Courses - MATH Graduate Level
520. Finite Element Method (3)
Prerequisite: MATH 323, MATH 361A, MATH 364A. Recommended: MATH 470.
Variational forms and weak solutions of partial differential equations, Galerkin method, construction of elements, numerical algorithms for matrix equations and for one-dimensional and two-dimensional problems. Convergence analysis and error estimate. Numerical implementations of algorithms.
Letter grade only (A-F). (Lecture 3 hrs.)
521. Matrix Method in Data Analysis and Pattern Recognition (3)
Prerequisite: MATH 423 or MATH 576.
Vector spaces and linear transformations, optimal orthogonal projections, eigenvalues, eigenvectors, SVD, generalized SVD, Fourier and wavelet transforms, convolution, tangent distance. Implementations include object recognition, handwritten digit classification, digital image processing, feature extraction, image deblurring, text mining.
Letter grade only (A-F). (Lecture 3 hrs.)
540. Elements of Abstract Algebra (3)
Prerequisite: MATH 444.
Group theory including symmetric groups; group actions on sets; Sylow theorems and finitely generated abelian groups; ring theory including polynomial rings, division rings, Euclidean domains, principal ideal domains, and unique factorization domains.
Letter grade only (A-F). (Lecture 3 hours).
542. Algebraic Geometry (3)
Prerequisite: Consent of instructor.
An introduction to algebraic geometry: Algebraic sets; affine and projective varieties. Additional topics at the discretion of the instructor may include: Algebraic Curves; Intersection Theory; Invariant Theory; Computational Approaches.
Letter grade only (A-F). (Lecture 3 hrs.)
545. Topics in Abstract Algebra (3)
Prerequisite: MATH 540.
Selected topics in algebra that build upon the material of MATH 540. Content will vary by semester. May be taken for credit more than once with the consent of the graduate advisor
Letter grade only (A-F). (Lecture 3 hours).
550. Elements of Topology (3)
Prerequisite: MATH 361B.
Fundamentals of point-set topology: metric spaces and topological spaces; bases and neighborhoods; continuous functions; subspaces, product spaces and quotient spaces; separation properties, countability properties; compactness, compactification; connectedness; convergence of sequences; other topics, such as nets, filters and metrizability, as time permits.Letter grade only (A-F). (Lecture 3 hours).
555. Topics in Topology (3)
Prerequisite: MATH 550.
Selected topics in topology that build upon the material of MATH 550. Content will vary by semester. May be taken for credit more than once with the consent of the graduate advisor
Letter grade only (A-F). (Lecture 3 hours).
560. Functional Analysis (3)
Prerequisites: MATH 247, MATH 361B.
Linear spaces, metric and topological spaces, normed linear spaces; four principles of functional analysis: Hahn-Banach, Open Mapping, Uniform Boundedness, and Closed Graph theorems; adjoint spaces; normed space convergence, conjugate spaces, and operator spaces; Banach Fixed Point theorem; Hilbert spaces..
Letter grade only (A-F). (Lecture 3 hours).
561. Elements of Real Analysis (3)
Prerequisite: MATH 361B.
Theory of measure and integration, focusing on the Lebesgue integral on Euclidean space, particularly the real line. Modes of convergence. Fatou's Lemma, the monotone convergence theorem and the dominated convergence theorem. Fubini's theorem.
Letter grade only (A-F). (Lecture 3 hours)
562. Elements of Complex Analysis (3)
Prerequisite: MATH 361B.
Axiomatic development of real and complex numbers; elements of point set theory; differentiation and analytic functions, classical integral theorems; Taylor's series, singularities, Laurent series, calculus of residues.
Letter grade only (A-F). (Lecture 3 hours).
563. Applied Analysis (3)
Prerequisites: MATH 361B and either MATH 364A or MATH 370A.
Hilbert Spaces, Lp spaces, Distributions, Fourier Transforms, and applications to differential and integral equations from physics and engineering.
Letter grade only (A-F). (Lecture 3 hrs.)
564. Applied Nonlinear Ordinary Differential
Equations (3)
Prerequisites: MATH 361B; MATH 364A or MATH 370A.
Stability and asymptotic analysis, Perturbation methods, Phase plane analysis, Bifurcation, Chaos, Applications to science and engineering.
(Lecture 3 hrs.)
565. Topics in Real Analysis (3)
Prerequisite: MATH 561.
Selected topics in real analysis that build upon the material of MATH 561. Content will vary by semester. May be taken for credit more than once with the consent of the graduate advisor
Letter grade only (A-F). (Lecture 3 hours).
566. Topics in Complex Analysis (3)
Prerequisite: MATH 562.
Selected topics in real analysis that build upon the material of MATH 562. Content will vary by semester. May be taken for credit more than once with the consent of the graduate advisor
Letter grade only (A-F). (Lecture 3 hours).
570. Partial Differential Equations (3)
Prerequisites: MATH 364A and MATH 463.
Cauchy's problem; classification of second order equations; methods of solution of hyperbolic, parabolic, and elliptic equations.
Letter grade only (A-F). (Lecture 3 hrs.)
573. Advanced Scientific Computing (3)
Prerequisites: MATH 323 and MATH 364A or MATH 370A.
Analysis and implementation of numerical algorithms for linear systems, linear and nonlinear regression, differentiation, integration, optimization and fast convolution using FFT. Numerical solutions for differential equations.
Letter grade only (A-F).
574. Stochastic Calculus and Applications (3)
Prerequisites: MATH 361B, MATH 364A or MATH 370A, MATH 380.
Review of probability theory. Markov processes. Wiener processes. Stochastic integrals. Stochastic differential equations. Applications to Finance and Engineering.
(Lecture 3 hrs.)
575. Calculus of Variations (3)
Prerequisites: MATH 361B and either MATH 364A or MATH 370A
Solution methods for variational problems. First variation, Euler-Lagrange equation, variational principles, problems with constraints, boundary conditions, applications to physics and geometry. May include multiple integral problems, eigenvalue problems, convexity, and second variation.
Letter grade only (A-F). (Lecture 3 hrs.)
576. Numerical Analysis (3)
Prerequisites: MATH 323, MATH 361B, MATH 364A.
Advanced numerical methods. Introduction to error analysis, convergence, and stability of numerical algorithms. Topics may include solution of ordinary differential equations, partial differential equations, systems of linear and nonlinear equations, and optimization theory.
Letter grade only (A-F). (Lecture 3 hrs.)
577. Numerical Solution of Partial Differential Equations (3)
Prerequisite: MATH 423 or MATH 576 or consent of instructor.
Finite difference methods solving hyperbolic, parabolic, elliptic PDE'S; accuracy, convergence, and stability analysis. Selected initial-value boundary-value problems, characteristics, domain of dependence, matrix and von Neumann's methods of stability analysis. Solutions of large sparse linear systems. Finite element method.
(Lecture 3 hrs.)
578. Numerical Linear Algebra (3)
Prerequisites: MATH 247 and MATH 323 or consent of instructor.
Numerical solutions of linear systems, least squares problems, eigenvalue problems. Matrix factorization: LU, QR, SVD, iterative methods. Error analysis. Applications with attention to linear algebra problems arising in numerical solutions of partial differential equations. Numerical implementation of algorithms.
Letter grade only (A-F). (Lecture 3 hrs.)
579. Advanced Mathematical Modeling (3)
Prerequisites: MATH 247, MATH 323; MATH 364A or MATH 370A; one additional graduate level mathematics course, and consent of instructor.
Application of mathematics to develop models of phenomena in science, engineering, business, and other disciplines. Evaluation of benefits and limitations of mathematical modeling.
Letter grade only (A-F).
590. Selected Topics in Mathematics (3)
Prerequisite: Consent of Instructor
Specialized and advanced topics in mathematics.
May be repeated to a maximum of 9 units in different or same semester. (3 hours lecture)
695. Seminar in Mathematics (3)
Prerequisite: Consent of instructor.
Presentation and discussion of advanced work, including original research by faculty and students. Topics announced in the Schedule of Classes.
May be repeated to a maximum of 6 units. Letter grade only (A-F).
697. Directed Studies (1-6)
Prerequisite: Consent of instructor.
Research on a specific area in mathematics. Topics for study to be approved and directed by faculty advisor in the Department of Mathematics and Statistics.
Letter grade only (A-F).
698. Thesis or Project (1-6)
Prerequisite: Advancement to candidacy.
Formal report of research or project in mathematics.
May be repeated to a maximum of six units. Letter grade only (A-F).
