Professor Eldon Miller, Acting Chair
305 Hume Hall

Professors Alexander, Buskes, Hopkins, Labuda, Paterson, and Staton
Associate Professors Bowman, Cole, Kranz, and Reid
Assistant Professors Denley, K. Wu, and H. Wu

MASTER OF ARTS OR MASTER OF SCIENCE

Program
The programs for the master's degree in mathematics are designed to meet the needs of four groups: (1) students attracted to mathematics as a major scholarly pursuit, including students who plan eventually to work toward the doctorate in this field; (2) students preparing for the teaching of mathematics, particularly in high schools and community colleges; (3) students preparing for nonteaching professions or vocations, such as Civil Service, actuarial work, or statistical work, in which mathematics plays a principal part; (4) students who wish to supplement study in other fields with suitable courses in mathematics.

Prerequisite
The full four-semester sequence of calculus is prerequisite to all graduate courses. Prerequisite to a major graduate program is a background preparation in mathematics equivalent to the undergraduate major in the College of Liberal Arts; that is, courses through calculus, supplemented by at least 18 hours in mathematics on the higher level which is to include the advanced calculus sequence.
Foreign Language Requirement o A reading knowledge of French or German is desirable, especially for the first group named above, but is not a requirement for the master's degree.

MASTER OF SCIENCE

A candidate for the Master of Science degree must complete 30 graduate hours, including at least two of the following three sequences: Modern Algebra (MATH 525, 526); Theory of Functions of Real Variables (MATH 653, 654); and Theory of Functions of Complex Variables (MATH 655, 656). The candidate may satisfy the 30 semester hour requirement in one of three ways: 1) 30 hours of graduate mathematics; 2) 24 hours of graduate mathematics and an approved 6-hour minor; or 3) 24 hours of graduate mathematics and an approved master's thesis.

MASTER OF ARTS
A candidate for the Master of Arts degree must complete 30 graduate hours, including the first course from five of the following seven sequences: Topology (MATH 501, 502); Modern Algebra (MATH 525, 526); Applied Probability (MATH 573, 574); Statistics (MATH 575, 576); Theory of Functions of Real Variables (MATH 653, 654); Theory of Functions of Complex Variables (MATH 655, 656); and Graph Theory (MATH 681, 682). The M.A. candidate must complete the second course in two of these sequences. The candidate may satisfy the 30 semester hour requirement in one of three ways: 1) 30 hours of graduate mathematics; 2) 24 hours of graduate mathematics and an approved 6-hour minor; or 3) 24 hours of graduate mathematics and an approved master's thesis.

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DOCTOR OF PHILOSOPHY

Requirements
An advisory committee consisting of five members of the graduate faculty will be appointed for each graduate student who declares his or her intention to become a candidate for the degree. The candidate must complete a minimum of 48 course hours of graduate work, exclusive of the dissertation. This must include the sequences Modern Algebra (MATH 525, 526); Theory of Functions of Real Variables (MATH 653, 654); and Theory of Functions of Complex Variables (MATH 655, 656). Of the 48 course hours, 36 must be in courses open only to graduate students. Reading knowledge of one foreign language is required; French, Russian, or German is recommended. This requirement may be satisfied by the completion of six hours of an undergraduate language at the sophomore level or by making an appropriate score on the Graduate School Foreign Language Test of the Educational Testing Service.

Written exams will be administered covering the required sequences and one other approved sequence. In addition, the candidate must satisfy the advisory committee as to the extent of the candidate's research ability and activity, as well as the suitability and excellence of course work presented. Prospective students are advised to communicate and consult freely.

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Course Descriptions

Mathematics - MATH

501, 502. GENERAL TOPOLOGY. Metric spaces, continuity, separation axioms, connectedness, compactness, and other related topics. Prerequisite: MATH 556. (3, 3).

513, 514. THEORY OF NUMBERS I, II. Congruences; divisibility; properties of prime numbers; arithmetical functions; quadratic forms; quadratic residues. (3, 3).

519. MATRICES. Basic matrix theory, eigenvalues, eigenvectors, normal and Hermitian matrices, similarity, Sylvester's Law of Inertia, normal forms, functions of matrices. (3).

520. LINEAR ALGEBRA. An introduction to vector spaces and linear transformations; eigenvalues and the spectral theorem. (3).

525, 526. MODERN ALGEBRA I, II. General properties of groups, rings, and fields; introduction to ideal theory. (3, 3).

533. TOPICS IN EUCLIDEAN GEOMETRY. A study of incidence geometry; distance and congruence; separation; angular measure; congruences between triangles; inequalities; parallel postulate; similarities between triangles. (3).

537. NON-EUCLIDEAN GEOMETRY. Brief review of the foundation of Euclidean plane geometry with special emphasis given the Fifth Postulate; hyperbolic plane geometry; elliptic plane geometry. (3).

540. HISTORY OF MATHEMATICS. Development of mathematics, especially algebra, geometry, and analysis; lives and works of Euclid, Pythagoras, Cardan, Descartes, Newton, Euler, and Gauss. Prerequisite: Math 305 or consent of instructor. (3).

545. SELECTED TOPICS IN MATHEMATICS FOR SECONDARY SCHOOL TEACHERS. High school subjects from an advanced point of view; their relation to the more advanced subjects. (3).

555, 556. ADVANCED CALCULUS I, II. Limits, continuity, power series, partial differentiation, multiple definite integrals, improper integrals, line integrals; applications. (3, 3). Prerequisite: MATH 305 or consent of instructor.

567, 568. INTRODUCTION TO FUNCTIONAL ANALYSIS. Metric spaces, Normed linear spaces and linear operators. Prerequisite: 556 or consent of instructor. (3, 3).

569. THEORY OF INTEGRALS. Continuity, quasi-continuity, measure, variation, Stieltjes integrals, Lebesgue integrals. (3).

571. FINITE DIFFERENCES. Principles of differencing, summation, and the standard interpolation formulas and procedures. (3).

572. INTRODUCTION TO PROBABILITY AND STATISTICS. Emphasis on standard statistical methods and the application of probability to statistical problems. Prerequisite: MATH 264. (3).

573. APPLIED PROBABILITY. Emphasis on understanding the theory of probability and knowing how to apply it. Proofs are given only when they are simple and illuminating. Among topics covered are joint, marginal, and conditional distributions, conditional and unconditional moments, independence, the weak law of large numbers, Tchebycheff's inequality, Central Limit Theorem. Prerequisite: MATH 264. (3).

574. PROBABILITY. Topics introduced in MATH 573 will be covered at a more sophisticated mathematical level. Additional topics will include the Borel-Cantelli Lemma, the Strong Law of Large Numbers, characteristic functions (Fourier transforms). Prerequisite: MATH 573. (3).

575, 576. MATHEMATICAL STATISTICS I, II. Mathematical treatment of statistical and moment characteristics; frequence distribution; least squares; correlation; sampling theory. Prerequisite: MATH 262. (3, 3).

577. APPLIED STOCHASTIC PROCESSES. Emphasis on the application of the theory of stochastic processes to problems in engineering, physics, and economics. Discrete and continuous time Markov processes, Brownian Motion, Ergodic theory for Stationary processes. Prerequisite: MATH 573 or consent of instructor. (3).

578. STOCHASTIC PROCESSES. Topics will include General Diffusions, Martingales, and Stochastic Differential Equations. (3).

590. TECHNIQUES IN TEACHING COLLEGE MATHEMATICS. Directed studies of methods in the presentation of college mathematics topics, teaching and testing techniques. Z grade. This course is required of all teaching assistants, each semester, and may not be used for credit toward a degree. Prerequisite: departmental consent. (1-3).

597. SPECIAL PROBLEMS. (1-3).

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631. FOUNDATIONS OF GEOMETRY. Development of Euclidean geometry in two and three dimensions using the axiomatic method; introduction to high dimensional Euclidean geometry and to non-Euclidean geometrics. (3).

639. PROJECTIVE GEOMETRY. Fundamental propositions of projective geometry from synthetic and analytic point of view; principle of duality; poles and polars; cross ratios; theorems of Desargues, Pascal, Brianchon; involutions. (3).

647. TOPICS IN MODERN MATHEMATICS. Survey of the more recent developments in pure and applied mathematics. Prerequisite: consent of instructor. (3).

649. CONTINUED FRACTIONS. Arithmetic theory; analytic theory; applications to Lyapunov theory. Prerequisite: consent of instructor. (3).

653, 654. THEORY OF FUNCTIONS OF REAL VARIABLES. The number system; sets, convergence; measure and integration; differentiation; variation; absolute continuity. (3, 3).

655, 656. THEORY OF FUNCTIONS OF COMPLEX VARIABLES I, II. Complex functions; mappings, integration theory, entire functions; topics of current interest. (3, 3).

661, 662. NUMERICAL ANALYSIS I, II. Numerical linear algebra, error analysis, computation of eigenvalues and eigenvectors, finite differences, techniques for ordinary and partial differential equations, stability and convergence analysis. (3, 3).

663. SPECIAL FUNCTIONS. Advanced study of gamma functions; hypergeometric functions; generating function; theory and application of cylinder functions and spherical harmonics. (3).

667, 668. FUNCTIONAL ANALYSIS I, II. Linear spaces; operators and functionals. (3, 3).

669. PARTIAL DIFFERENTIAL EQUATIONS I. Classical theories of wave and heat equations. Prerequisite: MATH 353 or MATH 555. (3).

670. PARTIAL DIFFERENTIAL EQUATIONS II. Hilbert space methods for boundary value problems. Prerequisite: MATH 669. (3).

673, 674. ADVANCED PROBABILITY. Current topics in probability are treated at an advanced mathematical level. Measure theoretic foundations, infinitely divisible laws, stable laws, and multidimensional central limit theorem, strong laws, law of the integrated logarithm. Prerequisite: MATH 654 (or may be taken concurrently). (3, 3).

675. ADVANCED MATHEMATICAL STATISTICS I. Univariate distribution functions and their characteristics; moment generating functions and semi-invariants; Pearson's system; Gram-Charlier series; inversion theorems. (3).

676. ADVANCED MATHEMATICAL STATISTICS II. Multivariate distributions and regression systems; multiple and partial correlation; sampling theory; statistical hypotheses; power and efficiency of tests. (3).

677, 678. ADVANCED STOCHASTIC PROCESSES. Special topics in the mathematical theory of stochastic processes. Separability, Martingales, stochastic integrals, the Wiener process, Gaussian processes, random walk, Ornstein-Uhlenbeck process, semi-group theory for diffusions. Prerequisite: MATH 674. (3, 3).

681, 682. GRAPH THEORY I, II. Topics in graph theory including trees, connectivity, coverings, planarity, colorability, directed graphs. (3, 3).

697. THESIS. (1-12).

700. SEMINAR IN TOPOLOGY. Prerequisite: consent of instructor. (May be repeated for credit). (3).

710. SEMINAR IN ALGEBRA. Prerequisite: consent of instructor. (May be repeated for credit). (3).

750. SEMINAR IN ANALYSIS. Prerequisite: consent of instructor. (May be repeated for credit). (3).

780. SEMINAR IN GRAPH THEORY. Prerequisite: consent of instructor. (May be repeated for credit up to a maximum of 9 hours). (3).

797. DISSERTATION. (1-18).

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