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MATHEMATICAL SCIENCES (MASC)

William A. Feldman, Department Chair, 538 Hotz Hall, (479) 575-3351

Mark Arnold, Coordinator of Graduate Studiesm, 538 Hotz Hall, (479) 575-3351

Distinguished Professor Schein • Professors Akeroyd, Brewer, Cochran, Duncan, Feldman, Khavinson, Luecking, Madison, Tabachnikov, Tubbs • Associate Professors Arnold, Goodman-Strauss, Johnson, Meaux, Meek, Monroe, Ryan • Assistant Professors Capogna, Lanzani, Petris, Woodland

Degrees Conferred:

M.S., Ph.D. in Mathematics (MATH)

M.A. in Secondary Mathematics (SMTH)

M.S. in Statistics (STAT)
(See Statistics)

Prerequisites to Degree Program: Prospective candidates for the Master of Science degree in mathematics are expected to have completed a program equivalent to that required by the Department for a B.S. degree, as set forth in the current catalog of the Fulbright College of Arts and Sciences. Deficiencies may be removed either by taking the appropriate undergraduate courses or by examination.

The degree of Master of Science is intended for collegiate teachers of mathematics, non-teaching professional mathematicians, and those who desire to continue advanced study.

Requirements for the Master of Science Degree: This degree is offered under two separate options, a general option and a computational mathematics option. The general option is intended for students who plan to be collegiate teachers of mathematics, continue advanced study in mathematics, or obtain a broad background for preparation as a non-teaching professional mathematician. The computational mathematics option is intended for students who intend to specialize in computational and applied mathematics in preparation for professional employment in an inter-disciplinary or computationally intensive environment.

The program of a candidate will be determined in conference with the candidate's graduate adviser. A comprehensive examination must be passed by each candidate for the Master of Science degree. It should be taken near the end of the last semester of residence. Normally this examination will be given during the thirteenth week of each semester. At least four weeks prior to the scheduled date students must notify the Department of their intention to take the examination. No student may take the comprehensive examination more than three times. MATH 4513, 5013, 5033 and 504V are not applicable to the Master of Science degree in mathematics. The program will include at least two semesters of one hour credit in Mathematics Seminar, MATH 510V.

The candidate for the general option must complete a minimum of 32 semester hours of approved graduate work. Students may include up to nine semester hours of graduate work in courses outside the Department. All selected courses are subject to the approval of the Graduate Committee. The comprehensive examination for the general option will include material covered in six semester hours of graduate courses in each of: (1) abstract algebra, (2) topology, (3) real or complex analysis, and (4) an area chosen by the candidate and approved by the Graduate Committee. When there is a choice in the above list of topics, students shall make their choice not less than four weeks before the date of the examination.

The candidate for the computational mathematics option must complete a minimum of 32 semester hours of approved graduate work. Students must include at least six but not more than twelve semester hours of graduate work in courses outside of mathematics. All selected courses are subject to the approval of the Graduate Committee. The comprehensive examination for the computational mathematics option will include material covered in six semester hours of graduate courses in each of: (1) numerical analysis, (2) applied mathematics, (3) analysis or algebra, and (4) an area other than mathematics chosen by the student and approved by the Graduate Committee.

Requirements for the Master of Arts Degree with a Major in Secondary Mathematics: This program is designed for secondary school teachers of mathematics. It requires 32 semester hours of graduate work. All requirements for certification must be fulfilled before the degree will be awarded. Prospective candidates for the Master of Arts degree in secondary mathematics are expected to have earned credit in courses equivalent to MATH 2574, 3083, 3113, and 3773. Deficiencies may be removed either by taking the appropriate courses or by examination.

The candidate's program must include MATH 4513, 5123, two semesters of one hour credit in MATH 510V, and one of the following courses: MATH 5133, 5303, 5313, 5503, 5523 or 5703. Not more than 12 semester hours of credit toward this degree will be allowed from the following categories: (1) "Institute type" mathematics courses and (2) graduate courses in education. All courses selected to apply on this degree must be approved by the student's adviser in accordance with the above requirements. Recommended courses include MATH 4103, 4253, 4353, 4363, 4523, and either STAT 3013 or 5103.

Each person receiving the Master of Arts degree in secondary mathematics must pass a written examination covering: (1) algebra, MATH 5123, (2) advanced calculus, MATH 4513, (3) geometry, and one other area of mathematics to be approved by the candidate's adviser. The examination schedule is the same as for the Master of Science degree. No student will be allowed to take the examination more than three times.

Requirements for the Doctor of Philosophy Degree: Candidates for the degree of Doctor of Philosophy with a major in mathematics will be required to earn not less than 60 semester hours of course credit beyond the bachelor's degree in mathematics and closely related fields. The number of hours and the courses for each student will be determined by the advisory committee. The candidate must fulfill the course requirements for the Master of Science degree in mathematics.

The basic requirement for the Ph.D. degree is the preparation of an acceptable dissertation. This dissertation must demonstrate the candidate's ability to do independent, original, and significant work in mathematics. It is required that this dissertation possess the degree of excellence of research papers ordinarily published in the leading mathematical journals.

A comprehensive examination is given each year during the weeks preceding the beginning of the fall and spring semesters. This examination is taken by all students in the graduate program who have completed the requirements for the M.S. degree, and who have not been admitted to candidacy for the Ph.D. degree. The examination serves as both a qualifying and candidacy examination. The prospective candidate for the Ph.D. will be allowed to take the examination at most three times. Two failures to qualify eliminates the student from the graduate program in mathematics.

Each candidate will be required to pass proficiency examinations in one of the languages French, German, and Russian, which are given by the Department of Foreign Languages.

In addition to extending knowledge by personal reading and research, a doctoral graduate in mathematics will normally communicate knowledge to others. Therefore each student in the Ph.D. program is required to acquire the equivalent of one semester of full-time experience in teaching; this requirement may be fulfilled by part-time experience over several semesters. Typically, teaching assistantship appointments will satisfy this requirement, but other similar experience may qualify as approved by the Department.

COURSES: MATHEMATICS (MATH)

MATH4103 Finite Dimensional Vector Spaces (IR) Linear functionals, matrix representation of linear transformations, scalar product, spectral representation of linear transformations. Prerequisite: MATH 3083.

MATH4113 Introduction to Abstract Algebra II (FA) Topics in abstract algebra including finite abelian groups, linear groups, factorization in cummutative rings, quadratic field extensions, Gaussian integers, Wedderburn's theorem, and multilinear algebra. Prerequisite: MATH 3113.

MATH4203 Linear Programming and Game Theory (IR) Solution sets, duality, and pivoting in linear programming. Feasible solutions and the simplex method. The transportation problem. Matrix games. Prerequisite: MATH 3083 and proficiency in a high-level computer language.

MATH4253 Symbolic Logic I (FA) Rigorous analyses of the concepts of proof, consistency, equivalence, validity, implication, and truth. Full coverage of truth-functional logic and quantification theory (predicate calculus). Discussion of the nature and limits of mechanical procedures (algorithms) for proving theorems in logic and mathematics. Informal accounts of the basic facts about infinite sets. (Same as PHIL 4253)

MATH4263 Symbolic Logic II (SP) Topics include: soundness and completeness of propositional logic, soundness and completeness of quantification theory, the elements of model theory and recursion theory, G]odel's incompleteness theorems, and the limitative theorems of Tarski and Church. (Same as PHIL 4263) Prerequisite: MATH 4253 or PHIL 4253.

MATH4353 Numerical Linear Algebra (SP) Numerical methods for problems of linear algebra, including the solution of very large systems, eigenvalues, and eigenvectors. Prerequisite: MATH 3083 and programming experience.

MATH4363 Numerical Analysis (FA) General iterative techniques, error analysis, root finding, interpolation, approximation, numerical integration, numerical solution of differential equations. Prerequisite: MATH 4513 and programming experience.

MATH4503 Differential Geometry and Vector Calculus (IR) Topics include: Vector differential and integral calculus, Stokes' Theorem in 3-space, classical differential geometry in 3-space (curves, surfaces), differential forms, general Stokes' Theorem, applications to hydrodynamics, and electromagnetism. Prerequisite: MATH 3083 and MATH 4513.

MATH4513 Advanced Calculus I (FA) The real and complex number systems, basic set theory and topology, sequences and series, continuity, differentiation, Taylor's theorem. Emphasis is placed on careful mathematical reasoning. Prerequisite: MATH 2574 and MATH 3083.

MATH4523 Advanced Calculus II (SP) The Riemann-Stieltjes integral, uniform convergence of functions, Fourier series, implicit function theorem, Jacobians, and derivatives of higher order. Prerequisite: MATH 4513.

MATH5013 Topics in Algebra for Teachers (IR) Topics from abstract and linear algebra of current interest to teachers. May be repeated. Prerequisite: graduate standing.

MATH5033 Topics in Analysis for Teachers (IR) Topics related to calculus of current interest to secondary school teachers. May be repeated. Prerequisite: graduate standing.

MATH504V Special Topics for Teachers (1-6) (IR) Current topics in mathematics of interest to secondary school teachers. May be repeated. Prerequisite: graduate standing.

MATH510V Mathematical Seminar (1-3) (FA) Members of the faculty and advanced students meet for presentation and discussion of topics. Prerequisite: graduate standing.

MATH5123 Algebra I (SP) What the beginning graduate student should know about algebra: groups, rings, fields, modules, algebras, categories, homological algebra, Galois Theory. Prerequisite: MATH 3113.

MATH5133 Algebra II (FA) Continuation of 5123. Prerequisite: MATH 5123.

MATH5303 Ordinary Differential Equations (FA) Existence, uniqueness, stability, qualitative behavior, and numerical solutions. Prerequisite: MATH 3404 and MATH 4513 and programming experience.

MATH5313 Partial Differential Equations (SP) Classification, boundary value problems, applications, numerical solutions. Prerequisite: MATH 3423 and MATH 4513.

MATH5503 Theory of Functions of a Real Variable I (FA) Real number system, Lebesque measure, Lebesque integral, convergence theorems, differentiation of monotone functions, absolute continuity and the fundamental theorem of calculus L^P spaces, Holder and Minkowski inequalities, bounded linear functionals on the L^P spaces. Prerequisite: MATH 4523.

MATH5513 Theory of Functions of a Real Variable II (SP) Measure and integration on abstract measure spaces, signed measures, Hahn decomposition, Radon-Nikdoym theorem, Lebesque decomposition, measures on algebras and their extensions, product measures, Fubini's theorem. Prerequisite: MATH 5503.

MATH5523 Theory of Functions of a Complex Variable I (FA) Complex numbers, analytic functions, power series, complex integration, Cauchy's Theorem and integral formula, maximum principle, singularities, Laurent series, Mibius maps. Prerequisite: MATH 4513.

MATH5533 Theory of Functions of a Complex Variable II (SP) Riemann Mapping Theorem, analytic continuation, harmonic functions, entire functions. Prerequisite: MATH 5523.

MATH5703 Foundations of Topology (FA) Metric and general topological spaces, separation axioms, Urysohn's lemma, Tietze extension theorem, connectedness, compactness, and the Tychonoff theorem. Prerequisite: MATH 4513.

MATH5713 Algebraic Topology (FA) Homotopy, singular and relative homology, excision theorem, the Mayer-Vietoris sequence, Beti numbers, and the Euler characteristic. Prerequisite: MATH 5703.

MATH600V Master's Thesis (1-6) (FA, SP, SU) Prerequisite: graduate standing.

MATH610V Directed Readings (1-6) (IR)

MATH619V Topics in Algebra (1-6) (FA, SP, SU) Current research interests in algebra. May be repeated.

MATH659V Topics in Analysis (1-6) (FA, SP, SU) Current research interests in analysis. May be repeated.

MATH679V Topics in Topology (1-6) (FA, SP, SU) Current research interest in topology. May be repeated.

MATH700V Doctoral Dissertation (1-6) (FA, SP, SU)