Mathematics

MATHEMATICS

Chair: Gerald M. Armstrong
Associate Chair: Gurcharan S. Gill
Graduate Coordinator: William E. Lang
292 TMCB
Provo, UT 84602-6542
(801) 378-2062

THE PROGRAM OF STUDIES

The Department of Mathematics has approximately forty graduate students, most of whom are supported by teaching assistantships for which they receive tuition support as well as a stipend for providing teaching support in college algebra and calculus.

Three degrees are offered through the Department of Mathematics: Mathematics—MS; Mathematics—MA; and Mathematics—PhD.

MS and MA students study mathematics courses in preparation for careers in business, industry, government, or education. Other students use a master's degree in mathematics in preparation for a doctoral degree in mathematics or a closely related discipline or a discipline where technical competence is appreciated. Master's students graduate in an average of two years.

The department supports from ten to twelve PhD students. Designed for gifted and dedicated students, the program requires about four years past a master's degree. The department has special strength in the areas of applied mathematics, algebraic geometry, analytic number theory, geometric topology and group theory, and linear analysis.

Mathematics—MS

The master of science is designed to prepare students for positions in business and industry. It also provides preparation for further graduate study leading to a doctoral degree.

Information for Degree—Thesis and Nonthesis Programs. Graduate mathematics courses: approved graduate mathematics courses include all classes numbered 500 and above with the exceptions of 501, 502, 529R, and 629. Faculty sponsor: the graduate coordinator will assign each student a faculty sponsor on admission to the graduate program. Students should communicate with the sponsor as soon as they arrive on campus.

Admission and Entry.

• Semesters of entry and application deadlines: fall, winter, spring, summer, March 1 (U.S. and international).
• Entrance examinations: GRE general test and subject test in mathematics. Every international applicant whose native language is not English is required to submit Test of English as a Foreign Language (TOEFL) scores.
• Prerequisite: credit at least equivalent to BYU requirements for a baccalaureate degree in mathematics; a year's sequence in abstract algebra; and a year's sequence in advanced calculus.

Requirements for Degree—Thesis Program.

• Credit hours (30): minimum 24 course work hours in approved graduate mathematics including 12 hours in courses numbered 600 or above and 6 thesis hours (Math 699R).
• Examination: pass a written master's examination. The examination should be passed by the end of the student's first semester of the second year, after which two more attempts are allowed.
• Thesis.
• Oral defense of thesis.

Requirements for Degree—
Nonthesis Program.

• Credit hours:

Traditional Mathematics Option (32): minimum 30 course work hours in approved graduate mathematics including 18 hours in courses numbered 600 or above and 2 project hours (698R).

Minor Option (35): minimum 24 course work hours in approved graduate mathematics including 6 hours in courses numbered 600 or above, 9 hours in an approved minor, and 2 project hours (698R).

Applied Option (38): minimum 24 course work hours in approved graduate mathematics including 6 hours in courses numbered 600 or above, 12 hours in areas related to applications of mathematics, and 2 project hours (698R). The 12 hours of application must be approved by the graduate coordinator.

• Project and presentation: write a paper on an area of advanced mathematics and give a 45-minute presentation based on the paper.
• Examination: pass a written master's examination. The examination should be passed no later than the end of the first semester of the second year, after which two more attempts are allowed.

Mathematics—MA

The MA curriculum is designed to prepare students for teaching mathematics in a secondary school, or for a doctoral program in mathematics education.

Admission and Entry.

• Semesters of entry and application deadlines: fall, winter, spring, summer, March 1 (U.S. and international).
• Entrance examination: GRE general test and subject test are recommended (required for international applicants).
• Prerequisite: credit at least equivalent to current BYU requirements for a BA degree in education with a teaching major in mathematics or a BA degree in mathematics.

Requirements for Degree—Thesis Program.

• Credit hours (30): minimum 24 course work hours and 6 thesis hours (Math 699R).
• State teacher certification (required certification courses may not be part of the graduate program).
• Examination: pass a written master's examination. The examination should be passed by the end of the student's first semester of the second year, after which two more attempts are allowed.
• Required courses: Math 316, 372 (if not taken as undergraduate classes), 629; any two-semester 500 or 600 sequence.
• Minor (optional): any approved minor.
• Thesis.
• Oral defense of thesis.

Requirements for Degree—
Nonthesis Program.

• Credit hours (32): minimum 30 course work hours and 2 project hours (698R).
• State teacher certification (required certification courses may not be part of the graduate program.)
• Examination: pass a written master's examination. The examination should be passed by the end of the student's first semester of the second year, after which two more attempts are allowed.
• Required courses: Math 316, 372 (if not taken as undergraduate classes), 629; any two-semester 500 or 600 sequence.
• Minor (optional): any approved minor
• Project and presentation: write a paper on an area of advanced mathematics or mathematics education and give a 45-minute presentation based on the paper.

Mathematics—PhD

The doctoral program prepares students for a career in research and teaching at the university level or in basic research in a nonacademic setting.

Admission and Entry.

• Semesters of entry and application deadlines: fall, winter, spring, summer, March 1 (U.S. and international).
• Entrance examinations: GRE general test and GRE subject test in mathematics. Every international applicant whose native language is not English is required to take the Test of English as a Foreign Language (TOEFL).
• Prerequisite: undergraduate degree in mathematics or its equivalent; one year of mathematical analysis (or advanced calculus); one year of abstract algebra, including linear algebra.

Requirements for Degree.

• Credit hours (54): minimum 36 course work hours in mathematics courses numbered 600 or above with a grade of B or better in each, plus 18 dissertation hours (Math 799R).
• Required courses: complete at least 3 hours each in algebra, analysis, applied mathematics, and geometry/ topology.
• Examinations:

Written Examinations: at the beginning of the second year, pass examinations in three of the four areas of algebra, analysis, applied mathematics, and geometry/topology. Four hours are allotted to each examination. A failed examination may be repeated once at the beginning of the winter semester of the student's second year, after which permission must be obtained from the department graduate committee to retake the examination. Passed examinations need not be repeated. Syllabi are available for each examination.

Oral Examination: pass an oral qualifying examination covering the background necessary for research in a specific area. The student, having chosen a research area and having a dissertation advisor approved, will, with the advisor, outline suitable examination topics. These topics must be approved by an examination committee of three (including advisor) appointed by the department graduate committee, which conducts the examination.

Defense of Dissertation: a final oral defense of the dissertation is conducted by a faculty committee consisting of the student's research advisor, two other readers of the dissertation (one of whom may be an outside examiner) and two other members of the faculty.

• Language requirement: demonstrate proficiency in two approved foreign languages that are currently in major use in the mathematical literature. At present the approved languages are French, German, Russian, and Italian. Another language in certain cases may be substituted for one of these if the department graduate committee approves. The committee will consider the current usage of the language in the student's specialty area. The examinations are offered by the Mathematics Department twice a year. They are designed to test a student's ability to translate, with the aid of a dictionary, mathematical literature into scientifically correct English.
• Dissertation.

FINANCIAL ASSISTANCE

Most of the graduate students in mathematics are supported by teaching assistantships. The usual load for a TA is two 3-hour sections (6 hours) both fall and winter semesters. The usual load for a PhD candidate acting as a TA is two 3-hour sections for one semester and one 3-hour section the second semester (if the student is making adequate progress on the qualifying exams. Current TAs receive from $9,000 to $11,000 per academic year as well as tuition support.

RESOURCES AND OPPORTUNITIES

Faculty research interests currently include: algebraic geometry; combinational group theory; geometric group theory; geometric topology; linear algebra; mathematics education; matrix analysis; number theory; and partial differential equations.

For a more detailed description of the graduate program requirements, send for a copy of the department's bulletin.

COURSE DESCRIPTIONS

Class Schedule

501. Real Numbers. (3)

Prerequisite: Math 371.

Extensive examination of various axiomatic descriptions of the real numbers and interrelationships among these descriptions.

502. Set Theory. (3)

Prerequisite: Math 371.

Zermelo-Fraenkel axioms for set theory, the axiom of choice, ordinal and cardinal numbers, and algebra of sets.

511. Numerical Methods for Partial Differential Equations. (3)

Prerequisite: Math 311, 343; 313 or 434. Recommended: Math 323.

Finite difference and finite volume methods for partial differential equations. Stability, consistency, and convergence theory.

512. Numerical Analysis. (3)

Prerequisite: Math 311, 343, or instructor's consent.

Numerical matrix algebra, orthogonalization and least squares methods, unsymmetric and symmetric eigenvalue problems, iterative methods, Lanczos methods, advanced solvers for partial differential equations.

513R. Advanced Topics in Applied Mathematics. (3)

Prerequisite: instructor's consent.

521, 522. Methods of Applied Mathematics. (3 ea.)

Prerequisite: Math 343, 434.

Survey of current methods, continuous and discrete, including linear algebra, estimation, differential equations of equilibrium, eigenvalue and initial value problems; finite element, spectral, transform and difference methods; Fourier series, the Fourier matrix, fast Fourier transform; convolution.

529R. Topics in Mathematics Education. (3)

Prerequisite: instructor's consent.

Current research and curriculum in mathematics education nationally and internationally; research techniques and interpretation.

530. Calculus of Variations. (3)

Prerequisite: Math 343, 434. Recommended: Math 323, 541.

Euler-Lagrange equation, sufficient conditions, Hamilton's principle of least action, Dirichlet's principle; applications to mechanics, geometry, economics, eigenvalue problems, direct methods.

532. Complex Analysis. (3)

Prerequisite: Math 332 or instructor's consent.

Theory of complex analysis at the beginning graduate level. Topics: Cauchy integral equations, Riemann surfaces, Picard's theorem, etc.

541, 542. Real Analysis. (3 ea.)

Prerequisite: Math 315, 343, 344 for 541; Math 541 for 542.

Rigorous treatment of differentiation and integration theory, Lebesque measure, Banach spaces.

543. Advanced Probability. (3)

Prerequisite: multivariable calculus. Recommended: Stat 341 or 520.

Combinatorial methods, random walk, Markov chains, stochastic processes.

547. Partial Differential Equations. (3)

Prerequisite: Math 344, 434.

Topics from elliptic equations, heat equations; wave equations, stability, Fourier methods, energy methods, existence of solutions, etc.

551, 552. Introduction to Topology. (3 ea.)

Prerequisite: Math 315 for 551; Math 551 for 552.

Axiomatic treatment of linearly ordered spaces, metric spaces, arcs, and Jordan curves; types of connectedness.

585. Matrix Analysis. (3)

Prerequisite: Math 343.

Special classes of matrices, canonical forms, matrix and vector norms, localization of eigenvalues, matrix functions, applications.

621, 622. Matrix Theory. (3 ea.)

Prerequisite: Math 585

Zero-one matrices, spectra of graphs, Laplacian matrix, irreducible and primitive matrices, cycle expansion of the determinant, matrix completion problems, permanents, generalized matrix functions.

629. Teaching Mathematics in Secondary Schools. (3)

631, 632. Complex Analysis. (3 ea.)

Prerequisite: Math 332, 542 for 631; Math 631 for 632.

634, 635. Theory of Ordinary Differential Equations. (3 ea.)

Prerequisite: Math 434.

641, 642. Functions of Real and Complex Variables. (3 ea.)

Prerequisite: Math 542 or instructor's consent for 641; Math 641 for 642.

643R. Special Topics in Analysis. (3)

Prerequisite: Math 642.

Continued fractions, stochastic processes, generalized functions, etc.

644. Harmonic Analysis. (3)

Prerequisite: Math 532, 542.

Harmonic analysis on the torus and in Euclidean space; pointwise and norm convergence of Fourier series and functional-analytic aspects of Fourier transforms emphasized.

647, 648. Theory of Partial Differential Equations. (3 ea.)

Prerequisite: Math 323, 542 for 647; Math 647 for 648.

651, 652. General Topology 1, 2. (3 ea.)

Prerequisite: Math 552.

653R. Special Topics in Geometry. (3)

Prerequisite: Math 672.

Topics from n-dimensional projective and algebraic geometry, foundations, transformations, curves and surfaces, forms and sheaf theory.

655. Algebraic Topology 1. (3)

Prerequisite: instructor's consent.

656. Algebraic Topology 2. (3)

Prerequisite: Math 655.

661, 662. Functional Analysis. (3 ea.)

Prerequisite: Math 641 for 661; Math 661 for 662.

671, 672. Algebra. (3 ea.)

Prerequisite: Math 372 for 671; Math 671 for 672.

675R. Special Topics in Algebra. (3)

Prerequisite: Math 672.

676. Commutative Algebra. (3)

Prerequisite: Math 671, 672.

Commutative rings, modules, tensor products, localization, primary decomposition, Noetherian and Artinian rings, application to algebraic geometry and algebraic number theory.

677. Homological Algebra. (3)

Prerequisite: Math 671, 672.

Chain complexes, derived functors, cohomology of groups, ext and tor, spectral sequences, etc. Application to algebraic geometry and algebraic number theory.

687R. Topics in Analytic Number Theory. (3)

Prerequisite: Math 387, 372, 532, and instructor's consent.

Current topics of research interest.

688R. Topics in Algebraic Number Theory. (3)

Prerequisite: Math 372, 387, and instructor's consent.

Current topics of research interest.

695R. Readings in Mathematics. (1-2)

699R. Master's Thesis. (1-9)

751R. Advanced Special Topics in Topology. (3)

Prerequisite: instructor's consent and Math 651, 652.

Current topics in topology of research interest.

780R. Seminar in Algebraic Geometry. (3)

Topics selected from current research literature.

799R. Doctoral Dissertation. (Arr.)

FACULTY 

ARMSTRONG, GERALD M., Associate Professor. PhD, University of Wisconsin, Madison, 1971. Real Analysis.

BAKER, ROGER C., Professor. PhD, University of London, 1971. Number Theory.

BARRETT, WAYNE WALTON, Professor. PhD, New York University, 1975. Matrix Theory; Graph Theory; Combinatorics.

BATES, PETER W., Professor. PhD, University of Utah, 1976. Nonlinear Partial Differential Equations; Dynamical Systems.

CANNON, JAMES W., Professor. PhD, University of Utah, 1969. Geometric Topology; Geometric Group Theory.

CHAHAL, JASBIR S., Associate Professor. PhD, Johns Hopkins University, 1979. Number Theory.

CLARK, DAVID A., Assistant Professor. PhD, McGill University, 1992. Number Theory.

CONNER, GREGORY R., Assistant Professor. PhD, University of Utah, 1992. Geometric Group Theory; Combinatorial Group Theory; Topology.

CRAWLEY, PETER L., Professor. PhD, California Institute of Technology, 1961. Infinite Groups.

FEARNLEY, LAWRENCE, Professor. PhD, University of London, 1970. Topology.

FORCADE, RODNEY W., Professor. PhD, University of Washington, 1971. Combinatorics.

GARBE, DOUGLAS G., Associate Professor. PhD, University of Texas, Austin, 1973. Mathematics Education.

GARNER, LYNN E., Professor. PhD, University of Oregon, 1968. Geometry; Commutative Algebra; Number Theory; Calculus Reform; Technology in Education.

GILL, GURCHARAN S., Professor. PhD, University of Utah, 1965. Functional Analysis.

GRANT, CHRISTOPHER P., Assistant Professor. PhD, University of Utah, 1991. Partial Differential Equations.

HANSEN, RICHARD A., Professor. PhD, University of Utah, 1965. Numerical Analysis.

HUMPHRIES, STEPHEN P., Associate Professor. PhD, University of Wales, 1983. Low-Dimensional Topology; Classical Groups.

JAMISON, RONALD D., Professor. PhD, University of Utah, 1965. Ordinary Differential Equations; Applied Mathematics; Mathematics Education.

JARVIS, TYLER J., Assistant Professor. PhD, Princeton University, 1994. Algebraic Geometry.

LAMOREAUX, JACK W., Professor. PhD, University of Utah, 1967. Topology.

LANG, WILLIAM E., Professor. PhD, Harvard University, 1978. Algebraic Geometry.

LAWLOR, GARY, Assistant Professor. PhD, Stanford University, 1988. Minimal Surfaces.

LU, KENING, Associate Professor. PhD, Michigan State University, 1988. Applied Mathematics; Nonlinear Partial Differential Equations; Dynamical Systems.

LUNDQUIST, MICHAEL, Associate Professor. PhD, Clemson University, 1990. Matrix Theory.

MCKAY, STEVEN M., Assistant Professor. PhD, Colorado State University, 1990. Numerical Analysis.

OUYANG, TIANCHENG, Assistant Professor. PhD, University of Minnesota, 1989. Partial Differential Equations.

PETERSON, BLAKE W., Assistant Professor. PhD, Washington State University, 1993. Mathematics Education.

POLLINGTON, ANDREW D., Professor. PhD, University of London, 1978. Number Theory.

ROBINSON, DONALD W., Professor. PhD, Case Institute of Technology, 1956. Linear Algebra.

SKARDA, R. VENCIL, Associate Professor. PhD, California Institute of Technology, 1965. Functional Analysis.

SMITH, WILLIAM V., Professor. PhD, University of Utah, 1978. Spectral Theory.

SNOW, DONALD RAY, Professor. PhD, Stanford University, 1965. Calculus of Variations; Functional Equations; Combinatorics.

SPEISER, ROBERT DAVID, Professor. PhD, Cornell University, 1970. Algebraic Geometry; Commutative Algebra.

TOLMAN, L. KIRK, Associate Professor. PhD, University of New Mexico, 1972. Graph Theory; Combinatorics; Ordered Rings and Fields; Differential Equations.

WALTER, CHARLES N., Associate Professor. PhD, University of New Mexico, 1970. Algebraic Geometry; Ordered Fields; Mathematics Education.

WILLIAMS, STEVEN R., Associate Professor., PhD, University of Wisconsin, Madison, 1989. Mathematics Education.

WRIGHT, DAVID G., Professor. PhD, University of Wisconsin, Madison, 1973. Geometric Topology.

WYNN, JAN EUGENE, Associate Professor. PhD, Colorado State University, 1972. Pad Approximations.

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