The course descriptions below are organized by subfield. Essentially the same information, ordered instead by course number, is available on the Fall 2008 University Course Catalog Descriptions page.
The organization of courses into categories was last changed on 14 December 2007, and the title/description records were last changed on 27 August 2008.
MATH 503 — Foundations of Mathematics (3 units)
Description: [Taught Spring semester in even-numbered years] Topics in set theory such as functions, relations, transfinite induction and recursion, cardinal and ordinal arithmetic; related topics such as axiomatic systems, the development of the real number system, recursive functions and philosophy of Mathematics. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 403. Usually offered: Spring.
MATH 504 — History of Mathematics (3 units)
Description: The development of mathematics from ancient times through the 17th century, with emphasis on problem solving. The study of selected topics from each field is extended to the 20th century. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 404. Prerequisite(s): not applicable to M.A., M.S., or Ph.D. degrees for math majors except for the M.A. teaching option. Usually offered: Fall.
MATH 511A — Algebra (3 units)
Description: Structure of groups, rings, modules, algebras; Galois theory. Usually offered: Fall.
MATH 511B — Algebra (3 units)
Description: Structure of groups, rings, modules, algebras; Galois theory. Prerequisite(s): MATH 511A. Usually offered: Spring.
MATH 513 — Linear Algebra (3 units)
Description: Vector spaces, linear transformations and matrices, determinants, eigenvalues and diagonalization, bilinear forms, orthogonal and unitary transformations, Jordan canonical form. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 413. Usually offered: Fall, Spring.
MATH 514A — Algebraic Number Theory (3 units)
Description: [Taught Fall semester in odd-numbered years] Dedekind domains, complete fields, class groups and class numbers, Dirichlet unit theorem, algebraic function fields. Prerequisite(s): MATH 511B. Usually offered: Fall.
MATH 514B — Algebraic Number Theory (3 units)
Description: [Taught Fall semester in even numbered years] Dedekind domains, complete fields, class groups and class numbers, Dirichlet unit theorem, algebraic function fields. Prerequisite(s): MATH 514A. Usually offered: Spring.
MATH 515A — Introduction to Abstract Algebra (3 units)
Description: Introduction to groups, rings, and fields. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 415A. Usually offered: Fall.
MATH 515B — Second Course in Abstract Algebra (3 units)
Description: A continuation of MATH 415A/515A. Topics may include finite groups, matrix groups, Galois theory, linear and multilinear algebra, finite fields and coding theory. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 415B. Usually offered: Spring.
MATH 517A — Group Theory (3 units)
Description: [Taught Fall semester in even numbered years] Selections from such topics as finite groups, abelian groups, characters and representations. Prerequisite(s): MATH 511B. Usually offered: Fall.
MATH 517B — Group Theory (3 units)
Description: [Taught Spring semester in odd-numbered years] Selections from such topics as finite groups, abelian groups, characters and representations. Prerequisite(s): MATH 517A. Usually offered: Spring.
MATH 539 — Algebraic Coding Theory (3 units)
Description: [Taught Spring semester in even-numbered years] Construction and properties of error correcting codes; encoding and decoding procedures and information rate for various codes. Prerequisite(s): MATH 415A. Usually offered: Spring.
MATH 546 — Theory of Numbers (3 units)
Description: [Taught Spring semester in odd-numbered years]. Divisibility properties of primes, congruences, quadratic residues, number-theoretic functions, primality, factoring, applications to crytopgraphy, introduction to algebraic numbers. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 446. Usually offered: Spring.
MATH578 — TITLE/DESCRIPTION RECORD MISSING
MATH 508 — Harmonic Analysis (3 units)
Description: [Taught Spring semester in even-numbered years]. Fast Fourier transforms, classical Fourier analysis, related group theory done concretely. Graduate-level requirements include writing a paper dealing with almost periodic functions and their fourier series and solve a set of problems dealing with this topic. May be convened with: MATH 408. Usually offered: Spring.
MATH 520A — Complex Analysis (3 units)
Description: Analyticity, Cauchy's integral formula, residues, infinite products, conformal mapping, Dirichlet problem, Riemann mapping theorem. Usually offered: Fall.
MATH 520B — Complex Analysis (3 units)
Description: Rudiments of Riemann surfaces. Usually offered: Spring.
MATH 523A — Real Analysis (3 units)
Description: Lebesgue measure and integration, differentiation, Radon-Nikodym theorem, Lp spaces, applications. Usually offered: Fall.
MATH 523B — Real Analysis (3 units)
Description: Lebesgue measure and integration, differentiation, Radon-Nikodym theorem, Lp spaces, applications. Prerequisite(s): MATH 523A. Usually offered: Spring.
MATH 525A — Real Analysis of One Variable (3 units)
Description: Continuity and differentiation of functions of one variable. Riemann integration, sequences and series of functions and uniform convergence. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 425A. Usually offered: Fall.
MATH 525B — Real Analysis of Several Variables (3 units)
Description: Continuity and differentiation in higher dimensions, curves and surfaces; change of coordinates; theorems of Green, Gauss and Stokes; inverse and implicit function theorems. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 425B. Usually offered: Spring.
MATH 527A — Principles of Analysis (3 units)
Description: Metric spaces, basic properties of normed linear spaces, distributions, the Lebesgue integral and Lebesque spaces, convergence theorems; applications chosen by the instructor. Prerequisite(s): MATH 410, MATH 424, and a differential equations course. Usually offered: Fall.
MATH 527B — Principles of Analysis (3 units)
Description: Metric spaces, basic properties of normed linear spaces, distributions, the Lebesque intergral and Lebesque spaces, convergence theorems; applications chosen by the instructor. Prerequisite(s): MATH 527A. Usually offered: Spring.
MATH 528A — Banach and Hilbert Spaces (3 units)
Description: Introduction to the theory of normed spaces, Banach spaces and Hilbert spaces, operators on Banach spaces, spectral theory of operators on Hilbert spaces, applications. Prerequisite(s): MATH 527B or MATH 523A. Usually offered: Fall.
MATH 528B — Banach and Hilbert Spaces (3 units)
Description: Introduction to the theory of normed spaces, Banach spaces and Hilbert spaces, operators on Banach spaces, spectral theory of operators on Hilbert spaces, applications. Prerequisite(s): MATH 528A. Usually offered: Spring.
MATH 553A — Partial Differential Equations (3 units)
Description: [Taught Fall semester in odd-numbered years] Theory and examples of linear equations; characteristics, well-posed problems, regularity, variational properties, asymptotics. Topics in nonlinear equations, such as shock waves, diffusion waves, and estimates in Sobolev spaces. Prerequisite(s): MATH 523B or MATH 527B or MATH 583B. Usually offered: Fall.
MATH 553B — Partial Differential Equations (3 units)
Description: [Taught Spring semester in even-numbered years] Theory and examples of linear equations; characteristics, well-posed problems, regularity, variational properties, asymptotics. Topics in nonlinear equations, such as shock waves, diffusion waves, and estimates in Sobolev spaces. Prerequisite(s): MATH 553A. Usually offered: Spring.
MATH 557A — Dynamical Systems and Chaos (3 units)
Description: Qualitative theory of dynamical systems, phase space analysis, bifurcation, period doubling, universal scaling, onset of chaos. Applications drawn from atmospheric physics, biology, ecology, fluid mechanics and optics. Prerequisite(s): MATH 454 or (MATH 254 and MATH 422): and MATH 421 or MATH 424. Usually offered: Fall.
MATH 557B — Dynamical Systems and Chaos (3 units)
Description: Qualitative theory of dynamical systems, phase space analysis, bifurcation, period doubling, universal scaling, onset of chaos. Applications drawn from atmospheric physics, biology, ecology, fluid mechanics and optics. Prerequisite(s): MATH 557A. Usually offered: Spring.
MATH 521 — Complex Variables with Applications (3 units)
Description: Complex numbers, analytic functions, harmonic functions, elementary functions, complex integration, Cauchy's integral theorem, series representations for analytic functions, residue theory, conformal mapping, applications to steady-state temperature and oscillating systems. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 421. Usually offered: Fall.
MATH 524 — Theory of Complex Variables (3 units)
Description: Complex numbers, complex-valued functions, analytic functions, elementary functions, series, residues and poles, mapping by elementary functions, conformal mapping, the Schwarz-Christoffel transformation, integral formulas of Poisson type. Graduate-level requirements include more extensive problem sets or advanced project. May be convened with: MATH 424. Usually offered: Spring.
MATH530 — TITLE/DESCRIPTION RECORD MISSING
MATH 531 — Algebraic Topology (3 units)
Description: [Taught Fall semester in odd-numbered years] Poincare duality, fixed point theorems, characteristic classes, classification of principal bundles, homology of fiber bundles, higher homotopy groups, low dimensional manifolds. Prerequisite(s): MATH 534B. Usually offered: Fall.
MATH 534A — Topology-Geometry (3 units)
Description: Point set topology, the fundamental group, calculus on manifolds. Homology, de Rham cohomology, other topics. Examples will be emphasized. Usually offered: Fall.
MATH 534B — Topology-Geometry (3 units)
Description: Point set topology, the fundamental group, calculus on manifolds. Homology, de Rham cohomology, other topics. Examples will be emphasized. Prerequisite(s): MATH 534A. Usually offered: Spring.
MATH 536A — Algebraic Geometry (3 units)
Description: [Taught Fall semester in even numbered years] Affine and projective varieties, morphisms and rational maps. Dimension, degree and smoothness. Basic coherent sheaf theory and Cech cohomology. Line bundles, Riemann-Roch theorem. Prerequisite(s): MATH 520A, MATH 534A. Usually offered: Fall.
MATH 536B — Algebraic Geometry (3 units)
Description: [Taught Spring semester in odd-numbered years] Affine and projective varieties, morphisms and rational maps. Dimension, degree and smoothness. Basic coherent sheaf theory and Cech cohomology. Line bundles, Riemann-Roch theorem. Prerequisite(s): MATH 536A. Usually offered: Spring.
MATH 537A — Global Differential Geometry (3 units)
Description: [Taught Fall semester in odd-numbered years] Surfaces in R3, structure equations, curvature. Gauss-Bonnet theorem, parallel transport, geodesics, calculus of variations, Jacobi fields and conjugate points, topology and curvature; Riemannian geometry, connections, curvature tensor, Riemannian submanifolds and submersions, symmetric spaces, vector bundles. Morse theory, symplectic geometry. Prerequisite(s): MATH 534A, MATH 534B. Usually offered: Fall.
MATH 537B — Global Differential Geometry (3 units)
Description: [Taught Spring semester in even-numbered years] Surfaces in R3, structure equations, curvature. Gauss-Bonnet theorem, parallel transport, geodesics, calculus of variations, Jacobi fields and conjugate points, topology and curvature; Riemannian geometry, connections, curvature tensor, Riemannian submanifolds and submersions, symmetric spaces, vector bundles. Morse theory, symplectic geometry. Prerequisite(s): MATH 537A. Usually offered: Spring.
MATH 559A — Lie Groups and Lie Algebras (3 units)
Description: [Taught Fall semester in even-numbered years]. Correspondence between Lie groups and Lie algebras, structure and representation theory, applications to topology and geometry of homogeneous spaces, applications to harmonic analysis. Prerequisite(s): MATH 511A, MATH 523A, MATH 534A, MATH 534B or consent of instructor. Usually offered: Fall.
MATH 559B — Lie Groups and Lie Algebras (3 units)
Description: [Taught Spring semester in odd-numbered years]. Correspondence between Lie groups and Lie algebras, structure and representation theory, applications to topology and geometry of homogeneous spaces, applications to harmonic analysis. Prerequisite(s): MATH 559A. Usually offered: Spring.
MATH563A — TITLE/DESCRIPTION RECORD MISSING
MATH563B — TITLE/DESCRIPTION RECORD MISSING
MATH 564 — Theory of Probability (3 units)
Description: Probability spaces, random variables, weak law of large numbers, central limit theorem, various discrete and continuous probability distributions. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 464. Usually offered: Fall.
MATH 565A — Stochastic Processes (3 units)
Description: [Taught Spring semester in odd-numbered years]. Stochastic Processes in continuous time: Levy processes, Martingales, Markov processes, introduction to stochastic integrals. Prerequisite(s): strong probability background. Usually offered: Spring.
MATH 565B — Stochastic Processes (3 units)
Description: [Taught Fall semester in even-number years]. Stochastic processes in continuous time; Levy processes, martingales, Markov processes, introduction to stochastic integrals. Prerequisite(s): MATH 565A. Usually offered: Fall.
MATH 568 — Applied Stochastic Processes (3 units)
Description: Graduate-level requirements include more extensive problem sets or advanced projects. Usually offered: Spring.
MATH561 — TITLE/DESCRIPTION RECORD MISSING
MATH562 — TITLE/DESCRIPTION RECORD MISSING
MATH 566 — Theory of Statistics (3 units)
Description: Sampling theory. Point estimation. Limiting distributions. Testing Hypotheses. Confidence intervals. Large sample methods. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 466. Usually offered: Spring.
MATH 567A — Theoretical Statistics (3 units)
Description: [Taught Spring semester in even-numbered years] Basic decision theory. Bayes' rules for estimation. Admissibility and completeness. The minimax theorem. Sufficiency. Exponential families of distributions. Complete sufficient statistics. Invariant decision problems. Location and scale parameters. Theory of nonparametric statistics. Hypothesis testing. Neyman-Pearson lemma. UMP and UMPU tests. Two-sided tests. Two-sample tests. Confidence sets. Multiple decision problems. Prerequisite(s): MATH 466. Usually offered: Spring.
MATH 567B — Theoretical Statistics (3 units)
Description: [Taught Fall semester in even-numbered years] Basic decision theory. Bayes' rules for estimation. Admissibility and completeness. The minimax theorem. Sufficiency. Exponential families of distributions. Complete sufficient statistics. Invariant decision problems. Location and scale parameters. Theory of nonparametric statistics. Hypothesis testing. Neyman-Pearson lemma. UMP and UMPU tests. Two-sided tests. Two-sample tests. Confidence sets. Multiple decision problems. Prerequisite(s): MATH 567A. Usually offered: Fall.
MATH569 — TITLE/DESCRIPTION RECORD MISSING
MATH570 — TITLE/DESCRIPTION RECORD MISSING
MATH571 — TITLE/DESCRIPTION RECORD MISSING
MATH 522 — Advanced Applied Analysis (3 units)
Description: Graduate-level requirements include more extensive problem sets or advanced projects. Usually offered: Fall.
MATH550 — TITLE/DESCRIPTION RECORD MISSING
MATH 554 — Ordinary Differential Equations (3 units)
Description: [Taught Fall semester in even-numbered years]. General theory of linear systems, Foquet theory. Local theory of nonlinear systems, stable manifold and Hartman-Grobman theorems. Poincare-Bendixson theory, limit cycles, Poincare maps. Bifurcation theory, including the Hopf theorem. Prerequisite(s): MATH 413 or consent of instructor. Usually offered: Fall.
MATH 556 — Applied Partial Differential Equations (3 units)
Description: Properties of partial differential equations and techniques for their solution: Fourier methods, Green's functions, numerical methods. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 456. Usually offered: Spring.
MATH 557A — Dynamical Systems and Chaos (3 units)
Description: Qualitative theory of dynamical systems, phase space analysis, bifurcation, period doubling, universal scaling, onset of chaos. Applications drawn from atmospheric physics, biology, ecology, fluid mechanics and optics. Prerequisite(s): MATH 454 or (MATH 254 and MATH 422): and MATH 421 or MATH 424. Usually offered: Fall.
MATH 557B — Dynamical Systems and Chaos (3 units)
Description: Qualitative theory of dynamical systems, phase space analysis, bifurcation, period doubling, universal scaling, onset of chaos. Applications drawn from atmospheric physics, biology, ecology, fluid mechanics and optics. Prerequisite(s): MATH 557A. Usually offered: Spring.
MATH 575A — Numerical Analysis (3 units)
Description: Error analysis, solution of linear systems and nonlinear equations, eigenvalue interpolation and approximation, numerical integration, initial and boundary value problems for ordinary differential equations, optimization. Usually offered: Fall.
MATH 575B — Numerical Analysis (3 units)
Description: Error analysis, solution of linear systems and nonlinear equations, eigenvalue interpolation and approximation, numerical integration, initial and boundary value problems for ordinary differential equations, optimization. Prerequisite(s): MATH 575A. Usually offered: Spring.
MATH 576A — Numerical Analysis PDE (3 units)
Description: [Taught Fall semester in even numbered years] Finite difference, finite element, and spectral discretization methods; semidiscrete, matrix, and Fourier analysis. Prerequisite(s): MATH 413, MATH 456, MATH 575B. Usually offered: Fall.
MATH 576B — Numerical Analysis PDE (3 units)
Description: [Taught Spring semester in odd-numbered years] Well-posedness, numerical boundary conditions, nonlinear instability, time-split algorithms, special methods for stiff and singular problems. Prerequisite(s): MATH 576A. Usually offered: Spring.
MATH 582 — Applied Complex Analysis (3 units)
Description: [Taught Spring semester in odd numbered years] Representations of special functions, asymptotic methods for integrals and linear differential equations in the complex domain, applications of conformal mapping, Wiener-Hopf techniques. Prerequisite(s): MATH 424. Usually offered: Spring.
MATH 583A — Principles and Methods of Applied Mathematics (3 units)
Description: Boundary value problems; Green's functions, distributions, Fourier transforms, the classical partial differential equations (Laplace, heat, wave) of mathematical physics. Linear operators, spectral theory, integral equations, Fredholm theory. Usually offered: Fall.
MATH 583B — Principles and Methods of Applied Mathematics (3 units)
Description: Boundary value problems; Green's functions, distributions, Fourier transforms, the classical partial differential equations (Laplace, heat, wave) of mathematical physics. Linear operators, spectral theory, integral equations, Fredholm theory. Prerequisite(s): MATH 583A. Usually offered: Spring.
MATH 585 — Mathematical Modeling (3 units)
Description: Development, analysis, and evaluation of mathematical models for physical, biological, social, and technical problems; both analytical and numerical solution techniques are required. Graduate-level requirements include more advanced projects. May be convened with: MATH 485. Usually offered: Spring.
MATH 587 — Perturbation Methods in Applied Mathematics (3 units)
Description: [Taught Fall semester in odd-numbered years] Regular and singular perturbations, boundary layer theory, multiscale and averaging methods for nonlinear waves and oscillators. Prerequisite(s): MATH 422; MATH 421 or MATH 454. Usually offered: Fall.
MATH 501A — Symbolic Logic (3 units)
Description: Graduate-level requirements include an in-depth research project on a central theme or topic of the course. Identical to: PHIL 501A. Usually offered: Fall.
MATH 501B — Symbolic Logic (3 units)
Description: Graduate-level requirements include an in-depth research project on a central theme or topic of the course. Identical to: PHIL 501B. Usually offered: Spring.
MATH 502 — Mathematical Logic (3 units)
Description: [Taught Fall semester in odd-numbered years] Sentential calculus, predicate calculus; consistency, independence, completeness, and the decision problem. Designed to be of interest to majors in mathematics or philosophy. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 402. Usually offered: Fall.
MATH 543 — Theory of Graphs and Networks (3 units)
Description: Graduate-level requirements include more extensive problem sets or advanced projects. Usually offered: Fall.
MATH 547 — Combinatorial Mathematics (3 units)
Description: [Taught Spring semester in odd-numbered years]. Enumeration and construction of arrangements and designs; generating functions; principle of inclusion-exclusion; recurrence relations; a variety of applications. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 447. Usually offered: Spring.
MATH 579 — Game Theory and Mathematical Programming (3 units)
Description: [Taught Spring semester in even-numbered years] Linear inequalities, games of strategy, minimax theorem, optimal strategies, duality theorems, simplex method, nonzero sum games, applications to economics and decision making, Nash theorems. Graduate-level requirements include more extensive problem sets or advanced projects. May be convened with: MATH 479. Usually offered: Spring.
MATH 535A — The Mathematics of Computer Graphics (3 units)
Description: The Mathematical aspects of computer graphics, including scan conversion methods, projective geometry and geometric transformations, the construction and rendering of curves and surfaces, and color models. Usually offered: Fall.
MATH581 — TITLE/DESCRIPTION RECORD MISSING
MATH 589 — Software Tools for Computational Science and Engineering (3 units)
Description: Techniques and tools useful at the interface between mathematical and technical computing on the one hand, and the Internet on the other. Topics include scripting languages such as Perl and Tcl/Tk, graphics file formats, the mathematics of raster and vector graphics, and standard libraries and applications for numerical and symbolic computing. Also, the fundamentals of computer networking from a user's point of view. Prerequisite(s): C SC 352 and ability to program in at least one modern high-level language. Usually offered: Spring.