Course Descriptions

*Current with 2007-2008 BYU Catalogs.

102. Quantitative Reasoning. (3:3:0) F, W, Sp, Su For students who do not need developmental algebra for subsequent courses.

Practicing and applying quantitative reasoning: personal finance, consumer statistics, etc.

110. College Algebra. (3:3:0) F, W, Sp, Su Independent Study also. Prerequisite: Math 97 or equivalent.

Functions, polynomials, theory of equations, exponential and logarithmic functions, matrices, determinants, systems of linear equations, permutations, combinations, binomial theorem.

111. Trigonometry. (2:2:0) F, W, Sp, Su Independent Study also. Prerequisite: Math 110 or equivalent.

Circular functions, triangle relationships, identities, inverse trig functions, trigonometric equations, vectors, complex numbers, DeMoivre's theorem.

112. Calculus 1. (4:5:0) F, W, Sp, Su Honors also. Prerequisite: Math 110 and 111 or equivalents.

Differential and integral calculus: limits; continuity; the derivative and applications; extrema; the definite integral; fundamental theorem of calculus; L’Hôpital's rule.

113. Calculus 2. (4:5:0) F, W, Sp, Su Honors also. Prerequisite: Math 112 or equivalent.

Techniques and applications of integration; sequences, series, convergence tests, power series; parametric equations; polar coordinates.

119. Introduction to Calculus. (4:4:1) For students in the College of Biology and Agriculture and the Marriott School of Management. Independent Study also. F, W, Sp, Su Prerequisite: Math 110 or equivalent.

Introduction to plane analytic geometry and calculus.

190. Fundamentals of Mathematics. (3:3:0) F, W, Su Prerequisite: Math 112 or concurrent enrollment with instructor's consent.

Achieving maturity in mathematical communication. Introduction to mathematical proof; methods of proof; analysis of proof; induction; logical reasoning.

214. Calculus of Several Variables. (3:3:0) F, W, Sp, Su Prerequisite: Math 113; 343 or concurrent enrollment.

Partial differentiation, the Jacobian matrix, and integral theorems of vector calculus.

300. History of Mathematics. (3:3:0) F, W, Su Independent Study also. Prerequisite: Math 113.

Development of mathematics, emphasizing underlying principles and motivations.

302. Mathematics for Engineering 1. (4:4:0) F, W Prerequisite: Math 113 and passing grade on required preparatory exam taken during first week of class. (Practice exams available on class Web site.)

Multivariable calculus, linear algebra, and numerical methods.

303. Mathematics for Engineering 2. (4:4:0) F, W Prerequisite: Math 302; or Math 214 and 343.

ODEs, Laplace transforms, Fourier series, PDEs, numerical methods.

315, 316. Theory of Analysis. (3:3:0 ea.) 315: F, W; 316: F, W Prerequisite: for 315: Math 113, 190, 343; for 316: Math 315.

Rigorous treatment of calculus of single and several variables. Topics include uniform continuity, metric spaces, Riemann integral, implicit function theorem, and integral theorems of vector calculus.

331. Probability Theory (Math - EC En 370) (3:3:0) F Prerequisite: Math 343.

Axiomatic prob ability theory, conditional probability, discrete / continuous random variables, expectation, conditional expectation, moments, functions of random variables, multivariate distributions, laws of large numbers, central limit theorem.

332. Introduction to Complex Analysis. (3:3:0) F, W Prerequisite: Math 214 or 316.

Complex algebra, analytic functions, integration in the complex plane, infinite series, theory of residues, conformal mapping.

334. Ordinary Differential Equations. (3:3:0) F, W, Sp, Su Prerequisite: Math 113, 343.

Methods and theory of ordinary differential equations.

343. Elementary Linear Algebra. (3:3:0) F, W, Sp, Su. Independent Study also. Prerequisite: Math 112 or 119.

Linear systems, matrices, vectors and vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.

347. Introduction to Partial Differential Equations. (3:3:0) W, Su Prerequisite: Math 303 or 334.

Boundary value problems; transform methods; Fourier series; Bessel functions; Legendre polynomials.

350. Combinatorics. (3:3:0) Prerequisite: Math 343, 371.

Permutations, combinations, recurrence relations, applications.

355. Graph Theory. (3:3:0) Prerequisite: Math 343.

Maps, graphs and digraphs, coloring problems, applications.

362. Survey of Geometry. (3:3:0) F, W, Su. Prerequisite: Math 112, 190.

Logical structure of Euclidean, non-Euclidean, and finite geometries.

371, 372. Abstract Algebra. (3:3:0 ea.) 371: F, W, Sp; 372: W Prerequisite: for 371: Math 190, 343; for 372: Math 371.

Groups, rings, fields, vector spaces, linear transformations, matrices, field extensions, etc.

387. Number Theory. (3:3:0) Prerequisite: Math 343, 371.

Foundations; congruences; quadratic reciprocity; unique factorization, prime distribution or Diophantine equations.

391R. Seminar in Mathematics. (1:1:0) F

Topics from classical problems of antiquity, combinatorial mathematics, graph theory, real functions, number theory, functional equations.

399R. Academic Internship. (1–9:9:0 ea.) On dem.

On-the-job experience.

410. Introduction to Numerical Methods. (3:3:0) F, W Prerequisite: Math 214, 343.

Root finding, interpolation, curve fitting, numerical differentiation and integration, multiple integrals, direct solvers for linear systems, least squares, rational approximations, fourier and other orthogonal methods.

411. Numerical Methods. (3:3:0) W Prerequisite: Math 334, 410.

Iterative solvers for linear systems, eigenvalue, eigenvector approximations, numerical solutions to nonlinear systems, numerical techniques for initial and boundary value problems, elementary solvers for PDEs.

451. Introduction to Topology. (3:3:0) F, W Prerequisite: Math 315.

Developing topological concepts, beginning from a linear setting. Developing proofs or counterexamples from axioms to a structured seqence of topological propositions using only notes provided.

460R. Topics in Geometry. (3:3:0 ea.) On dem. Prerequisite: Math 343, 362; or equivalents.

Topics selected from the various aspects of synthetic, analytic, algebraic, and differential geometry.

480. Mathematical Models. (3:3:0) On dem. Prerequisite: Math 214, 334, 343, 410.

Construction, solution, and interpretation of discrete and continuous models applied to problems in the physical, natural, and social sciences.

495R. Readings in Mathematics. (1–2:0:3 ea.) F, W, Sp, Su Prerequisite: instructor's consent.

Directed readings beyond the scope of usual undergraduate courses.

499R. Senior Thesis. (1–3:0:3 ea.) F, W, Sp, Su

510. Numerical Methods for Linear Algebra. (3:3:0) F Prerequisite: Math 343, 410, or equivalents.

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

511. Numerical Methods for Partial Differential Equations. (3:3:0) W Prerequisite: Math 303 or 347; 410; or equivalents.

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

513R. Advanced Topics in Applied Mathematics. (3:3:0 ea.) On dem. Prerequisite: instructor's consent.

521, 522. Methods of Applied Mathematics 1, 2. (3:3:0 ea.) On dem. Prerequisite: Math 334, 343; or equivalents.

Possible topics include: variational, integral, and partial differential equations; spectral and transform methods; nonlinear waves; Green's functions; scaling and asymptotic analysis; perturbation theory; continuum mechanics.

532. Complex Analysis. (3:3:0) Prerequisite: Math 332 or instructor's consent.

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

534. Introduction to Dynamical Systems 1. (3:3:0) Prerequisite: Math 315, 334; or equivalents.

Discrete dynamical systems; iterations of maps on the line and the plane; bifurcation theory; chaos, Julia sets, and fractals. Computational experimentation.

541, 542. Real Analysis. (3:3:0 ea.) F, W Prerequisite: Math 315, 343; 214 or 316; or equivalents.

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

543, 544. Advanced Probability 1, 2. (3:3:0 ea.) On dem. Prerequisite: Math 214 or equivalent. Recommended: Math 315, 316, Stat 441; or equivalents.

Probability theory and its applicatiions. Topics include random variables, independence and conditioning, laws of large numbers, random walks, martingales, Markov chains, renewal processes, ergodic theorems, Brownian motion, and stochastic integration.

547, 548. Partial Differential Equations 1, 2. (3:3:0 ea.) On dem. Prerequisite: Math 214, 334; or equivalents. Recommended: Math 315, 316; or equivalents.

Topics include the method of characteristics, elliptic equations, potential theory, parabolic equations and systems, maximum principles, linear and nonlinear waves, Hamilton-Jacobi equations, Fourier transforms, Green's functions, distributions, and energy methods.

553. Foundations of Topology 1. (3:3:0) F, W Prerequisite: Math 451 or instructor's consent.

Naïve set theory, topological spaces, product spaces, subspaces, continuous functions, connectedness, compactness, countability, separation axioms, metrization, complete metric spaces, function spaces, and Baire spaces.

554. Foundations of Topology 2. (3:3:0) F, W Prerequisite: Math 553 or instructor's consent.

Fundamental group, retractions and fixed points, homotopy types, separation theorems, classification of surfaces, Seifert–van Kampen Theorem, classification of covering spaces, and applications to group theory.

561, 562. Introduction to Algebraic Geometry. (3:3:0) Prerequisite: Math 671 or concurrent enrollment.

Projective varieties, curves, surfaces, differential forms, and divisors.

565. Differential Geometry. (3:3:0) Prerequisite: Math 214, 315; or equivalents. Recommended: Math 316 or equivalent.

Curves and surfaces, including the first and second fundamental forms, Gauss map, curvatures, geodesics, minimal surfaces, and the Gauss-Bonnet Theorem.

570. Matrix Analysis. (3:3:0) Prerequisite: Math 302 or 343; or equivalents.

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

586. Introduction to Algebraic Number Theory. (3:3:0) F or W Prerequisite: Math 372 or equivalent; instructor's consent.

Algebraic integers, different and discriminant; decomposition of primes; class group; Dirichlet unit theorem; Dedekind zeta-function; cyclotomic fields; valuations; completions.

587. Introduction to Analytic Number Theory. (3:3:0) F or W Prerequisite: Math 332 or equivalent; instructor's consent.

Arithmetical functions; distribution of primes; Dirichlet characters; Dirichlet's theorem; Gauss sums; primitive roots; Dirichlet L-functions; Riemann zeta-function; prime number theorem; partitions.

621, 622. Matrix Theory. (3 ea.) Prerequisite: Math 570.

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

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 315, 334.

641. Functions of Real and Complex Variables 1. (3) Prerequisite: Math 542 or instructor’s consent.

Fundamentals of measure and integration, Borel measures, product measures, LP spaces, introduction to functional analysis, Radon Nikodym theorem, differentiation theory, Fourier transforms.

642. Functions of Real and Complex Variables 2. (3) Prerequisite: Math 641.

Advanced topics chosen by the instructor, such as but not limited to probability, Haar measures, Fourier analysis on locally compact groups, Sobolev spaces, ergodic theory, differentiation theory of Radon measures, area and co-area formulas, etc.

643R. Special Topics in Analysis. (3) Prerequisite: Math 642 or instructor’s consent.

Advanced topics in analysis drawn from pure and applied mathematics. Possible topics include nonlinear partial differential equations, nonlinear functional analysis, asymptotic analysis, wavelets, numerical analysis, and analysis applied to biological and medical systems.

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.

645, 646. Functional Analysis. (3 ea.) Prerequisite: Math 641 for 645; Math 645 for 646.

647, 648. Theory of Partial Differential Equations. (3 ea.) Prerequisite: Math 347, 542 for 647; Math 647 for 648.

651, 652. Topology 1, 2. (3 ea.) Prerequisite: for 651: Math 553, 554; for 652: Math 651.

Advanced topics in topology. Topics may include, but are not limited to, piecewise linear topology, 3-manifold theory, homotopy theory, differential topology, Riemannian geometry, and geometric group theory.

655. Algebraic Topology 1. (3) Prerequisite: instructor’s consent.

656. Algebraic Topology 2. (3) Prerequisite: Math 655.

663, 664. Algebraic Geometry. (3 ea.) Prerequisite: Math 672; Math 676 or concurrent enrollment.

Varieties, sheaves, and schemes; their cohomology and classification; applications.

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.

686R. Topics in Algebraic Number Theory. (3) Prerequisite: Math 372, 387, and instructor’s consent.

Current topics of research interest.

687R. Topics in Analytic Number Theory. (3) Prerequisite: Math 387, 372, 532, and instructor’s consent.

Current topics of research interest.

695R. Readings in Mathematics. (1–2)

698R. Master’s Project. (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.

799R. Doctoral Dissertation. (Arr.)