# Course Catalog 2012-2013      Mathematics

MATH 101 - SINGLE VARIABLE CALCULUS I
Credits: 3
Differentiation, extrema, Newton's method, integration, fundamental theorem of calculus, area, volume, natural logarithm, exponential, arc length, surface area, Simpson's rule, L'Hopital's rule. May substitute MATH 111-112 or take MATH 101 after completing MATH 111.
College: School of Natural Sciences
Department: Mathematics

MATH 102 - SINGLE VARIABLE CALCULUS II
Credits: 3
Continuation of MATH 101. Includes further techniques of integration, as well as infinite sequences and series, tests for convergence, power series, radius of convergence, polar coordinates, parametric equations, and arc length.
College: School of Natural Sciences
Department: Mathematics

MATH 111 - FUNDAMENTAL THEOREM OF CALCULUS
Credits: 3
Study of calculus, forming with MATH 112 a slower-paced version of MATH 101/102. Contains less detail in the coverage of infinite series. Students may take MATH 111/112 followed by MATH 102, or MATH 111 followed by MATH 101/102.
College: School of Natural Sciences
Department: Mathematics

MATH 112 - CALCULUS AND ITS APPLICATIONS
Credits: 3
Continuation of the study of calculus from MATH 111.
College: School of Natural Sciences
Department: Mathematics

MATH 211 - ORDINARY DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA
Credits: 3
Study of ordinary differential equations (e.g., solutions to separable and linear first-order equations and to higher-order linear equations with constant coefficients, the properties of solutions to differential equations, and numerical solution methods) and linear algebra (e.g., vector spaces and solutions to algebraic linear equations, dimension, eigenvalues, and eigenvectors of a matrix), as well as the application of linear algebra to first-order systems of differential equations and the qualitative theory of nonlinear systems and phase portraits. Use of the computers in Owlnet as part of each homework assignment required.
College: School of Natural Sciences
Department: Mathematics

MATH 212 - MULTIVARIABLE CALCULUS
Credits: 3
Study of gradient, divergence, and curl, Lagrange multipliers, multiple integrals, as well as line integrals, conservative vector fields, Green's theorem, Stokes's theorem, and Gauss's theorem. May substitute Math 221 and 222. Equivalency: MATH 222.
College: School of Natural Sciences
Department: Mathematics

MATH 221 - HONORS CALCULUS III
Credits: 3
This course and MATH 222 include the material of MATH 212 and much more. Topology of Rn, calculus for functions of several variables, linear and multilinear algebra, theory of determinants, inner product spaces, integration on manifolds.
College: School of Natural Sciences
Department: Mathematics

MATH 222 - HONORS CALCULUS IV
Credits: 3
See MATH 221. A student may not receive credit for both MATH 222 and MATH 212. Equivalency: MATH 212.
College: School of Natural Sciences
Department: Mathematics

MATH 300 - TOPICS IN UNDERGRADUATE MATH
Credits: 3
Treatment of topics in undergraduate mathematics with an emphasis on writing of clear, cogent compete mathematical proofs. Topics vary by year. May be repeated for credit with permission of department.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 102

MATH 302 - ELEMENTS OF ANALYSIS
Credits: 3
Introductory treatment of topics in analysis and topology, with the real line as a central example. Techniques include connected and compact sets, sequences and subsequences, continuity, and uniform approximation. Clear, cogent, and complete mathematical arguments are emphasized.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 102

MATH 304 - ELEMENTS OF KNOT THEORY
Credits: 3
Introduction to the mathematical theory of knots. Techniques to distinguish knots from one another. Reidemeister moves, mod-p colorings, knot determinants, knot polynomials, Seifert surfaces, Euler characteristic, knot groups, and untying knots in four dimensions. We will also discuss open problems in knot theory. Recommended Corequisite(s): MATH 221 OR MATH 354 OR MATH 355.
College: School of Natural Sciences
Department: Mathematics

MATH 310 - MATHEMATICS OF MUSIC
Credits: 2
This course will survey the many places in which mathematics plays a role in music. The topics and mathematical tools used are diverse: simple number theory in musical scales and tunings, differential equations in the physics of musical instruments, and symmetry in music explained in terms of mathematical group theory. Throughout the course we will use Csound, a compiler language for creating sound, to give concrete examples.
College: School of Music
Department: Mathematics
Pre-requisites: MATH 211 OR MATH 213

MATH 321 - INTRODUCTION TO ANALYSIS I
Credits: 3
A thorough treatment of basic methods of analysis such as metric spaces, compactness, sequences and series of functions. Also further topics in analysis, such as Hilbert spaces, Fourier series, Sturm- Liouville theory.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 221 or MATH 300 or MATH 302 or permission of department

MATH 322 - INTRODUCTION TO ANALYSIS II
Credits: 3
See MATH 321. Includes proofs of the basic results for multivariable calculus (MATH 321 provides proofs for single-variable calculus).
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 321 or permission of instructor

MATH 354 - HONORS LINEAR ALGEBRA
Credits: 3
Course URL: http://math.rice.edu/~cochran/
Vector spaces, linear transformations and matrices, theory of systems of linear equations, determinants, eigenvalues and diagonalizability, inner product spaces; and optional material chosen from: dual vector spaces, spectral theorem for self-adjoint operators, Jordan canonical form. Content is similar to that of MATH 355, but with more emphasis on theory. The course will include instruction on how to construct mathematical proofs. This course is appropriate for potential Mathematics majors and others interested in learning how to construct rigorous mathematical arguments. Credit may not be received for both MATH 354 and MATH 355. Equivalency: MATH 355. Recommend a 200-level math class.
College: School of Natural Sciences
Department: Mathematics

MATH 355 - LINEAR ALGEBRA
Credits: 3
Linear transformations and matrices, solution of linear equations, eigenvalues and eigenvectors, quadratic forms, Jordan canonical form. Credit may not be received for both MATH 354 and MATH 355. Equivalency: MATH 354.
College: School of Natural Sciences
Department: Mathematics

MATH 356 - ABSTRACT ALGEBRA I
Credits: 3
Group theory: normal subgroups, factor groups, Abelian groups, permutations, matrix groups, and group action.
College: School of Natural Sciences
Department: Mathematics

MATH 365 - NUMBER THEORY
Credits: 3
Properties of numbers depending mainly on the notion of divisibility. Continued fractions.
College: School of Natural Sciences
Department: Mathematics

MATH 366 - GEOMETRY
Credits: 3
Topics chosen from Euclidean, spherical, hyperbolic, and projective geometry, with emphasis on the similarities and differences found in various geometries. Isometries and other transformations are studied and used throughout. The history of the development of geometric ideas is discussed. This course is strongly recommended for prospective high school teachers.
College: School of Natural Sciences
Department: Mathematics

MATH 368 - TOPICS IN COMBINATORICS
Credits: 3
Study of combinatorics and discrete mathematics. Topics that may be covered include graph theory, Ramsey theory, finite geometries, combinatorial enumeration, combinatorial games.
College: School of Natural Sciences
Department: Mathematics

MATH 370 - CALCULUS ON MANIFOLDS
Credits: 3
Differentiation and integration on manifolds: calculus on ℜn, exterior differentiation, differential forms, vector fields, Stokes' theorem.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: (MATH 212 OR MATH 222) AND (MATH 211 OR MATH 355 OR MATH 354)

MATH 371 - LIE THEORY
Credits: 3
Study of classical groups as symmetries of Euclidean spaces. Geometry of complex numbers and quaternions, rotations and reflections of ℜn, the orthogonal, unitary and sympletic groups. Tangent spaces to matrix groups, Lie algebras and exponential map. If time permits: the structure of Lie algebras and the matrix logarithm. Recommended Corequisite(s): MATH 354 or MATH 355.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: (MATH 211 AND MATH 212) OR MATH 221

MATH 373 - ELLIPTIC CURVES
Credits: 3
Elliptic curves are central to modern number theory and instrumental in the proof of Fermat's Last Theorem. Topics will include: the addition law, solutions over the rational numbers, explicit computations, applications to factorization and cryptography; if time permits, infinite series attached to elliptic curves and the Birch-Swinnerton-Dyer conjecture. Recommended Prerequisite(s): 200 Level Math Course
College: School of Natural Sciences
Department: Mathematics

MATH 374 - INTRODUCTION TO REPRESENTATION THEORY
Credits: 3
First course in representation theory, with an emphasis on concrete examples, especially the symmetric group. Topics include representations of finite groups, characters, classification, symmetric functions, Young symmetrizers, and Schur-Weyl duality. Prior experience with proofs is necessary; some familiarity with linear or abstract algebra would be helpful, but can be acquired along the way.
College: School of Natural Sciences
Department: Mathematics

MATH 381 - INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
Credits: 3
Laplace transform: inverse transform, applications to constant coefficient differential equations. Boundary value problems: Fourier series, Bessel functions, Legendre polynomials.
College: School of Natural Sciences
Department: Mathematics

MATH 382 - COMPLEX ANALYSIS
Credits: 3
Study of the Cauchy integral theorem, Taylor series, residues, as well as the evaluation of integrals by means of residues, conformal mapping, and application to two-dimensional fluid flow. May not receive credit for this course and MATH 427. Equivalency: MATH 427.
College: School of Natural Sciences
Department: Mathematics

Credits: 1
Lectures by undergraduate students on mathematical topics not usually covered in other courses. Presentation of one lecture and attendance at all sessions required.
College: School of Natural Sciences
Department: Mathematics

MATH 401 - DIFFERENTIAL GEOMETRY
Credits: 3
Study of the differential geometry of curves and surfaces in R3. Includes an introduction to the concept of curvature and thorough treatment of the Gauss-Bonnet theorem.
College: School of Natural Sciences
Department: Mathematics

MATH 402 - DIFFERENTIAL GEOMETRY
Credits: 3
Introduction to Riemannian geometry. Content varies from year to year. Content varies from year to year. Graduate/Undergraduate Equivalency: MATH 500. Recommended Prerequisite: MATH 401.
College: School of Natural Sciences
Department: Mathematics

MATH 410 - CALCULUS OF VARIATIONS
Credits: 3
Study of classical and modern theories about functions having some integral expression which is maximal, minimal, or critical. Geodesics, brachistochrone problem, minimal surfaces, and numerous applications to physics. Euler-Lagrange equations, 1st and 2nd variations, Hamilton's Principle.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 101 AND MATH 102 AND (MATH 211 OR MATH 212 OR MATH 221 OR MATH 222)

MATH 423 - PARTIAL DIFFERENTIAL EQUATIONS I
Credits: 3
First order of partial differential equations. The method of characteristics. Analysis of the solutions of the wave equation, heat equation and Laplace's equation. Integral relations and Green's functions. Potential theory, Dirichlet and Neumann problems. Asymptotic methods: the method of stationary phase, geometrical optics, regular and singular perturbation methods. Cross-list: CAAM 423.
College: School of Natural Sciences
Department: Mathematics

MATH 424 - PARTIAL DIFFERENTIAL EQUATIONS
Credits: 3
Continuation of MATH 423.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 423

MATH 425- INTEGRATION THEORY
Credits: 3
College: School of Natural Sciences
Department: Mathematics

MATH 426 - TOPICS IN REAL ANALYSIS
Credits: 3
Content varies from year to year. May include Fourier series, harmonic analysis, probability theory, advanced topics in measure theory, ergodic theory, and elliptic integrals. Equivalency: MATH 516.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 425

MATH 427 - COMPLEX ANALYSIS
Credits: 3
Study of the Cauchy-Riemann equation, power series, Cauchy's integral formula, residue calculus, and conformal mappings. Emphasis on the theory. Credit may not be received for both MATH 382 and MATH 427. Equivalency: MATH 382, Graduate/Undergraduate Equivalency: MATH 517.
College: School of Natural Sciences
Department: Mathematics

MATH 428 - TOPICS IN COMPLEX ANALYSIS
Credits: 3
Special topics include Riemann mapping theorem, Runge's Theorem, elliptic function theory, prime number theorem, Riemann surfaces, et al. Graduate/Undergraduate Equivalency: MATH 518.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 382 OR MATH 427

MATH 435 - DYNAMICAL SYSTEMS
Credits: 3
Existence and uniqueness for solutions of ordinary differential equations and difference equations, linear systems, nonlinear systems, stability, periodic solutions, bifurcation theory. Theory and theoretical examples are complemented by computational, model driven examples from bioilogical and physical sciences. Cross-list: CAAM 435. Recommended Prerequisite(s): (CAAM 210 AND MATH 212) AND (CAAM 335 OR MATH 355) AND (CAAM 401 OR MATH 321).
College: School of Natural Sciences
Department: Mathematics

MATH 443 - GENERAL TOPOLOGY
Credits: 3
Study of basic point set topology. Includes a treatment of cardinality and well ordering, as well as metrization. Graduate/Undergraduate Equivalency: MATH 538.
College: School of Natural Sciences
Department: Mathematics

MATH 444 - GEOMETRIC TOPOLOGY
Credits: 3
Introduction to algebraic methods in topology and differential topology. Elementary homotopy theory. Covering spaces. Graduate/Undergraduate Equivalency: MATH 539.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 443

MATH 445 - ALGEBRAIC TOPOLOGY
Credits: 3
Introduction to the theory of homology. Includes simplicial complexes, cell complexes and cellular homology and cohomology, as well as manifolds, and Poincare duality. Graduate/Undergraduate Equivalency: MATH 540.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 444

MATH 463 - ABSTRACT ALGEBRA II
Credits: 3
Ring theory: ideals, polynomials, factorization. Advanced linear algebra: quadratic forms, canonical forms. Field theory: extensions, Galois theory, soluability in radicals. Continuation of MATH 356. Graduate/Undergraduate Equivalency: MATH 563.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 356 or permission of instructor

MATH 464 - ABSTRACT ALGEBRA III
Credits: 3
Continuation of MATH 463. Tensor and exterior algebra, introductory commutative algebra, structure of modules, and elements of homological algebra. Additional advanced topics may include representations of finite groups and affine algebraic geometry. Graduate/Undergraduate Equivalency: MATH 564.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 463 or permission of instructor

MATH 465 - TOPICS IN ALGEBRA: An Introduction to Algebraic Geometry
Credits: 3
Varieties as solution sets of systems of polynomial equations, varieties in projective space, rational and regular functions, maps of varieties, local properties and singularities. Graduate/Undergraduate Equivalency: MATH 565.
College: School of Natural Sciences
Department: Mathematics

MATH 466 - TOPICS IN ALGEBRA II
Credits: 3
College: School of Natural Sciences
Department: Mathematics

MATH 468 - POTPOURRI
Credits: 3
This course deals with miscellaneous special topics not covered in other courses.
College: School of Natural Sciences
Department: Mathematics

MATH 471 - MATHEMATICS OF APERIODIC ORDER
Credits: 3
Mathematical models of quasicrystals, whose discovery in the early 1980's led to a paradigm shift in materials science. Topics include: classical theory of ordered structures (i.e., lattices modeling crystals), Delone subsets and tilings of Euclidean space, aperiodically ordered structures generated by inflation or cut-and-project schemes. Recommended Prerequisite(s): MATH 356.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 321 OR MATH 355

MATH 472 - ASYMPTOTIC ANALYSIS: INTRODUCTION TO PERTURBATION METHODS
Credits: 3
Order symbols, asymptotic expansions, asymptotic solutions to algebraic and transcendental equations, asymptotic solutions of ordinary differential equations, matched asymptotic expansions, multiple scales, WKB and related methods, asymptotic approximations of integrals, stationary phase/steepest descent methods.
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 211 AND MATH 382

Credits: Hours Variable
College: School of Natural Sciences
Department: Mathematics

MATH 498 - RESEARCH THEMES IN THE MATHEMATICAL SCIENCES
Credits: Hours Variable
A seminar course that will cover selected themes of general research in the mathematical sciences from the perspectives of mathematics, computational and applied mathematics and statistics. The course may be repeated multiple times for credit. Cross-list: CAAM 498, STAT 498, Graduate/Undergraduate Equivalency: MATH 698.
College: School of Natural Sciences
Department: Mathematics

MATH 499 - MATHEMATICAL SCIENCES VIGRE SEMINAR
Credits: Hours Variable
College: School of Natural Sciences
Department: Mathematics

MATH 500 - DIFFERENTIAL GEOMETRY
Credits: 3
College: School of Natural Sciences
Department: Mathematics

MATH 501 - TOPICS IN DIFFERENTIAL GEOMETRY
Credits: 3
Topic to be announced.
College: School of Natural Sciences
Department: Mathematics

MATH 502 - TOPICS IN DIFFERENTIAL GEOMETRY
Credits: 3
Topic to be announced.
College: School of Natural Sciences
Department: Mathematics

MATH 515 - INTEGRATION THEORY
Credits: 3
College: School of Natural Sciences
Department: Mathematics

MATH 516 - TOPICS IN REAL ANALYSIS
Credits: 3
Cross-list: MATH 426.
College: School of Natural Sciences
Department: Mathematics

MATH 517 - COMPLEX ANALYSIS
Credits: 3
College: School of Natural Sciences
Department: Mathematics

MATH 518 - TOPICS IN COMPLEX ANALYSIS
Credits: 3
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 382 OR MATH 427

MATH 521 - ADVANCED TOPICS IN REAL ANALYSIS
Credits: 3
Topic to be announced.
College: School of Natural Sciences
Department: Mathematics

MATH 522 - TOPICS IN ANALYSIS
Credits: 3
Topic to be announced.
College: School of Natural Sciences
Department: Mathematics

MATH 523 - FUNCTIONAL ANALYSIS
Credits: 3
Banach spaces: review of L^p spaces, linear operators, dual space, Hahn-Banach theorem, weak topologies, Banach-Alaoglu theorem, compact and bounded operators, closed graph theorem; Hilbert spaces: self-adjoint and unitary operators (including spectral theorem), symmetric operators and self-adjoint extensions; if time allows, distributions and Sobolev spaces.
College: School of Natural Sciences
Department: Mathematics

MATH 527 - ERGODIC THEORY AND TOPOLOGICAL DYNAMICS
Credits: 3
Topic to be announced.
College: School of Natural Sciences
Department: Mathematics

MATH 528 - ERGODIC THEORY AND TOPOLOGICAL DYNAMICS
Credits: 3
College: School of Natural Sciences
Department: Mathematics

MATH 538 - GENERAL TOPOLOGY
Credits: 3
College: School of Natural Sciences
Department: Mathematics

MATH 539 - GEOMETRIC TOPOLOGY
Credits: 3
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 443

MATH 540 - ALGEBRAIC TOPOLOGY
Credits: 3
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 539

MATH 541 - TOPICS IN TOPOLOGY
Credits: 3
Topic to be announced.
College: School of Natural Sciences
Department: Mathematics

MATH 542 - TOPICS IN ADVANCED TOPOLOGY
Credits: 3
Topic to be announced.
College: School of Natural Sciences
Department: Mathematics

MATH 543 - TOPICS IN LOW DIMENSIONAL TOPOLOGY
Credits:Variable
College: School of Natural Sciences
Department: Mathematics

MATH 563 - ABSTRACT ALGEBRA II
Credits: 3
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 356

MATH 564 - ABSTRACT ALGEBRA III
Credits: 3
College: School of Natural Sciences
Department: Mathematics
Pre-requisites: MATH 563

MATH 565 - TOPICS IN ALGEBRA: INTRODUCTION TO ALGEBRAIC GEOMETRY
Credits: 3
Varieties as solution sets of systems of polynomial equations, varieties in projective space, rational and regular functions, maps of varieties, local properties and singularities. Graduate/Undergraduate Equivalency: MATH 465.
College: School of Natural Sciences
Department: Mathematics

MATH 566 - TOPICS IN ALGEBRA II
Credits: 3
College: School of Natural Sciences
Department: Mathematics

MATH 567 - TOPICS IN ALGEBRAIC GEOMETRY
Credits: 3
Possible topics include rational points on algebraic varieties, moduli spaces, deformation theory, and Hodge structures. Recommended prerequisite(s): Mathematics 463 - 464.
College: School of Natural Sciences
Department: Mathematics

MATH 590 - CURRENT MATHEMATICS SEMINAR
Credits: 1
Lectures on topics of recent research in mathematics delivered by mathematics graduate students and faculty.
College: School of Natural Sciences
Department: Mathematics

MATH 591 - GRADUATE TEACHING SEMINAR
Credits: 1
Discussion on teaching issues and practice lectures by participants as preparation for classroom teaching of mathematics.
College: School of Natural Sciences
Department: Mathematics

MATH 680 - MATHEMATICS COLLOQUIUM
Credits:1
Presentations of research topics in mathematics and related fields
College: School of Natural Sciences
Department: Mathematics

MATH 681 - TOPOLOGY SEMINAR
Credits:1
Presentations of research topics in topology and related fields
College: School of Natural Sciences
Department: Mathematics

MATH 682 - ALGEBRAIC GEOMETRY SEMINAR
Credits:1
Presentations of research topics in algebraic geometry and related fields
College: School of Natural Sciences
Department: Mathematics

MATH 683 - GEOMETRY AND ANALYSIS SEMINAR
Credits:1
Presentations of research topics in geometric analysis, mathematical physics, dynamics and related fields
College: School of Natural Sciences
Department: Mathematics

Credits: Variable
College: School of Natural Sciences
Department: Mathematics

MATH 698 - RESEARCH THEMES IN THE MATHEMATICAL SCIENCES
Credits: Hours Variable
A seminar course that will cover selected theme of general research in the mathematical sciences from the perspectives of mathematics, computational and applied mathematics and statistics. The cousre may be repeated multiple times for credit. Cross-list: CAAM 698, STAT 698, Graduate/Undergraduate Equivalency: MATH 498.
College: School of Natural Sciences
Department: Mathematics

MATH 699 - MATHEMATICAL SCIENCES VIGRE SEMINAR
Credits: Hours Variable
College: School of Natural Sciences
Department: Mathematics

MATH 777 - VISITING RESEARCH TRAINEE
Credits: 0
College: School of Natural Sciences
Department: Mathematics

MATH 800 - THESIS AND RESEARCH
Credits: Hours Variable
College: School of Natural Sciences
Department: Mathematics
##### Updated November 6, 2012
Department of Mathematics, Rice University, 6100 Main St, Houston, TX 77005
Mailing Address: MS-136, P.O. Box 1892, Houston, Texas 77251-1892
Telephone (713) 348-4829     Fax (713) 348-5231