MATHEMATICS /135 Smith; C. W. Stark; S.J. Sumners; H. K. Voelklein; T. Walsh. AssistantProfessors: F. G. Garvan;S. A. McCullough; P. Sin. The Department of Mathematics offers the degrees of Doctor of Philosophy, Master of Science and Master of Arts, and the Master of Arts in Teaching and Master of Science in Teaching, each with a major in mathematics. The Department also has an accelerated bachelor's/ master's program designed for superior undergraduate students who have the ability to pursue such a plan of study leading to the Master of Science or Master of Arts degree. The main feature of the program is that up to 21 semester hours of approved graduate level mathematics courses may be used as dual credit for both the undergraduate and the graduate degree. All other requirements for both the bachelor's degree and the master's degree must be met. For admission requirements for this program, see the Associate Chair for the Undergraduate Program. There are opportunities for concentrated study in a number of specific areas of pure and applied mathematics at both the master's and doctoral levels. The faculty directs studies and research in algebra, number theory, analysis, geometry, topology, logic, differential equations, dynami- cal systems, control theory, probability theory, mathemati- cal systems theory, numerical analysis, approximation theory, combinatorial analysis, graph theory, computer applications, biomathematics, and mathematical physics. In addition to the requirements of the Graduate School, the minimum prerequisite for admission to the program of graduate studies in mathematics is the completion, with an average grade of B or better, of at least 24 credits of undergraduate mathematics, including a full year of calcu- lus and three semesters of appropriate work beyond the calculus. The most appropriate courses for this purpose are advanced calculus, abstract algebra, and linear algebra. Students lacking partof the requirements will be required to make up the deficiency early in their graduate work. Prerequisites to individual courses should be determined before registration by consultation with the instructor con- cerned. Some of the courses listed are offered only as needed. Since times of offering courses are estimated a year in advance, certain changes may be made if needs are known by the department. The courses MAA 5228, 5229, MAS 5311, and 5312 are required for all advanced degree programs in mathematics. The requirements for the master's degree include 32 semester hours of course work and a comprehensive writ- ten examination. A thesis is not required. There are two master's specializations available, one in pure mathematics and one in applied mathematics. A student normally takes two years to complete either program. The requirements for a doctoral degree include 36 hours of 6000-level course work in mathematics. No hours of teaching, colloquium, dissertation, or individual work will count toward this requirement. The doctoral student must pass a written and oral com- prehensive preliminary examination administered by the Department to become a candidate for the degree. The doctoral student must pass reading knowledge examina- tions in one of the following foreign languages: French, German, or Russian. The dissertation is an important requirement for the doctoral degree in mathematics. The topic for the disserta- tion may be chosen from a number of areas of current research in pure and applied mathematics. Every graduate student is expected to attend the regular colloquium. Details concerning all requirements for graduate degrees in mathematics may be obtained by writing the Mathemat- ics Department Graduate Selection Committee. MAA 5104-Advanced Calculus for Engineers and Physical Scientists I (3) MAA 5105-Advanced Calculus for Engineers and Physical Scientists 11 (3) Prereq: MAA 5104. MAA 5228-Modern Analysis 1 (3) Prereq:MAC3313. Topology of metric spaces, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, sequences and series of functions, the Stone-Weierstrass theorem, functions of several variables. Stokes' theorem, the Lebesgue theory. MAA 5229-Modern Analysis 11 (3) Prereq: MAA 5228. MAA 5404-Introduction to Complex Variables for Engineers and Physical Scientists (3) MAA 5506-Introduction to Functional Analysis I (3) MAA 5507-Introduction to Functional Analysis 11 (3) Prereq: MAA 5506. MAA 6236-Mathematical Analysis for Statisticians (4) Coreq: STA 6326. Numerical sequences and series, limits, continuity, differentiation, integration, series of functions. Applications to probability and statistics stressed. MAA 6406--Complex Analysis I (3) Prereq: MAA 5229. Rapid survey of properties of complex numbers, linear transformations, geometric forms and necessary concepts from topology. Complex integration. Cauchy's theorem and its corollaries. Taylor's series and the implicit function theorem in complex form. Conformality and the Riemann-Caratheodory mapping theorem. Theorems of Bloch, Schottky, and the big and little theorems of Picard. Harmonicity and Dirichlet's problems. MAA 6407--Complex Analysis II (3) Prereq: MAA 6406. MAA 6616-General Theory of Measure and Integration I (3) Prereq: MAA 5229. Borel sets, measurable functions and the monotone class theorem. Measures and their extension theorems, Lebesgue integral, convergence theorems. Product measures and Fubini's theorem, the Radon-Nikodym theorem and differentia- tion. Elementary Hilbert and Banach space theory, LP-spaces. MAA 6617-General Theory of Measure and Integration II (3) Prereq: MAA 6616. MAA 7526-Advanced Topics in Functional Analysis 1 (3; max: 6) Prereq: MAA 6617, MAS 6332. Algebraic and topological approach to current material and methods in analysis. MAA 7527-Advanced Topics in Functional Analysis II (3; max: 6) Prereq: MAA 7526. MAD 5405-Numerical Methods of Differential Equations (3) Prereq: MAS 3113, 3114, or 4105; and programming language. Numerical methods for the solution of differential equations. MAD 6206-Combinatorial Theory I (3) Matching theory, Ramsey's theorem, lattice theory, Mobius inversion, generating functions. Polya's theorem, matroids, applications, block designs, graph theory. MAD 6207-Combinatorial Theory 11 (3) Prereq: MAD 6206. MAD 6406-Numerical Linear Algebra (3) Prereq: MAS 3114, 4105, or4124; andprogramming language. Topics most useful in applications with emphasis on numerical techniques: systems of linear equations, positive definite and toeplitz systems, least squares problems, singular value decomposition, and eigenval- ues. Numerical stability and efficiency of algorithms as well as effectof perturbations on the problem. Companion to MAD 6407. MAD 6407-Numerical Analysis (3) Prereq:MAA 4212,5105, or 5229;andprogramminglanguage. Numerical techniques to solve systems of nonlinear equations to approximate functions, to compute derivatives, to evaluate integral, and to integrate sys- tems of differential equations. Introduction to numerical tech- niques for partial differential equations. Companion to MAD 6406. MAD 7396-Topics in Combinatorial Theory 1 (3; max 6) Prereq: MAS 5312. Topics chosen from among graph theory, coding theory, matroid theory, finite geometries, projective geometry, difference methods, and Latin squares.