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# Mathematics and Statistics

## Courses - MATH Graduate Level

#### 520. Finite Element Method (3)

Prerequisite: MATH 323, 361A, 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 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.)

#### 540A. Abstract Algebra I (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 hrs.)

#### 540B. Abstract Algebra II (3)

Prerequisite: MATH 540A.
Modules; Field extensions; Finite fields; Splitting fields, Galois theory. Commutative ring theory including chain conditions and primary ideals. Topics of current interest.
Letter grade only (A-F). (Lecture 3 hrs.)

#### 541. Elliptic Curves (3)

Prerequisites: MATH 341, 444 and consent of instructor. Recommended: MATH 461 and 540A.
Fermat's method of descent; finite fields; Weierstrass normal form; integer, rational points on elliptic curves; group structures of rational points; Mordell's Theorem; computation examples. May include congruent numbers, Certicom's public cryptography challenges, Lenstra's factorization method, Birch/Swinnerton-Dyer Conjecture
Letter grade only (A-F). (Lecture 3 hrs.)

#### 542. Algebraic Geometry (3)

Prerequisite: MATH 540A or 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.)

#### 550A. Topology I (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, connectedness; convergence of sequences, nets and filters.
Letter grade only (A-F). (Lecture 3 hrs.)

#### 550B. Topology II (3)

Prerequisite: MATH 550A.
Further topics in point-set topology: local compactness, paracompactness, compactifications; metrizability; Baire category theorem; homotopy and the fundamental group. Topics may also include uniform spaces, function spaces, topological groups or topics from algebraic topology.
Letter grade only (A-F). (Lecture 3 hrs.)

#### 560A. Functional Analysis I (3)

Prerequisites: MATH 247, 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 hrs.)

#### 560B. Functional Analysis II (3)

Prerequisite: MATH 560A or consent of instructor.
Spectral theory of operators on normed spaces; special operators; elementary theory of Banach algebras; selected topics from applied functional analysis.
(Lecture 3 hrs.)

#### 561A. Real Analysis I (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 hrs.)

#### 561B. Real Analysis II (3)

Prerequisite: MATH 561A or consent of instructor.
Lp spaces of functions. Holder's inequality. Minkowski's inequality. Norm convergence, weak convergence and duality in Lp. Further topics from convergence of Fourier series, measure-theoretic probability, the Radon-Nikodym theorem; other topics depending on time and interest.
Letter grade only (A-F). (Lecture 3 hrs.)

#### 562A. Complex Analysis I (3)

Prerequisite: MATH 361B. Recommended: MATH 461.
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 hrs.)

#### 562B. Complex Analysis II (3)

Prerequisite: MATH 562A.
Multiple-valued functions, Riemann surfaces; analytic continuation; maximum modulus theorem; conformal mapping with applications, integral functions; gamma function, zeta function, special functions.
Letter grade only (A-F). (Lecture 3 hrs.)

#### 563. Applied Analysis (3)

Prerequisites: MATH 361B and either 364A or 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; 364A or 370A.
Stability and asymptotic analysis, Perturbation methods, Phase plane analysis, Bifurcation, Chaos, Applications to science and engineering.
(Lecture 3 hrs.)

#### 570. Partial Differential Equations (3)

Prerequisites: MATH 364A and 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./473. Scientific Computing (3)

Prerequisites: MATH 323 and either 364A or 370A. (Undergraduates register in MATH 473; graduates in MATH 573.)
Introduction to programming languages, implementations of numerical algorithms for solution of linear algebraic equations, interpolation and extrapolation, integration and evaluation of functions, root finding and nonlinear equations, fast Fourier transforms, minimization and maximization of functions, numerical solutions of differential equations.
Letter grade only (A-F). (Lecture 3 hrs.) Not open for credit to students with credit in MATH 473.

#### 574. Stochastic Calculus and Applications (3)

Prerequisites: MATH 361B, 364A or 370A, 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 364A or 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, 361B, 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 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./479. Mathematical Modeling (3)

Prerequisites: MATH 247, 364A or 370A, 323; one additional upper-division mathematics course, or consent of instructor. (Undergraduates register in MATH 479; graduates in MATH 579.)
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). (Lecture 3 hrs.)

#### 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 (1-3)

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).