MATHEMATICS (MTH–Arts and Science; Department of Mathematics and Statistics)

Note: A service course is for specific non-mathematics and non-statistics majors. Mathematics and statistics majors should not take service courses.

407/507 Mathematical Structures Through Inquiry (3)

MTH 407 is open only to middle childhood education majors; MTH 507 is open only to preK-9 teachers. Study of the structure of mathematical systems, especially number systems, developed through student-centered inquiry: pattern recognition, hypothesis formation, hypothesis testing, and proof. Prerequisite: nine semester hours of MTH/STA courses including MTH 217 and 218 or permission of instructor.

408/508 Mathematical Problem Solving with Technology (3)

For current and prospective mathematics teachers; built around problem solving experiences. Heuristics for problem solving are developed, and students solve problems in a variety of mathematical areas. Various technologies, including computers and calculators, are used as tools for problem solving. Only for students in licensure or MAT programs.

410/510 Topics In Geometry (3)

A course in an area of geometry; for example: affine and metric geometry, differential geometry, advanced analytic geometry, non-Euclidean geometries, finite geometries.

411/511 Foundations of Geometry (3)

Careful examination of underlying ideas of Euclidean geometry and some non-Euclidean geometries, including projective, metric, and finite. Various approaches include transformations and synthetic treatments. Prerequisite: MTH 222.

420/520 Topics in Algebra (1-4; maximum 8)

Topics selected from an area of modern or linear algebra. Prerequisite: permission of instructor. Offered infrequently.

421/521 Introduction to Abstract Algebra (4)

Elementary theory of groups, rings, integral domains, fields, homomorphisms, and quotient structures. Prerequisite: MTH 222 or 231 and Calculus III.

422/522 Matrices and Linear Algebra (4)

Fields and an introduction to Galois theory. Linear algebra, matrix algebra, determinants, an introduction to modules, and canonical forms. Prerequisite: MTH 222 and 421/521 or 621 or permission of instructor.

425/525 Number Theory (3)

Study of patterns that arise when whole numbers are added, multiplied, subtracted, and factored. A variety of ideas from algebra, geometry, calculus, and set theory contribute to the solution of such problems, and number theory provides surprising connections among these ideas. Once thought to be ‘pure’ mathematics, without applications, number theory is now highly valued in industry and government for its use in encoding and decoding secure transmissions of information. Prerequisite: MTH 421 or permission of instructor.

432/532 Optimization (3)

Optimization of functions of several variables, Lagrange multiplier method, Kuhn-Tucker conditions, linear programming, mathematical programming. Prerequisite: MTH 222 and Calculus III or equivalents or permission of instructor.

435/535 Mathematical Modeling Seminar (3)

Applications of mathematics to real-world situations in a variety of projects. Emphasizes integrating a wide range of mathematical techniques, making oral and written presentations of results, and using both software packages and computer programming for problem solving. Prerequisite: MTH 347 or a 400-level MTH/STA course, or permission of instructor.

437/537 Game Theory and Related Topics (3)

Two-person games with applications. N-person cooperative games with side payments. Various solution concepts for games with applications to social and environmental problems. Power indices for voting games including multi-candidate elections. Related topics such as utility theory, decision theory, measurement theory, fair division or partition function games. Prerequisite: MTH 222 or 231 or permission of instructor.

438/538 Theory and Applications of Graphs (3)

Basic structural properties, trees, connectivity, traversability (Eulerian Tours and Hamiltonian Cycles), vertex and edge colorings, cliques, planarity, and directed graphs. Applications to finding algorithms for shortest path problem, minimum weight tree problem, optimal assignment problem, and other scheduling and transportation problems. Prerequisite: MTH 222 or 231 or permission of instructor.

439/539 Combinatorics (3)

Counting methods: permutations, combinations, generating functions, recurrence relations, inclusion-exclusion. Incidence structures: block designs, Latin squares, finite geometries. Prerequisite: MTH 222 or 231 or permission of instructor.

440/540 Topics in Analysis (1-4; maximum 8)

Topics selected from an area of analysis. Prerequisite: permission of instructor. Offered infrequently.

441/541 Real Analysis (3,3)

Continuity, differentiation, convergence, series and integration, in both one and several variables. Prerequisite: (441/541) MTH 222 or 231 and Calculus III; (442/542) MTH 222 and 441/541.

442/542 Real Analysis (3,3)

Continuity, differentiation, convergence, series and integration, in both one and several variables. Prerequisite: (441/541) MTH 222 or 231 and Calculus III; (442/542) MTH 222 and 441/541.

447/547 Topics in Mathematical Finance (3)

Mathematical methods in options pricing; options and their combinations, arbitrage and put-call parity, stock and option trees, risk neutral pricing, geometric Brownian motion for stock models and derivation of the Black-Sholes formula; and as time allows, additional topics such as futures, forwards, swaps and bond models. Prerequisite: Calculus II and an introduction to statistics such as STA 301 or DSC 205.

451/551 Introduction to Complex Variables (3)

Algebra and geometry of complex numbers, elementary functions of a complex variable including integrals, power series, residues and poles, conformal mapping, and their applications. Prerequisite: MTH 222 or 231 and Calculus III.

453/553 Numerical Analysis (3,3)

Errors and error propagation, theory of polynomial equations, difference equations, solutions of nonlinear equations, programming of algorithms, numerical methods for polynomial equations. Prerequisite: MTH 222 and Calculus III and knowledge of computer programming.

454/554 Numerical Analysis (3,3)

Interpolating polynomials, approximation methods, difference methods, numerical differentiation and integration, numerical solution of differential equations. Prerequisite: MTH 347 and 453/553.

470/570 Topics in Combinatorics/Graph Theory (3)

The single topic covered and the prerequisite varies; consult the instructor. Typical topics include parallel computation networks, error-correcting codes, probabilistic methods in combinatorics, and combinatorics of finite sets. Prerequisite: permission of instructor. Summer only; offered infrequently.

483/583 Introduction to Mathematical Logic (3)

Survey of topics that bear upon the nature of pure mathematics and logic. Special attention given to first-order mathematical logic with related discussions of such topics as mathematical linguistics, theory of effective computability, nonstandard analysis, and foundations of mathematics. Prerequisite: MTH 421 or 441 or permission of instructor.

485/585 Mathematical Recreations (2-3; maximum 3)

Topics in mathematics of a recreational nature, including algebraic puzzles, magic squares, network problems, and mathematical games. Other topics selected from computational tricks, geometric and topological problems, logic puzzles, and cryptography. Prerequisite: MTH 411 or 421 or permission of instructor.

486/586 Introduction to Set Theory (3)

The Recursion Theorem, Cardinality, Cardinal Numbers, Well-orderings, Ordinals, The Axiom of Replacement, Transfinite Induction and Recursion, Ordinal Arithmetic, The Axiom of Choice, Cardinal Exponentiation, Ultrafilters, Stationary Sets. Prerequisites: MTH 222 and Calculus III.

491/591 Introduction to Topology (3)

Elementary set theory and cardinality, metric spaces and topological spaces, sequence convergence, complete metric spaces, Baire Category Theorem, continuity, uniform continuity, bases for a topology space, first and second countability, relationships among separable, Lindelof and second countable properties, product topology, separation axioms, Urysohn’s Lemma, Tietze Extension Theorem, compactness, characterizations of compactness in metric spaces, Tychonoff Theorem, local compactness, connectedness. Prerequisite: MTH 222 and Calculus III.

492/592 Topics in Topology (3)

Content selected to include some of the following: topology of surfaces (Klein bottle, Moebius strip, torus, etc.), fixed point theorems, vector fields, networks, homotopy, homology, knot theory. Prerequisite: MTH 491/591 or permission of instructor. Summer only; offered infrequently.

600 Topics in Advanced Mathematics (1-4; maximum 10)

Prerequisite: permission of department chair.

604 Discrete Mathematics for Secondary School Teachers (3)

For high school teachers. Selected topics, such as: algorithms, Boolean algebra, combinatorics, difference equations, functions, graphs, and networks. For students in mathematics and statistics programs, credit may only be applied to the degree Master of Arts in Teaching. Prerequisite: licensure in secondary school mathematics or permission of instructor. Summer only; offered every third summer.

605 Calculus for Secondary School Teachers (3)

For high school teachers. A return to the main topics of calculus with more emphasis on theory, applications, and historical development than in the usual introductory course. For students in mathematics and statistics programs, credit may only be applied to the degree Master of Arts in Teaching. Prerequisite: licensure in secondary school mathematics or permission of instructor. Summer only.

606 Geometry for Secondary School Teachers (3)

For high school teachers. Re-examination of traditional material of secondary-school geometry from an advanced viewpoint. Recent developments on content and methods are included. For students in mathematics and statistics programs, credit may be applied only to the degree Master of Arts in Teaching. Prerequisite: licensure in secondary school mathematics or permission of instructor. Summer only.

607 Algebra for Secondary School Teachers (3)

For high school teachers. An in-depth development of selected topics with their applications and history. Theory of equations, number theory, number systems, complex numbers, systems of equations, matrices, determinants, algebraic structures. For students in mathematics and statistics programs, credit may be applied only to the degree Master of Arts in Teaching. Prerequisite: licensure in secondary school mathematics or permission of instructor. Summer only; offered every third summer.

620 Topics in Algebra (1-4; maximum 8)

Topics selected from an area of algebra. Prerequisite: permission of department chair. Offered infrequently.

621 Abstract Algebra I (4)

Sylow theory, composition series, polynomial rings. Galois theory of fields, modules over a principal ideal domain and their application. Prerequisite: MTH 421/521 or permission of department chair.

622 Abstract Algebra II (3)

Continued study of structures from MTH 621 together with algebras, tensor products, radicals, chain conditions and dimension, within one of the frameworks: commutative algebra, artinian rings, homological algebra, or Lie algebras. Prerequisite: MTH 621.

630 Topics in Operations Research (1-4; maximum 8)

Special topics selected from game theory, combinatorics, graph theory, optimization, computer algorithms, and other subjects under general heading of operations research. Prerequisite: permission of instructor.

632 Advanced Optimization (3)

Separation of convex sets, linear programming duality, convex analysis and Lagrangian duality, Fritz John and Kuhn-Tucker conditions, penalty function methods. Prerequisite: MTH 432/532 and 441/541 or permission of instructor.

638 Advanced Graph Theory (3)

Advanced treatment of graph theory. Extremal problems, probabilistic, algebraic, and topological aspects of graph theory, analysis of graph algorithms, Ramsey theory. Prerequisite: MTH 438/538 or permission of instructor.

641 Functions of a Real Variable (4)

Lebesgue measure, Lebesgue integration, differentiation, general measures and integration, Radon- Nikodym theorem, Fubini theorem, classical Lp spaces, Banach spaces. Prerequisite: MTH 491/591.

651 Functions of a Complex Variable (4)

Complex number system, analytic functions, complex integration and calculus of residues, representation, analytic continuation, Riemann mapping theorem. Prerequisite: MTH 441/541 and 451/551.

690 Advanced Topics in Topology (1-4; maximum 8)

Contents selected from: algebraic topology, differential topology, topological algebra, uniform spaces and proximity spaces, generalized metric spaces, dimension theory. Prerequisite: MTH 491/591 or permission of instructor. Offered infrequently.

691 Topology (4)

Topological spaces, separation, product and quotient spaces, connectedness, compactness, metric spaces, convergence. Prerequisite: MTH 491/591.

698 Seminar in the Teaching of First-Year Mathematics and Statistics (1)

Required of all newly appointed graduate assistants, this seminar deals with practical problems encountered in teaching algebra, trigonometry, statistics, and calculus. Credit does not count toward a graduate degree in mathematics or statistics. Offered on credit/no-credit basis only. Prerequisite: graduate standing and teaching responsibilities in mathematics or statistics. Summer only.

700 Research for Master’s Thesis (1-12; minimum 6, maximum 12)