OEO Office of Equal Opportunity - Department of Mathematics and ...

250 ARTS AND SCIENCES
472./572. Fourier Analysis and Wavelets. (3)
Discrete Fourier and Wavelet Transform. Fourier series and
integrals. Expansions in series of orthogond wavelets and
other functions. Multiresolution and time/frequency analysis.
Applications to signal processing and statistics.
Prerequisite: 314, 321 or 401 or permission of the instructor.
{Offered upon demand}
499. Individual Study. (1-3 to a maximum of 6)
Guided study, under the supervision of a faculty member, of
selected topics not covered in regular courses
Footnote:
1 See Restrictions earlier in Mathematics and Statistics.
IV. Graduate Courses
**501./401. Advanced Calculus I. (4)
Rigorous treatment of calculus in one variable. Definition and
topology of real numbers, sequences, limits, functions, continuity,
differentiation and integration. Students will learn how
to read, understand and construct mathematical proofs.
Prerequisite: 264 and two courses at the 300+ level.
**502./402. Advanced Calculus II. (3)
Generalization of 401/501 to several variables and metric
spaces: sequences, limits, compactness and continuity on
metric spaces; interchange of limit operations; series, power
series; partial derivatives; fixed point, implicit and inverse
function theorems; multiple integrals.
Prerequisite: 501.
504. Introductory Numerical Analysis: Numerical Linear
Algebra. (3)
(Also offered as CS 575.) Direct and iterative methods of the
solution of linear systems of equations and least squares
problems. Error analysis and numerical stability. The eigenvalue
problem. Descent methods for function minimization,
time permitting.
Prerequisites: 464, 514, some knowledge of programming.
{Spring}
505. Introductory Numerical Analysis: Approximation
and Differential Equations. (3)
(Also offered as CS 576.) Numerical approximation of functions.
Interpolation by polynomials, splines and trigonometric
functions. Numerical integration and solution of ordinary
differential equations. An introduction to finite difference and
finite element methods, time permitting.
Prerequisites: 316 or 401 and some knowledge of programming.
{Fall}
510. Introduction to Analysis I. (3)
Real number fields, sets and mappings. Basic point set topology,
sequences, series, convergence issues. Continuous functions,
differentiation, Riemann integral. General topology and
applications: Weierstrass and Stone-Weierstrass approximation
theorems, elements of Founier Analysis (time permitting).
Prerequisites: 321, 401. {Fall}
511. Introduction to Analysis II. (3)
Continuation of 510. Differentiation in R n . Inverse and implicit
function theorems, integration in R n , differential forms and
Stokes theorem.
Prerequisite: 510. {Spring}
512./462. Introduction to Ordinary Differential
Equations. (3)
Linear systems. Existence and uniqueness theorems, flows,
linearized stability for critical points, stable manifold theorem.
Gradient and Hamiltonian systems. Limit sets, attractors,
periodic orbits, Floquet theory and the Poincare Map.
Introduction to perturbation theory.
Prerequisites: 314, or 321, 316, 401. {Fall}
513./463. Introduction to Partial Differential Equations. (3)
Classification of partial differential equations; properly posed
problems; separation of variables, eigenfunctions and
Green’s functions; brief survey of numerical methods and
variational principles.
Prerequisites: 312, 313, 314 or 321, one of 311 or 402.
{Spring}
514./464. Applied Matrix Theory. (3)
Determinants; theory of linear equations; matrix analysis of
differential equations; eigenvalues, eigenvectors and canonical
forms; variational principles; generalized inverses.
Prerequisite: 314 or 321. {Fall}
519. Selected Topics in Number Theory. (3, no limit) ∆
520. Abstract Algebra I. (3)
Theory of groups, permutation groups, Sylow theorems.
Introduction to ring theory, polynomial rings. Principal ideal
domains.
Prerequisite: 322. {Fall}
521. Abstract Algebra II. (3)
Continuation of 520. Module theory, field theory, Galois
theory.
Prerequisites: 321, 520. {Spring}
530. Algebraic Geometry I. (3)
Basic theory of complex affine and projective varieties.
Smooth and singular points, dimension, regular and rational
mappings between varieties, Chow’s theorem.
Prerequisites: 431, 521, 561. {Alternate Falls}
531. Algebraic Geometry II. (3)
Continuation of 530. Degree of a variety and linear systems.
Detailed study of curves and surfaces.
Prerequisite: 530. {Alternate Springs}
532. Algebraic Topology I. (3)
Introduction to homology and cohomology theories. Homotopy
theory, CW complexes.
Prerequisites: 431, 521. {Alternate Falls}
533. Algebraic Topology II. (3)
Continuation of 532. Duality theorems, universal coefficients,
spectral sequence.
Prerequisite: 532. {Alternate Springs}
534./434. Introduction to Differential Geometry. (3)
Elementary theory of surfaces, differential forms, integral
geometry, Riemannian geometry.
Prerequisite: 311 or 402. {Offered upon demand}
535. Foundations of Topology. (3)
Basic point set topology. Separation axioms, metric spaces,
topological manifolds, fundamental group and covering
spaces.
Prerequisite: 401.
536. Introduction to Differentiable Manifolds. (3)
Concept of a manifold, differential structures, vector bundles,
tangent and cotangent bundles, embedding, immersions and
submersions, transversality, Stokes’ theorem.
Prerequisite: 511. {Alternate Springs}
537. Riemannian Geometry I. (3)
Theory of connections, curvature, Riemannian metrics, Hopf-
Rinow theorem, geodesics. Riemannian submanifolds.
Prerequisite: 536. {Alternate Falls}
538. Riemannian Geometry II. (3)
Continuation of MATH 537 with emphasis on adding more
structures. Riemannian submersions, Bochner theorems
with relation to topology of manifolds, Riemannian Foliations,
Complex and Kaehler geometry, Sasakian and contact
geometry.
Prerequisite: 537.
539. Selected Topics in Geometry and Topology. (3,
no limit) ∆
UNM CATALOG 2006–2007 Symbols, page 611.