MATH --  Mathematics

MATH 400 Vectors and Matrices (3) Prerequisite: MATH 221 or equivalent. Not open to students in the CMPS or Engineering Colleges. Credit will be granted for only one of the following: MATH 240, MATH 400, or MATH 461. The essentials of matrix theory needed in the management, social and biological sciences. Main topics: systems of linear equations, linear independence, rank, orthogonal transformations,eigenvalues, the principal axes theorem. Typical applications: linear models in economics and in statistics, Markov chains, age-specific population growth.

MATH 401 Applications of Linear Algebra (3) Prerequisite: MATH 240 or MATH 461. Various applications of linear algebra: theory of finite games, linear programming, matrix methods as applied to finite Markov chains, random walk, incidence matrices, graphs and directed graphs, networks, transportation problems.

MATH 402 Algebraic Structures (3) Prerequisite: MATH 240 or equivalent. Not open to mathematics graduate students. Credit will be granted for only one of the following: MATH 402 or MATH 403. For students having only limited experience with rigorous mathematical proofs. Parallels MATH 403. Students planning graduate work in mathematics should take MATH 403. Groups, rings, integral domains and fields, detailed study of several groups; properties of integers and polynomials. Emphasis is on the origin of the mathematical ideas studied and the logical structure of the subject.

MATH 403 Introduction to Abstract Algebra (3) Prerequisites: MATH 240 and MATH 241, or equivalent. Credit will be granted for only one of the following: MATH 402 or MATH 403. Integers; groups, rings, integral domains, fields.

MATH 404 Field Theory (3) Prerequisite: MATH 403. Algebraic and transcendental elements, Galois theory, constructions with straight-edge and compass, solutions of equations of low degrees, insolubility of the Quintic, Sylow theorems, fundamental theorem of finite Abelian groups.

MATH 405 Linear Algebra (3) Prerequisite: MATH 240 or MATH 461. An abstract treatment of finite dimensional vector spaces. Linear transformations and their invariants.

MATH 406 Introduction to Number Theory (3) Prerequisite: MATH 141 or permission of department. Integers, divisibility, prime numbers, unique factorization, congruences, quadratic reciprocity, Diophantine equations and arithmetic functions.

MATH 410 Advanced Calculus I (3) Prerequisites: MATH 240 and MATH 241 with a grade of C or better. Not open to students who have completed MATH 250. First semester of a year course. Subjects covered during the year are: sequences and series of numbers, continuity and differentiability of real valued functions of one variable, the Rieman integral, sequences of functions, and power series. Functions of several variables including partial derivatives, multiple integrals, line and surface integrals. The implicit function theorem.

MATH 411 Advanced Calculus II (3) Prerequisite: MATH 410. Not open to students who have completed MATH 250 and MATH 251. Continuation of MATH 410.

MATH 414 Differential Equations (3) Prerequisites: MATH 410; and MATH 240 or equivalent. Existence and uniqueness theorems for initial value problems. Linear theory: fundamental matrix solutions, variation of constants formula, Floquet theory for periodic linear systems. Asymptotic orbital and Lyapunov stability with phase plane diagrams. Boundary value theory and series solutions.

MATH 415 Introduction to Partial Differential Equations (3) Prerequisites: MATH 246; and {MATH 411 or MATH 251}. MATH 411 and MATH 415 may be taken concurrently. Credit will be granted for only one of the following: MATH 415 or MATH 462. First order equations, linear second order equations in two variables, one dimensional wave equation and the method of separation of variables, and other topics such as harmonic functions, the heat equation, and the wave equation in space.

MATH 417 Introduction to Fourier Analysis (3) Prerequisite: MATH 410. Fourier series. Fourier and Laplace transforms.

MATH 430 Euclidean and Non-Euclidean Geometries (3) Prerequisite: MATH 141. Hilbert's axioms for Euclidean geometry. Neutral geometry: the consistency of the hyperbolic parallel postulate and the inconsistency of the elliptic parallel postulate with neutral geometry. Models of hyerbolic geometry. Existence and properties of isometries.

MATH 432 Introduction to Point Set Topology (3) Prerequisite: MATH 410 or equivalent. Connectedness, compactness, transformations, homomorphisms; application of these concepts to various spaces, with particular attention to the Euclidean plane.

MATH 436 Differential Geometry of Curves and Surfaces I (3) Prerequisites: MATH 241; and either MATH 240 or MATH 461. Curves in the plane and Euclidean space, moving frames, surfaces in Euclidean space, orientability of surfaces; Gaussian and mean curvatures; surfaces of revolution, ruled surfaces, minimal surfaces, special curves on surfaces, "Theorema Egregium"; the intrinsic geometry of surfaces.

MATH 437 Differential Geometry of Curves and Surfaces II (3) Prerequisite: MATH 436. Differential forms, the Euler characteristic, Gauss-Bonnet theorem, the fundamental group; an outline of the topological classification of compact surfaces, vector fields, geodesics and Jacobi fields; classical calculus of variations, global differential geometry of surfaces, and elementary Riemann surface theory.

MATH 445 Elementary Mathematical Logic (3) Prerequisite: MATH 141. Credit will be granted for only one of the following: MATH 445 or MATH 450/CMSC 450. Elementary development of propositional and predicate logic, including semantics and deductive systems and with a discussion of completeness, incompleteness and the decision problem.

MATH 446 Axiomatic Set Theory (3) Prerequisite: MATH 403 or MATH 410. Development of a system of axiomatic set theory, choice principles, induction principles, ordinal arithmetic including discussion of cancellation laws, divisibility, canonical expansions, cardinal arithmetic including connections with the axiom of choice, Hartog's theorem, Konig's theorem, properties of regular, singular, and inaccessible cardinals.

MATH 447 Introduction to Mathematical Logic (3) Prerequisite: MATH 403 or MATH 410. Formal propositional logic, completeness, independence, decidability of the system, formal quantificational logic, first-order axiomatic theories, extended Godel completeness theorem, Lowenheim-Skolem theorem, model-theoretical applications.

MATH 450 Logic for Computer Science (3) Prerequisites: (CMSC 251 and MATH 141) (with grade of C or better). Also offered as CMSC 450. Credit will be granted for only one of the following: MATH 445 or MATH 450/CMSC 450. Elementary development of propositional and first-order logic accessible to the advanced undergraduate computer science student, including the resolution method in propositional logic and Herbrand's Unsatisfiability Theorem in first-order logic. Included are the concepts of truth, interpretation, validity, provability, soundness, completeness, incompleteness, decidability and semi-decidability.

MATH 452 Introduction to Dynamics and Chaos (3) Prerequisite: MATH 246. An introduction to mathematical dynamics and chaos. Orbits, bifurcations, Cantor sets and horseshoes, symbolic dynamics, fractal dimension, notions of stability, flows and chaos. Includes motivation and historical perspectives, as well as examples of fundamental maps studied in dynamics and applications of dynamics.

MATH 456 Cryptology (3) Prerequisite: Two 400-level MATH courses or two 400-level CMSC courses or permission of department. Also offered as CMSC 456. Credit will be granted for only one of the following: MATH 456 or CMSC 456. Importance in protecting data in communications between computers. The subject lies on the border between mathematics and computer science. Mathematical topics include number theory and probability, and computer science topics include complexity theory.

MATH 461 Linear Algebra for Scientists and Engineers (3) Prerequisites: MATH 141 and one MATH/STAT course for which MATH 141 is a prerequisite. This course cannot be used toward the upper level math requirements for MATH/STAT majors. Credit will be granted for only one of the following: MATH 240, MATH 400 or MATH 461. Basic concepts of linear algebra. This course is similar to MATH 240, but with more extensive coverage of the topics needed in applied linear algebra: change of basis, complex eigenvalues, diagonalization, the Jordan canonical form.

MATH 462 Partial Differential Equations for Scientists and Engineers (3) Prerequisites: MATH 241; and MATH 246. Credit will be granted for only one of the following: MATH 462 or MATH 415. Linear spaces and operators, orthogonality, Sturm-Liouville problems and eigenfunction expansions for ordinary differential equations, introduction to partial differential equations, including the heat equation, wave equation and Laplace's equation, boundary value problems, initial value problems, and initial-boundary value problems.

MATH 463 Complex Variables for Scientists and Engineers (3) Prerequisite: MATH 241 or equivalent. The algebra of complex numbers, analytic functions, mapping properties of the elementary functions. Cauchy integral formula. Theory of residues and application to evaluation of integrals. Conformal mapping.

MATH 464 Transform Methods for Scientists and Engineers (3) Prerequisite: MATH 246. Fourier series, Fourier and Laplace transforms. Evaluation of the complex inversion integral by the theory of residues. Applications to ordinary and partial differential equations of mathematical physics: solutions using transforms and separation of variables. Additional topics such as Bessel functions and calculus of variations.

MATH 472 Methods and Models in Applied Mathematics I (3) Prerequisite: {MATH 241; and MATH 246; and MATH 240; and PHYS 161 or PHYS 171} or permission of department. Recommended: one of: MATH 410, MATH 414, MATH 415, MATH 462, MATH 463 or PHYS 262, PHYS 273. Also offered as MAPL 472. Credit will be granted for only one of the following: MATH 472 and MAPL 472. Mathematical models in fluid dynamics and elasticity, both linear and non-linear partial differential equations, variational characterizations in eigenvalue problems, numerical algorithms. Additional optional topics as time permits. Some examples are Hamiltonian systems, Maxwell's equations, non-linear programming.

MATH 473 Methods and Models in Applied Mathematics II (3) Prerequisite: MATH 472 or permission of department. Also offered as MAPL 473. Credit will be granted for only one of the following: MATH 473 and MAPL 473. Continuation of the two semester sequence MATH 472 and MATH 473.

MATH 475 Combinatorics and Graph Theory (3) Prerequisites: MATH 240; and MATH 241. Also offered as CMSC 475. Credit will be granted for only one of the following: MATH 475 or CMSC 475. General enumeration methods, difference equations, generating functions. Elements of graph theory, matrix representations of graphs, applications of graph theory to transport networks, matching theory and graphical algorithms.

MATH 478 Selected Topics For Teachers of Mathematics (1-3) Prerequisite: one year of college mathematics or permission of department. (This course cannot be used toward the upper level math requirements for MATH/STAT majors).

MATH 498 Selected Topics in Mathematics (1-9) Honors students register for reading courses under this number. Repeatable to 9 credits if content differs. Topics of special interest to advanced undergraduate students will be offered occasionally under the general guidance of the departmental committee on undergraduate studies.

MATH 499 Honors Seminar (2) Prerequisite: permission of department. Not open to graduate students. Formerly MATH 398. Faculty supervised reports by students on mathematical literature. Both oral and written presentation on special topics of current interest.

MATH 598 Topics for Teachers Workshops (1-3) Prerequisite: current status as school teacher or permission of instructor. Workshops offered to school teachers for enrichment in various topics in modern mathematics.

MATH 600 Abstract Algebra I (3) Prerequisite: MATH 405 or equivalent. Groups with operators, homomorphism and isomorphism theorems, normal series, Sylow theorems, free groups, Abelian groups, rings, integral domains, fields, modules. If time permits, HOM (A,B), Tensor products, exterior algebra.

MATH 601 Abstract Algebra II (3) Prerequisite: MATH 600. Field theory, Galois theory, multilinear algebra. Further topics from: Dedekind domains, Noetherian domains, rings with minimum condition, homological algebra.

MATH 602 Homological Algebra (3) Prerequisite: MATH 600. Projective and injective modules, homological dimensions, derived functors, spectral sequence of a composite functor. Applications.

MATH 603 Commutative Algebra (3) Prerequisite: MATH 600. Ideal theory of Noetherian rings, valuations, localizations, complete local rings, Dedekind domains.

MATH 606 Algebraic Geometry I (3) Prerequisite: MATH 600 and MATH 601. Prime and primary ideals in Noetherian rings, Hilbert Nullstellensatz, places and valuations, prevarieties (in the sense of Serre), dimension, morphisms, singularities, varieties, schemes, rationality.

MATH 607 Algebraic Geometry II (3) Prerequisite: MATH 606. Topics in contemporary algebraic geometry chosen from among: theory of algebraic curves and surfaces, elliptic curves, Abelian varieties, theory of schemes, theory of zeta functions, formal cohomology, algebraic groups, reduction theory.

MATH 608 Selected Topics in Algebra (1-3) Prerequisite: permission of instructor.

MATH 620 Algebraic Number Theory I (3) Prerequisite: MATH 601. Algebraic numbers and algebraic integers, algebraic number fields of finite degree, ideals and units, fundamental theorem of algebraic number theory, theory of residue classes, Minkowski's theorem on linear forms, class numbers, Dirichlet's theorem on units, relative algebraic number fields, decomposition group, inertia group and ramification group of prime ideals with respect to a relatively Galois extension.

MATH 621 Algebraic Number Theory II (3) Prerequisites: MATH 600; and MATH 620 or equivalent. Valuation of a field, algebraic function fields, completion of a valuation field, ramification exponent and residue class degree, ramification theory, elements, differents, discriminants, product formula and characterization of fields by the formula, Gauss sum, class number formula of cyclotomic fields.

MATH 630 Real Analysis I (3) Prerequisite: MATH 411 or equivalent. Lebesque measure and the Lebesque integral on R, differentiation of functions of bounded variation, absolute continuity and fundamental theorem of calculus, Lp spaces on R, Riesz-Fisher theorem, bounded linear functionals on Lp, metric spaces, Baire category and uniform boundedness theorems.

MATH 631 Real Analysis II (3) Prerequisite: MATH 630. Abstract measure and integration theory, Radon-Nikodym theorem, Riesz Representation theorem, Lebesque decomposition, Fubini's theorem, Banach and Hilbert spaces, Banach-Steinhaus theorem, topological spaces, Arzela-Ascoli and Stone- Weierstrass theorems, compact sets and Tychonoff's theorem.

MATH 632 Functional Analysis (3) Prerequisite: MATH 631. Introduction to functional analysis and operator theory: normed linear spaces, basic principles of functional analysis, bounded linear operators on Hilbert spaces, spectral theory of selfadjoint operators, applications to differential and integral equations, additional topics as time permits.

MATH 634 Harmonic Analysis (3) Prerequisite: MATH 631. L1 theory: Fejer theorem, inversion theorem, ideal structure, Tauberian theorem. L2 theory: Plancherel-Parseval theorems, Paley-Wiener theorem. Lp theory: Hausdorff-Young theorem. Distribution theory: Bochner's theorem, Wiener continuous measures theorem, Malliavin theorem, Schwartz theory, almost periodic functions.

MATH 636 Representation Theory (3) Prerequisite: MATH 631. Introduction to representation theory of Lie groups and Lie algebras; initiation into non-abelian harmonic analysis through a detailed study of the most basic examples, such as unitary and orthogonal groups, the Heisenberg group, Euclidean motion groups, the special linear group. Additional topics from the theory of nilpotent Lie groups, semisimple Lie groups, p-adic groups or C*-algebras.

MATH 642 Dynamical Systems I (3) Prerequisites: MATH 432; and MATH 630 or equivalent. Foundations of topological dynamics, homeomorphisms, flows, periodic and recurrent points, transitivity and minimality, symbolic dynamics. Elements of ergodic theory, invariant measures and sets, ergodicity, ergodic theorems, mixing, spectral theory, flows and sections. Applications of dynamical systems to number theory, the Weyl theorem, the distribution of values of polynomials, Vander Waerden's theorem on arithmetic progressions.

MATH 643 Dynamical Systems II (3) Prerequisite: MATH 642 or equivalent. Entropy theory, variational principle for the entropy, expansiveness, measures with maximal entropy. Smooth systems on manifolds, diffeomorphisms and flows, periodic points, stable and unstable manifolds, homoclinic points, transversality, the Krupka-Smale theorem, Morse-Smale systems. Hyperbolicity, Anosov systems, distributions and foliations, strange attractors, Bowen's measure.

MATH 648 Selected Topics in Analysis (1-3) Prerequisite: permission of instructor.

MATH 655 Asymptotic Analysis and Special Functions I (3) Prerequisite: MATH 413 or MATH 463. Also offered as MAPL 655. Transcendental equations, Gamma function, orthogonal polynomials, Bessel functions, integral transforms, Watson's lemma, Laplace's method, stationary phase, analytic theory of ordinary differential equations, Liouville-Green (or WKBJ) approximation.

MATH 656 Asymptotic Analysis and Special Functions II (3) Prerequisite: MATH/MAPL 655. Also offered as MAPL 656. Steepest descents, coalescing saddle-points, singular integral equations, irregular singularities, Bessel, hypergeometic, and Legendre functions, Euler-Maclaurin formula, Darboux's method, turning points, phase shift.

MATH 660 Complex Analysis I (3) Prerequisite: MATH 410 or equivalent. Linear transformations, analytic functions, conformal mappings, Cauchy's theorem and applications, power series, partial fractions and factorization, elementary Riemann surfaces, Riemann's mapping theorem.

MATH 661 Complex Analysis II (3) Prerequisites: MATH 630; and MATH 660. Topics in conformal mappings, normal families, Picard's theorem, classes of univalent functions, extremal properties, variational methods, elliptic functions, Riemann surfaces.

MATH 668 Selected Topics in Complex Analysis (1-3) Repeatable if content differs. Prerequisite: permission of instructor. Material selected to suit interests and background of the students. Typical topics: Kaehler geometry, automorphic functions, several complex variables, symmetric spaces.

MATH 669 Selected Topics in Riemann Surfaces (1-3) Prerequisite: permission of instructor. Repeatable if content differs. Construction of Riemann surfaces, hyperbolic geometry, Fuchsian and Kleinian groups, potential theory, uniformisation spaces of meromorphic functions, line bundles, Picard variety, Riemann-Roch, Teichmueller theory.

MATH 670 Ordinary Differential Equations I (3) Prerequisites: MATH 405; and MATH 410 or the equivalent. Also offered as MAPL 670. Existence and uniqueness, linear systems usually with Floquet theory for periodic systems, linearization and stability, planar systems usually with Poincare-Bendixson theorem.

MATH 671 Ordinary Differential Equations II (3) Prerequisites: MATH 630; and MATH/MAPL 670 or the equivalent. Also offered as MAPL 671. The content of this course varies with the interests of the instructor and the class. Stability theory, control, time delay systems, Hamiltonian systems, bifurcation theory, and boundary value problems, and the like.

MATH 673 Partial Differential Equations I (3) Prerequisite: MATH 411 or equivalent. Also offered as MAPL 673. Credit will be granted for only one of the following: MATH 673 or MAPL 673. Analysis of boundary value problems for Laplace's equation, initial value problems for the heat and wave equations. Fundamental solutions, maximum principles, energy methods. First order nonlinear PDE, conservation laws. Characteristics, shock formation, weak solutions. Distributions, Fourier transform.

MATH 674 Partial Differential Equations II (3) Prerequisite: MATH/MAPL 673 or permission of instructor. Also offered as MAPL 674. Credit will be granted for only one of the following: MATH 674 or MAPL 674. Boundary value problems for elliptic partial differential equations via operator-theoretic methods. Hilbert spaces of functions. Duality, weak convergence. Sobolev spaces. Spectral theory of compact operators. Eigenfunction expansions.

MATH 680 Eigenvalue and Boundary Value Problems I (3) Prerequisites: MATH 405; and MATH 410 or equivalent. Also offered as MAPL 680. Operational methods applied to ordinary differential equations. Introduction to linear spaces, compact operators in Hilbert space, study of eigenvalues.

MATH 681 Eigenvalue and Boundary Value Problems II (3) Prerequisite: MATH/MAPL 680. Also offered as MAPL 681. Boundary value problems for linear paritial differential equations. Method of energy integrals applied to Laplace's equation, heat equations and the wave equations. Study of eigenvalues.

MATH 685 Modern Methods in Partial Differential Equations I (3) Prerequisites: MATH 630; and MATH 631. Also offered as MAPL 685. Spaces of distributions, Fourier transforms, concept of weak and strong solutions. Existence, uniqueness and regularity theory for elliptic and parabolic problems, methods of functional analysis.

MATH 686 Modern Methods in Partial Differential Equations II (3) Prerequisite: MATH/MAPL 685. Also offered as MAPL 686. Emphasis on nonlinear problems. Sobolev embedding theorems, methods of monotonicity, compactness, applications to elliptic, parabolic and hyperbolic problems.

MATH 695 Teaching Seminar (1) For MATH majors only. A course intended for first year teaching assistants. Topics include: everyday mechanics of teaching; teaching methods and styles; technology; course enrichment, diversity in the classroom; sexual harassment; teacher-student interactions; presentations by students.

MATH 710 Consistency Proofs in Set Theory (3) Prerequisites: MATH 446; and MATH 447. Consistency and independence of such fundamental principles of set theory as the laws of choice, of cardinal arithmetic of constructability and regularity. Godel's model of constructible sets, inner models, Cohen's generic models.

MATH 712 Mathematical Logic I (3) Sentential logic, first-order languages, models and formal deductions. Basic model theory including completeness and compactness theorems, other methods of constructing models, and applications such as non-standard analysis.

MATH 713 Mathematical Logic II (3) Prerequisite: MATH 712 or MATH 447. Incompleteness and undecidability results of Godel, Church, Tarski and others. Recursive function. Basic proof theory and axiomatic set theory.

MATH 715 Topics in Model Theory (3) Prerequisite: MATH 712. Topics in model theory drawn from the following areas, including recent developments: stability theory, classification theory, two-cardinal theorems, model-theoretic logics, models of arithmetic, homogeneous structures, applications to algebra and analysis.

MATH 716 Topics In Recursion Theory (3) Prerequisite: MATH 713. Topics in recursion theory drawn from the following areas: the lattice of r.e. sets, finite and infinite injury priority arguments, minimal degrees, automorphisms of the turing degrees, recursive ordinals, hyperarithmetical sets, the analytical hierarchy, E-recursion theory, bounded queries.

MATH 718 Selected Topics in Mathematical Logic (1-3) Prerequisite: permission of instructor.

MATH 730 Fundamental Concepts of Topology (3) Prerequisites: {MATH 410; and MATH 411; and MATH 403} or equivalent. Survey of basic point set topology, fundamental group, covering spaces, Van Kampen's theorem, simplicial complexes, simplicial homology, Euler characteristics and classification of surfaces.

MATH 734 Algebraic Topology (3) Prerequisite: MATH 403 or equivalent. Recommended: MATH 730. Singular homology and cohomology, cup products, Poincare duality, Eilenberg-Steenrod axioms, Whitehead and Hurewicz theorems, universal coefficient theorem, cellular homology.

MATH 740 Riemannian Geometry (3) Prerequisites: {MATH 405; and MATH 411} or equivalent. Manifolds, tangent vectors and differential forms, Riemannian metrics, connections, curvature, structure equations, geodesics, completeness, immersions, tensor algebra, Lie derivative.

MATH 742 Differential Topology (3) Prerequisites: {MATH 410; and MATH 411} or equivalent. Inverse and implicit function theorems, Sard's theorem, orientability, degrees, smooth vector bundles, imbeddings and immersions, transversality approximation theorems and applications, isotopy extension theorem, tubular neighborhoods.

MATH 744 Lie Groups I (3) Prerequisite: {MATH 403; MATH 405; MATH 411 and MATH 432} or equivalent. An introduction to the fundamentals of Lie groups, including some material on groups of matrices and Lie algebras.

MATH 745 Lie Groups II (3) Prerequisite: MATH 744. A continuation of Lie groups I in which some of the following topics will be emphasized: solvable Lie groups, compact Lie groups, classifications of semi-simple Lie groups, representation theory, homogeneous spaces.

MATH 748 Selected Topics in Geometry and Topology (1-3) Prerequisite: permission of instructor.

MATH 799 Master's Thesis Research (1-6)

MATH 899 Doctoral Dissertation Research (1-8)