Courses in Mathematics for Graduate Students

Upper Division Courses

The Graduate School will permit up to 10 hours of the 30 hours (10 courses) required for the Master's degree to be satisfied by 400 level courses. Most students in the Master's degree program in Mathematics start by taking at least some 400 level sequences.

• MT-A401: Elementary Theory of Probability (3). Prerequisite: MT-A244. Counting theory; axiomatic probability, random variables, expectation; limit theorems. Applications of the theory of probability to a variety of practical problems. (Fall semester.)
• MT-A402: Introductory Mathematical Statistics. Prerequisite MT-A401. Probability and random sampling; distributions of various statistics; statistical procedures, such as estimation of parameters, hypothesis testing, and simple linear regression. (Spring semester.)
• MT-A411: Elements of Modern Algebra (3): Prerequisite MT-A315. Elementary properties of the integers, sets and mappings, semi-groups, groups, rings, integral domains, division rings and fields. (Fall semester.)
• MT-A412: Linear Algebra (3): Prerequisite: MT-A411 and MT-A315. Advanced linear algebra, including linear transformations and duality, elementary canonical forms, rational and Jordan forms, inner product spaces, unitary operators, normal operators and spectral theory. (Spring semester.)
• MT-A421: Introduction to Analysis (3): Prerequisite: MT-A244 and MT-A315. Real number system, functions, sequences, limits, continuity, differentiation, integration and series. (Fall semester.)
• MT-A422: Metric Spaces (3). Prerequisite: MT-A421. Set theory, real line, topological spaces, separation properties, compactness, metric spaces, metrization. (Spring semester.)
• MT-A425: Theory of Numbers (3). Prerequisite: MT-A244. Fundamental concepts in number theory, with applications to solutions of diophantine equations of the first and second degree. (Occasionally.)
• MT-A441: Foundations of Geometry (3). Prerequisite: MT-A142. Historical background of the study of Euclidean geometry: development of two- dimensional Euclidean geometry from a selected set of postulates. (Occasionally.)
• MT-A447: Non-Euclidean Geometry (3). Prerequisite: MT-A142. The rise and development of the non-Euclidean geometries with intensive study of plane hyperbolic geometry. (Occasionally.)
• MT-A448: Differential Geometry (3). Prerequisite: MT-A244. Classical theory of smooth curves and surfaces in 3-spaces. Curvature and torsion of space curves, Gaussian curvature of surfaces, the Theorema Egregium of Gauss. (Occasionally.)
• MT-A451: Introduction to Complex Variables (3). Prerequisite MT-A244. Complex number system and its operations, limits and sequences, continuous functions and their properties, derivatives, conformal representation, curvilinear and complex integration. Cauchy integral theorems, power series and singularities. (Fall semester.)
• MT-A452: Complex Variables II (3). Prerequisite: MT-A451. This course is a continuation of MT-A451. Topics covered include series, residues and poles, conformal mapping, integral formulas, analytic continuation, Riemann surfaces. (Spring semester.)
• MT-A463: Graph Theory (3). Prerequisite: MT-A244. Basic definitions and concepts, undirected graphs (trees and graphs with cycles), directed graphs, and operation on graphs, Euler's formula, and surfaces. (Occasionally.)
• MT-A473: Fourier Series and Related Boundary Value Problems (3). Prerequisite: Consent of Instructor. Solution of boundary value problems of interest in physics and allied sciences in terms of series of orthogonal functions. (Occasionally.)
• MT-A493: Special Topics (1-4).
• MT-A498: Advanced Independent Study (0-6). Prior permission of sponsoring professor and chairperson required.

Graduate Courses

• MT-A501: Linear Algebra (3): Prerequisite: an advanced undergraduate course in modern algebra. Advanced linear algebra, including linear transformations and duality, elementary canonical forms, rational and Jordan forms, inner product spaces, unitary operators, normal operators and spectral theory. (Spring semester.)
• MT-A502: Metric Spaces (3). Prerequisite: an advanced undergraduate course in analysis. Set theory, real line, topological spaces, separation properties, compactness, metric spaces, metrization. (Spring semester.)
• MT-A505: Business Applications of Mathematics (3) Prerequisite: Permission of Department and/or Dean of the School of Business and Administration. Selected topics from: linear systems; finite mathematics; calculus of rational, exponential and logarithmic functions; business applications. (Fall and Spring semesters.)
• MT-A511: Algebra (3) Prerequisite: MT-A412 or MT-A501. Simple properties of groups, groups of transformations, subgroups, homomorphisms and isomorphisms, theorems of Schreier and Jordan-Hoelder, mappings into a group, rings, integral domains, fields, polynomials, direct sums and modules. (Fall semesters.)
• MT-A512: Algebra II (3) Prerequisite: MT-A511. Rings, fields, bases and degrees of extension fields, transcendental elements, normal fields and their structures. Galois theory, finite fields; solutions of equations by radicals, general equations of degree n. (Spring semester.)
• MT-A521: Real Analysis (3) Prerequisite: MT-A422 or MT-A502. The topology of the reals, Lebesgue and Borel measurable functions, properties of the Lebesgue integral, differential of the integral. (Fall semester.)
• MT-A522: Real Analysis II (3) Prerequisite: MT-A521. Compact and locally compact spaces, the Stone Weierstrauss theorem, measure theory, Radon Nikodym Theorem, the LP-spaces, product invariant measures. (Spring semester.)
• MT-A531: General Topology I (3) Prerequisite MT-A422 or MT-A502. Topological spaces, convergence, nets, product spaces, metrization, compact spaces, uniform spaces and function spaces. (Fall semester.)
• MT-A532: General Topology II (3) Prerequisite: MT-A531 Compact surfaces, fundamental groups, force groups and free products, Seifert-van Kampean theorem, covering spaces. (Spring semester.)
• MT-A593: Special Topics in Mathematics (1-3)
• MT-A595: Special Study for Examinations (0)
• MT-A598: Graduate Reading Course (1-3) Prior permission of instructor and chairperson required.
• MT-A599: Thesis Research (0-6)
• MT-A5CR-90: Master's Degree Study (0)
• MT-A611: Algebra III (3) Prerequisite: MT-A512. Categories and functors, properties of hom and tensor, projective and injective modules, chain conditions, decomposition and cancellation of modules, theorems of Maschke, Wedderburn and Artin-Wedderburn, tensor algebras. (Every other year.)
• MT-A618: Topics in Algebra (3) Prerequisite: MT-A512. Various topics are discussed to bring graduate students to the forefront of a research area in algebra. Times of offering in accordance with research interests of faculty. (Occasionally.)
• MT-A621: Lie Groups and Lie Algebras (3) Prerequisites: MT-A511, MT-A522 and MT-A531. Lie groups and Lie algebras, matrix groups, the Lie algebra of a Lie group, homogeneous spaces, solvable and nilpotent groups, semi-simple Lie groups. (Every other year.)
• MT-A622: Representation theory of Lie Groups (3) Prerequisite: MT-A621. Representation theory of Lie groups, irreducibility and complete reducibility, Cartan subalgebra and root space decomposition, root system and classification, coadjoint orbits, harmonic analysis on homogeneous spaces. (Every other year.)
• MT-A628: Topics in Analysis (3) Prerequisite: MT-A522. Various topics are discussed to bring graduate students to the forefront of a research area in analysis. Times of offering in accordance with research interests of faculty. (Occasionally.)
• MT-A631: Algebraic Topology (3) Prerequisite: MT-A532. Homotopy theory, homology theory, exact sequences, Mayer-Vietoris sequences, degrees of maps, cohomology, Kunneth formula, cup and cap products, applications to manifolds including Poincare-Lefshetz duality. (Every other year.)
• MT-A632: Topology of Manofolds (3) Prerequisite: MT-A631 Examples of manifolds, the tangent bundle, maps between bundles, embeddings, critical values, transversality, isotopies, vector bundles and bubular neighborhoods, cobordism, intersection numbers and Euler characteristics. May be taught in either the piecewise-linear or differentiable categories. (Every other year.)
• MT-A638: Topics in Topology (3) Prerequisite: MT-A532. Various topics are discussed to bring graduate students to the forefront of a research area in topology. Times of offering in accordance with research interests of faculty. (Occasionally.)
• MT-A641: Differential Geometry I (3) Prerequisite: MT-A532 and permission of the instructor. The theory of differentiable manifolds, topological manifolds, differential calculus of several variables, smooth manifolds and submanifolds, vector fields and ordinary differential equations, tensor fields, integration and De Rham cohomology. (Fall semester.)
• MT-A642: Differential Geometry II (3) Continuation of MT-A641. (Spring semester.)
• MT-A648: Topics in Geometry (3) Prerequisite: MT-A532. Various topics are discussed to bring graduate students to the forefront of a research area in geometry. Times of offering in accordance with research interests of faculty. (Occasionally.)
• MT-A695: Special Study for Examinations (0)
• MT-A698: Graduate Reading Course (1-3) Prior permission of instructor and chairperson required.
• MT-A699: Dissertation Research (0-6)
• MT-A6CR-99: Doctor of Philosophy Degree Study (0)

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