demonstrates the student’s ability to conduct sustained inquiry and communicate results effectively.

MATH 5290. Teaching Geometry through Problem Solving and Discovery Learning.

This course investigates the problem-solving heuristics embedded in the secondary school geometry curriculum and explores how to implement problem solving in geometry classrooms. This course also examines the unique “Hungarian style” method of discovery learning in mathematics, developed for students aged 12-18. The method referred to as the Pósa Method is similar to inquiry based learning with an emphasis on problem solving.

MATH 5299B. Thesis B.

This course provides students with the opportunity to conduct an independent, original research project under the supervision of a faculty advisor. Students formulate a research question, engage with relevant scholarly literature, and apply appropriate methods to investigate a focused topic within mathematics, mathematics education, or a closely related field. Emphasis is placed on rigorous reasoning, clear exposition, and adherence to disciplinary standards for scholarship. The course culminates in the completion of a written thesis that demonstrates the student’s ability to conduct sustained inquiry and communicate results effectively.

MATH 5301. Partial Differential Equations.

This course covers topics that include the derivation and classification of partial differential equations, vector and tensor methods, and first‑order equations. The course studies wave equations, vibrations, and normal modes, with emphasis on Fourier series and Fourier integrals. Classical solution techniques such as Cauchy’s method, characteristics, Green’s functions, potential theory, and boundary‑value problems are developed in detail. Additional methods, including Riemann–Volterra techniques, are introduced to analyze initial‑ and boundary‑value problems. Analytical reasoning and mathematical rigor are emphasized throughout. Prerequisite: MATH 3323 with a grade of "C" or better.

MATH 5303. History of Mathematics.

This course studies the development of mathematics and the accomplishments of mathematicians who contributed to its progress across cultures and historical periods. Topics include the emergence of major mathematical ideas, the historical development of algebra, geometry, calculus, and other significant areas, original and secondary historical sources, and the cultural and philosophical settings in which mathematics evolved. Emphasis is placed on historical interpretation, mathematical communication, understanding mathematics as an evolving human endeavor, and connections between earlier developments, contemporary mathematical thought, and the teaching and learning of mathematics. Cannot be used on a degree plan for M.S. degree.

MATH 5307. Modern Algebra.

This course provides a graduate level introduction to the fundamental structures of modern algebra. Students study groups, rings, fields, and related algebraic systems as needed with an emphasis on axiomatic development, structural properties, and proof techniques. Topics may include homomorphisms, quotient structures, polynomial rings, field extensions, and classical theorems central to algebraic reasoning. The course highlights methods for analyzing abstract structures, constructing examples, and formulating rigorous arguments. This course supports the development of advanced mathematical maturity and prepares students for further study in algebra and related disciplines. Material will be adapted to the needs of the class. Prerequisite: MATH 5384 with a grade of "B" or better.

MATH 5311. Foundations of Differential Equations.

This course provides a foundational study of differential equations, operator spaces, and related core mathematical topics. Students examine classical and modern approaches to these structures while exploring how recent developments contribute to contemporary mathematical analysis. Emphasis is placed on understanding theoretical frameworks, evaluating methods using clear mathematical reasoning, and investigating the logical relationships among differential operators and function spaces. Students are supported in conducting independent, evidence based inquiry that allows them to form their own interpretations grounded in mathematical argumentation. Prerequisite: MATH 5382 with a grade of "C" or better.

MATH 5312. Functions of a Complex Variable.

This course studies modern developments in the theory of functions of a complex variable. Topics may include analytic functions, contour integration, the Cauchy integral theorem and formula, Laurent series, residues, conformal mappings, and related core results of complex function theory. Additional topics may include Riemann surfaces, differential forms, and the Riemann-Roch theorem, with attention to how local complex-analytic methods connect to global structures in topology, geometry, and algebra. Emphasis is placed on rigorous reasoning, conceptual understanding, and the role of complex analysis in advanced mathematical theory. Prerequisite: MATH 5382 with a grade of "C" or better.

MATH 5313. Field Theory.

This course studies field theory with emphasis on separable extensions and Galois theory. Topics include field extensions, polynomial rings, splitting fields, algebraic and transcendental extensions, automorphism groups, and the structure of Galois correspondences. Related algebraic concepts such as rings, ideals, modules, and quotient structures may be developed as needed to support the theory. Emphasis is placed on rigorous proof, structural analysis, and the use of field-theoretic methods in advanced algebra and related areas. Prerequisite: MATH 5384 with a grade of "B" or better.

MATH 5314. Number Theory.

This course examines selected topics in number theory at the graduate level. Content may include elementary number theory, quadratic forms, and introductory concepts from algebraic or analytic number theory, with emphasis on fundamental structures, methods, and proofs. Topics are chosen to provide flexibility while maintaining mathematical rigor and coherence. The course develops techniques for analyzing integers and related algebraic objects, with attention to problem‑solving strategies and theoretical reasoning. Specific topics may vary depending on instructor expertise and student background, but the course is designed to strengthen students’ understanding of core ideas in number theory and prepare them for further study or research in mathematics. Prerequisite: MATH 5384 with a grade of "B" or better.

MATH 5317. Problems in Advanced Mathematics.

This course is open to graduate students on an individual basis by arrangement with the mathematics department. Students investigate topics appropriate to their level of mathematical preparation, applying advanced reasoning and analytical skills to selected problem sets or focused areas of study. The specific emphasis varies and may include theoretical exploration, problem solving techniques, or preparatory work for future coursework. The course supports mathematical development while maintaining flexibility in content, method, and depth. A substantial degree of mathematical maturity is expected, and the course may be repeated with different emphases. This course does not apply toward graduate degree credit.

MATH 5319. The Theory of Integration.

This course covers the theory of integration with special emphasis on Lebesgue integrals. Topics include the Lebesgue integral for bounded, finitely supported, and measurable functions, convergence theorems, differentiability of monotone functions, absolute continuity, Lp spaces, and Lp completeness. Additional attention may be given to convergence theorems, and the role of integration theory in modern analysis. Emphasis is placed on rigorous proof, abstract reasoning, and the analytical foundations needed for advanced work in real analysis, probability, and related mathematical fields.

MATH 5329. General Topology.

This course studies topological spaces and their properties at the graduate level. Topics include topological spaces and continuous functions, connectedness, compactness, separation axioms, product and quotient constructions, metrization, and CW complexes, together with selected related topics as time permits. Emphasis is placed on rigorous proof, careful use of definitions, construction of examples and counterexamples, and the role of topology in advanced study of analysis, geometry, and algebra.

MATH 5331. Metric Spaces.

This course provides an introduction to point set topology with a central focus on the structure and properties of metric spaces. Students study topics such as open and closed sets, convergence, continuity, compactness, completeness, and connectedness within the context of metric spaces. The course also includes a brief introduction to general topological spaces to illustrate how metric spaces fit into broader topological frameworks. Emphasis is placed on rigorous definitions, logical reasoning, and the development of proof based skills that support advanced mathematical study.

MATH 5336. Studies in Applied Mathematics.

This course examines selected topics in applied mathematics at the graduate level. Possible areas of study include optimization and control theory, numerical analysis, calculus of variations, boundary value problems, special functions, tensor analysis, and other subfields of applied mathematics. Topics are chosen to reflect current mathematical methods and applications, with emphasis on analytical techniques, modeling, and problem‑solving. The specific content may vary depending on instructor expertise and student preparation. This course may be repeated for credit when the topic emphasis differs, allowing students to engage with multiple areas of applied mathematics in depth. Prerequisite: Instructor approval.

MATH 5338. Advanced Independent Study in Mathematics or Statistics.

This course provides graduate students with an opportunity to pursue advanced independent study in mathematics or statistics under the supervision of a faculty member. Students investigate specialized topics aligned with faculty research interests through theoretical analysis, empirical investigation, or critical review and synthesis of existing scholarly literature. Emphasis is placed on independent inquiry, depth of understanding, and adherence to disciplinary standards of rigor and clarity. The specific subject matter and methods of study are determined collaboratively by the student and supervising faculty member. This course may be repeated once for credit when the topic emphasis differs. Prerequisite: Instructor approval.

MATH 5340. Scientific Computation.

This course studies computational methods used in science and mathematics through the analysis and implementation of algorithms in a computer algebra or scientific computing environment. Topics include symbolic, numerical, and graphical techniques for solving mathematical problems, algorithmic approaches to scientific computation, and applications drawn from mathematics, science, and engineering. Emphasis is placed on the mathematical structure of computational methods, the interpretation of computed results, and the effective use of computation as a tool for advanced problem solving.

MATH 5350. Combinatorics.

This course studies graduate-level combinatorics with emphasis on enumeration, structure, and proof. Topics may include permutations, combinations, Stirling numbers, chromatic numbers, Ramsey numbers, generating functions, Pólya theory, Latin squares, random block design, and related combinatorial structures. Additional topics may include recurrence relations, special sequences, discrete probability, extremal set theory, and graph-theoretic methods as they support the development of combinatorial reasoning. Emphasis is placed on rigorous proof, effective mathematical communication, and the use of combinatorial techniques in discrete mathematics and related areas. Prerequisite: Instructor Approval.

MATH 5355. Applied and Algorithmic Graph Theory.

This course studies graph theory with emphasis on the close relationship between theoretical and algorithmic aspects of the subject. Topics may include connectivity, trees, planarity, graph coloring, matchings, and network models, together with algorithms such as max-flow min-cut, maximum matching, shortest-path methods, and optimization algorithms for facility location problems in networks. Emphasis is placed on rigorous reasoning, algorithmic analysis, and the application of graph-theoretic methods to discrete structures, optimization, and network-based problems in mathematics and related fields. Prerequisite: MATH 5388 with a grade of 'C' or better.

MATH 5360. Mathematical Modeling.

This course introduces the principles and techniques of mathematical modeling with applications drawn primarily from the natural sciences. Students examine the process of translating real‑world phenomena into mathematical representations and analyzing those models using appropriate analytical and computational tools. Emphasis is placed on deterministic systems, stochastic models, and diffusion processes. The course explores model formulation, assumptions, solution methods, and interpretation of results, with attention to the strengths and limitations of different modeling approaches. Through selected examples, students develop skills in constructing, analyzing, and evaluating mathematical models as tools for understanding complex systems. Prerequisite: MATH 5301 with a grade of "C" or better.

MATH 5373. Theory of Functions of Real Variables.

This course covers measure theory with special emphasis on the Lebesgue measure. Topics include the outer measure, sigma-algebra of measurable sets, properties of measurable sets, Borell-Canteli lemma, non-measurable sets, Cantor-Lebesgue function, Lebesgue measurable functions, pointwise limits, simple approximations, and Littlewood’s three principles. Additional attention may be given to convergence theorems, and the role of approximation theory in modern analysis. Emphasis is placed on rigorous proof, abstract reasoning, and the analytical foundations needed for advanced work in real analysis, probability, and related mathematical fields. Prerequisite: Instructor approval.

MATH 5374. Numerical Linear Algebra.

This course introduces tools that mathematical scientists use with vectors and matrices. Topics include direct and iterative methods for solving linear systems, least-squares problems, eigenvalue problems, algorithmic stability under perturbations, and numerical methods for large, sparse systems. Additional topics may include Gaussian elimination, Jacobi and Gauss–Seidel methods, conjugate gradient methods, GMRES, Fast Fourier Transform techniques, and multigrid methods, with attention to both theoretical analysis and computational implementation. Emphasis is placed on balancing rigorous mathematical reasoning with practical numerical computation in applications from science, engineering, and applied mathematics.

MATH 5376F. Introduction to Probability Theory and Models.

This course introduces the definitions, constructions, theorems, and techniques used to build and analyze probability models. Students explore both the abstract foundations of probability theory and its application to a wide range of modeling contexts. Topics include conditional expectation, convergence of random variables, the weak and strong laws of large numbers, the central limit theorem, random walks, martingales, and Brownian motion. The course emphasizes the active construction and evaluation of probability models while developing the rigorous theoretical background needed to understand their behavior. Students engage with proofs, examples, and analytical methods that support deeper understanding of stochastic processes and probabilistic structures.

MATH 5381. Foundations of Set Theory.

This course provides a graduate-level study of set theory as a foundation for modern mathematics. Topics include sets, relations, functions, finite and infinite structures, cardinality, ordinals, transfinite methods, and basic principles of mathematical logic. Emphasis is placed on precise definitions, rigorous proof, and formal reasoning. The course develops the language and techniques used across advanced areas such as analysis, algebra, topology, and logic. It supports further graduate study and research in theoretical mathematics and foundations.

MATH 5382. Foundations of Real Analysis.

This course covers the foundations of mathematical analysis. Topics include the real number system, sequences, series, limits, and continuity of functions, together with methods of proof used in advanced mathematics. Additional attention may be given to completeness, compactness, inequalities, and the logical structure of rigorous analysis as students transition from computational calculus to proof-based reasoning. Emphasis is placed on precise definitions, careful argumentation, and the development of core concepts that support further study in analysis and related areas. Prerequisite: MATH 5381 with a grade of 'C' or better.

MATH 5384. Geometric Approach to Abstract Algebra.

This course studies the definitions and elementary properties of groups, rings, integral domains, fields, and vector spaces, with particular emphasis on the rings of integers, rational numbers, complex numbers, and polynomials. Attention is given to the structural relationships among these algebraic systems and to the interplay between algebra and geometry. Emphasis is placed on rigorous proof, abstraction, and the use of algebraic structures to analyze mathematical patterns, transformations, and foundational ideas that support advanced study in algebra and related areas. Prerequisite: MATH 5381 with a grade of 'C' or better.

MATH 5386. Knots and Surfaces, An Introduction to Low-Dimensional Topology.

This course introduces fundamental topics in low‑dimensional topology with emphasis on knot theory and the topology of surfaces. Students study knot polynomials and other knot invariants, as well as methods for distinguishing and classifying knots. The course also examines the topological classification of surfaces and associated invariants, including orientability, genus, and Euler characteristic. Emphasis is placed on understanding how invariants are defined and used to distinguish topological spaces. Through selected examples and proofs, students develop familiarity with core concepts, techniques, and reasoning in low‑dimensional topology, preparing them for further study in topology or related areas of mathematics.

MATH 5388. Discrete Mathematics.

This course provides a graduate-level study of discrete mathematics, emphasizing counting techniques, recurrence relations, discrete probability, and graph theory. Topics include permutations and combinations, inclusion-exclusion, generating functions, recurrence methods, discrete probabilistic models, and graph-theoretic structures. Emphasis is placed on rigorous reasoning, proof, and discrete modeling. The course highlights connections among combinatorics, probability, and graph theory and supports applications in mathematics, computation, and related quantitative fields. It prepares students for advanced coursework and research involving discrete structures and mathematical modeling.

MATH 5392. Survey of Geometries.

This course examines major geometric systems through an axiomatic and structural perspective. Topics include the axiomatic development of finite geometries, affine and projective geometry, Euclidean geometry, and selected non‑Euclidean geometries, with emphasis on their relationships to neutral geometry and Hilbert’s axioms. The course also may explore geometric transformations and an introduction to fractal geometry as extensions of classical geometric ideas. Students analyze how different axiom systems give rise to distinct geometric structures and investigate connections among algebraic, transformational, and metric approaches to geometry. Emphasis is placed on logical reasoning, proof techniques, and comparative analysis of geometric frameworks.

MATH 5393. Numerical Optimization.

This course introduces optimization methods used across mathematics, engineering, and applied sciences. Students study foundational concepts in optimization theory and examine numerical algorithms designed to locate points satisfying optimality conditions. Emphasis is placed on convergence analysis, algorithmic behavior, and practical implementation considerations. Topics include gradient based methods, unconstrained and constrained optimization techniques, line search and trust region strategies, and computational approaches for analyzing algorithm performance. Through examples, proofs, and computational exercises, students gain experience interpreting optimality conditions and selecting appropriate numerical techniques for a variety of mathematical problems.

MATH 5399A. Thesis.

This course provides students with the opportunity to conduct an independent, original research project under the supervision of a faculty advisor. Students formulate a research question, engage with relevant scholarly literature, and apply appropriate methods to investigate a focused topic within mathematics, mathematics education, or a closely related field. Emphasis is placed on rigorous reasoning, clear exposition, and adherence to disciplinary standards for scholarship. The course culminates in the completion of a written thesis that demonstrates the student’s ability to conduct sustained inquiry and communicate results effectively. This course represents a student’s initial thesis enrollment. No thesis credit is awarded until student has completed the thesis in Mathematics 5399B.

MATH 5399B. Thesis.

This course provides students with the opportunity to conduct an independent, original research project under the supervision of a faculty advisor. Students formulate a research question, engage with relevant scholarly literature, and apply appropriate methods to investigate a focused topic within mathematics, mathematics education, or a closely related field. Emphasis is placed on rigorous reasoning, clear exposition, and adherence to disciplinary standards for scholarship. The course culminates in the completion of a written thesis that demonstrates the student’s ability to conduct sustained inquiry and communicate results effectively. This represents a student’s continuing thesis enrollment. The student continues to enroll in this course until the thesis is submitted for binding.

MATH 5492. Experiencing the Hungarian Approach through Observation and Teaching Practicum.

This course provides a first-hand experience in putting the Hungarian style guided discovery into practice. As part of the course, students will spend one week at a mathematics camp for secondary students that is being run using the Hungarian style of teaching. Students will observe mathematics classes, discuss pedagogy with camp instructors, and design and teach their own lesson to camp participants.

MATH 5599B. Thesis.

This course provides students with the opportunity to conduct an independent, original research project under the supervision of a faculty advisor. Students formulate a research question, engage with relevant scholarly literature, and apply appropriate methods to investigate a focused topic within mathematics, mathematics education, or a closely related field. Emphasis is placed on rigorous reasoning, clear exposition, and adherence to disciplinary standards for scholarship. The course culminates in the completion of a written thesis that demonstrates the student’s ability to conduct sustained inquiry and communicate results effectively. This represents a student’s continuing thesis enrollment. The student continues to enroll in this course until the thesis is submitted for binding.

MATH 5999B. Thesis.

This course provides students with the opportunity to conduct an independent, original research project under the supervision of a faculty advisor. Students formulate a research question, engage with relevant scholarly literature, and apply appropriate methods to investigate a focused topic within mathematics, mathematics education, or a closely related field. Emphasis is placed on rigorous reasoning, clear exposition, and adherence to disciplinary standards for scholarship. The course culminates in the completion of a written thesis that demonstrates the student’s ability to conduct sustained inquiry and communicate results effectively. This represents a student’s continuing thesis enrollment. The student continues to enroll in this course until the thesis is submitted for binding.