Mathematics Courses (MAT) Courses for Advanced Undergraduates & Graduate Students 503 Problem Solving in Mathematics (3:3) Pr. grade of at least C in 191 and 303 or permission of instructor Hours count toward teacher licensure but do not count toward degree requirements for a mathematics major. Investigates the nature of problem solving, covers procedures involved in problem solving, develops individual problem solving skills, and collects a set of appropriate problems. Required for middle grades mathematics concentration. This course cannot be applied toward the requirements for the M.A. degree in Mathematics. 504 Foundations of Geometry for Teachers (3:3) Pr. grade of at least C in 292 or permission of instructor Hours count toward teacher licensure but do not count toward degree requirements for a mathematics major. Primarily for students seeking teacher certification. Includes logic and axiom systems, history, plane and solid Euclidean geometry, proof strategies, introduction to non-Euclidean geometries, and transformational geometry. This course cannot be applied toward the requirements for the M.A. degree in Mathematics. (Fall) 505 Foundations of Mathematics for Teachers (3:3) Pr. grade of at least C in 292 or 303 or permission of instructor Hours count toward teacher licensure but do not count toward degree requirements for a mathematics major. Primarily for students seeking teacher certification. Includes properties and algebra of real numbers; analytic geometry; polynomial, rational, exponential, logarithmic, and trigonometric functions; complex numbers; concept of limits of functions. This course cannot be applied toward the requirements for the M.A. degree in Mathematics. (Spring) 513 Historical Development of Mathematics (3:3) Pr. grade of at least C in 292 Study of the historical development of mathematics, not a history of persons involved in development. This course cannot be applied toward the requirements for the M.A. degree in Mathematics. (Fall) 514 Theory of Numbers (3:3) Pr. grade of at least C in 311 or permission of instructor An introductory course to both multiplicative and additive number theory. Divisibility, prime numbers, congruencies, linear and nonlinear Diophantine equations (including Pell’s equation), quadratic residues, number-theoretic functions, and other topics. 515 Mathematical Logic (3:3) Pr. grade of at least C in 253 or 311 or permission of instructor Formal languages, recursion, compactness, and effectiveness. First-order languages, truth, and models. Soundness and completeness theorems. Models of theories. (Odd Spring) 516 Polynomial Rings (3:3) Pr. grade of at least C in 311 Rings, integral domains, fields, division algorithm, factorization theorems, zeros of polynomials, greatest common divisor, relations between the zeros and the coefficients of a polynomial, formal derivatives, prime polynomials, Euclidean rings, the fundamental theorem of algebra. 517 Theory of Groups (3:3) Pr. grade of at least C in 311 Elementary properties of groups and homomorphisms, quotients and products of groups, the Sylow theorems, structure theory for finitely generated Abelian groups. 518 Set Theory and Transfinite Arithmetic (3:3) Pr. grade of at least C in 311 or 395 The axioms of set theory, operations on sets, relations and function, ordinal and cardinal numbers. 519 Intuitive Concepts in Topology (3:3) Pr. grade of at least C in 311 or 395 Basic concepts, vector fields, the Jordan curve theorem, surfaces, homology of complexes, continuity. 520 Non-Euclidean Geometry (3:3) Pr. grade of at least C in 311 or 395 Fifth postulate, hyperbolic geometries, elliptic geometries, consistency of non-Euclidean geometries, models for geometries, elements of inversion. 521 Projective Geometry (3:3) Pr. permission of instructor Transformation groups and projective, affine and metric geometries of the line, plane, and space. Homogeneous coordinates, principles of duality, involutions, cross-ratio, collineations, fixed points, conics, ideal and imaginary elements, models, and Euclidean specializations. 522 Hilbert Spaces and Spectral Theory (3:3) Pr. grade of at least C in MAT 395 Vector-spaces: basis, dimension, Hilbert spaces; pre-Hilbert spaces, norms, metrics, orthogonality, infinite sums. Linear subspaces; annihilators, closed and complete subspaces, convex sets. Continuous linear mappings; normed spaces. Banach spaces, Banach algebras, dual spaces. Reisz-Frechet theorem. Completion. Bilinear and seaquilinear maps. Adjoints. Operators in Hilbert space: isometric, unitary, self-adjoint, projection, and normal operations. Invariant subspaces. Continuous operators. Special theorems for a normal co-operator. 531 Combinatorial Analysis (3:3) Pr. grade of at least C in 253 or 295 or 311 or 395, or permission of instructor The pigeon-hole principle, permutations, combinations, generating functions, principle of inclusion and exclusion, distributions, partitions, recurrence relations. 532 Introductory Graph Theory (3:3) Pr. grade of at least C in 310 and any one of the courses 253, 295, 311, 395, 531 Basic concepts, graph coloring, trees, planar graphs, networks. 540 Complex Functions with Applications (3:3) Pr. grade of at least C in 293 The complex number system, holomorphic functions, power series, complex integration, representation theorems, the calculus of residues. 541, 542 Stochastic Processes (3:3), (3:3) Pr. grade of at least C in MAT 394 and either MAT 353 or STA 351, or equivalents Markov processes, Markov reward processes, queuing, decision making, graphs, and networks. Applications to performance, reliability, and availability modeling. 545 Differential Equations and Orthogonal Systems (3:3) Pr. grade of at least C in 293 and 390 or permission of instructor An introduction to Fourier series and orthogonal sets of functions, with applications to boundary value problems. 546 Partial Differential Equations with Applications (3:3) Pr. grade of at least C in 545 Fourier integrals, Bessel functions, Legendre polynomials and their applications. Existence and uniqueness of solutions to boundary value problems. 549 Topics in Applied Mathematics (3:3) Pr. grade of at least C in 293 and 390 or permission of instructor May be repeated for credit with approval of the Department Head. Selected topics of current interest in applied mathematics. 556 Advanced Discrete Mathematics (3:3) Pr. grade of at least C in 253 or permission of instructor Advanced topics in discrete mathematics and their uses in studying computer science. 591 Advanced Modern Algebra (3:3) Pr. grade of at least C in 311 Set theory: sets, mappings, integers. Group theory: normal subgroups, quotient groups, permutation groups, Sylow theorems. Ring theory: homomorphisms, ideals, quotient rings, integral domains, fields, Euclidean rings, polynomial rings. 592 Abstract Algebra (3:3) Pr. grade of at least C in 591 or 311 and permission of instructor Fields: extensions, transcendental elements, roots of polynomials, Euclidean constructions. Galois theory, solvability by radicals. 593, 594 Directed Study in Mathematics (1–3), (1–3) (Fall & Spring)
595, 596 Mathematical Analysis (3:3), (3:3) Pr. 395 or permission of instructor Real number axioms, metric spaces, sequences, series, continuity, differentiation, the Reimann-Stieltjes integral. Please refer to The Graduate School Bulletin for additional graduate-level courses. |
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