www. math.iastate.edu

For the undergraduate curriculum in liberal arts and sciences, major in mathematics, leading to the degree bachelor of science, see Liberal Arts and Sciences, Curriculum.

The program in mathematics offers training suitable for students planning to enter secondary school teaching, to work in mathematics and computation for industry or government, or to continue their studies in graduate school. The requirements for an undergraduate major in mathematics are designed so that the student may have opportunity for appropriate specialization to meet one or more of the foregoing objectives and, at the same time, obtain a thorough introduction to the mathematics underlying all of them.

Graduates understand a broad range of mathematical topics and are familiar with a broad range of mathematical models. They have skills for solving problems in diverse situations. They can construct rigorous arguments to demonstrate mathematical facts. They can communicate their mathematical methods to others and can justify their assumptions.

(a) Math 165, 166, 201, 265, 317, 301, 414, and either 266 or 267.

(b) 15 additional credits in mathematics courses at the 300 level or above.

(c) The courses used to satisfy a) and b) above must include one of the sequences 301, 302; 414, 415; 435, 436.

(d) In addition to the credits in (b), either Math 492 or 2 credits of C I/LAS 480C. (C I/LAS 480C is available only for students seeking secondary school certification).(e) Communication Proficiency requirement: The department requires a grade of C or better in each of English 150 and 250 (or 250H) and an upper-level communication skills requirement that may be met by writing an acceptable undergraduate thesis (Math 491) or by taking at least one of Engl 302, 305, 314 or Jl MC 201. A grade of C- or better is required.

The department strongly recommends that each student majoring in mathematics include in the program substantial supporting work beyond the minimum general education requirement of the college in one or more areas of application of mathematics, such as other mathematical sciences, engineering, natural science, or social science. In particular, it recommends that each student take Com S 207, 208; Phys 221, 222; and Stat 341, 342 (or Math 304). It also recommends that students contemplating graduate study in mathematics acquire a reading knowlege of French, German, or Russian. Credits earned in Math 104, 105, 140, 141, 142, 150, 151, 160, 181, 182, 195, 196, cannot be counted toward graduation by mathematics majors.

The department offers a minor in mathematics which may be earned by credit in Math 201, 265, (266 or 267), (307 or 317), and 301.

The department offers programs leading to a master of science or doctor of philosophy degree in mathematics or applied mathematics, as well as minor work for students whose major is in another department. The department also offers a program leading to the degree of master of school mathematics (M.S.M.).

Students desiring to undertake graduate work leading to the M.S. or Ph.D. degree should have at least 12 semester credits of work in mathematics beyond calculus. It is desirable that these credits include advanced calculus and abstract algebra.

The M.S. degree requires at least 30 credit hours and students must write a creative component or thesis and pass a comprehensive oral examination over their coursework and their creative component or thesis. See the department handbook for specific requirements.

The Ph.D. degree requires a student to take 54 hours of coursework in addition to research hours, pass written qualifying examinations, pass an oral preliminary exam, and perform an original research project culminating in a dissertation which is defended by an oral exam. Ph.D. candidates must have at least one year of supervised teaching experience. See the on-line Mathematics Graduate Handbook for specific requirements.

The M.S.M. degree is primarily for inservice secondary mathematics teachers. Students desiring to pursue the M.S.M degree should present some undergraduate work in mathematics beyond calculus. Candidates for the M.S.M. degree must write an approved creative component and pass a comprehensive oral examination over their course work and their creative component.

Courses open for nonmajor graduate credit: 301, 302, 304, 307, 314, 317, 331, 350, 365, 373, 385, 414, 415, 421, 426, 435, 436, 439, 465, 471, 481, 489.

** Math 301. Abstract Algebra I. ** (3-0) Cr. 3. F.S. * Prereq: 166 or 166H, 307 or 317, and 201. * Theory of groups. Homomorphisms. Quotient groups. Introduction to rings. Emphasis on writing proofs. Nonmajor graduate credit.

** Math 302. Abstract Algebra II. ** (3-0) Cr. 3. S. * Prereq: 301. * Theory of rings and fields. Introduction to Galois theory. Emphasis on writing proofs. Nonmajor graduate credit.

** Math 304. Introductory Combinatorics. ** (3-0) Cr. 3. F. * Prereq: 166 or 166H; 201 or experience with proofs. * Permutations, combinations, binomial coefficients, inclusion-exclusion principle, recurrence relations, generating functions. Additional topics selected from probability, random walks, and Markov chains. Nonmajor graduate credit.

** Math 307. Matrices and Linear Algebra. ** (3-0) Cr. 3. F.S.SS. * Prereq: 2 semesters of calculus. * Systems of linear equations, determinants, vector spaces, linear transformations, orthogonality, least-squares methods, eigenvalues and eigenvectors. Emphasis on methods and techniques. Only one of Math 307, 317 may be counted toward graduation. Nonmajor graduate credit.

** Math 314. Graphs and Networks. ** (3-0) Cr. 3. S. * Prereq: 166 or 166H; 201 or experience with proofs. * Structure and extremal properties of graphs. Topics are selected from: trees, networks, colorings, paths and cycles, connectivity, planarity, Ramsey theory, forbidden structures, enumeration, applications. Nonmajor graduate credit.

** Math 317. Theory of Linear Algebra. ** (4-0) Cr. 4. F.S. * Prereq: 166; credit or enrollment in 201. * Systems of linear equations, determinants, vector spaces, inner product spaces, linear transformations, eigenvalues and eigenvectors. Emphasis on writing proofs and results. Only one of Math 307, 317 may be counted toward graduation. Nonmajor graduate credit.

** Math 331. Topology. ** (3-0) Cr. 3. Alt. S., offered 2009. * Prereq: 307 or 317. * Topological properties of metric spaces, including Euclidean n-space, continuous functions, homeomorphisms, and topological invariants. Examples from surfaces, knots, links, and three-dimensional manifolds. Nonmajor graduate credit.

** Math 341. Introduction to the Theory of Probability and Statistics I. ** (Cross-listed with Stat). (3-0) Cr. 3. * Prereq: Math 265 (or 265H). * Probability; distribution functions and their properties; classical discrete and continuous distribution functions; moment generating functions, multivariate probability distributions and their properties; transformations of random variables; simulation of random variables and use of the R statistical package.

** Math 342. Introduction to the Theory of Probability and Statistics II. ** (Cross-listed with Stat). (3-0) Cr. 3. * Prereq: Stat 341; Math 307 or 317. * Sampling distributions; confidence intervals and hypothesis testing; theory of estimation and hypothesis tests; linear model theory, enumerative data.

** Math 350. Number Theory. ** (Cross-listed with Com S). (3-0) Cr. 3. S. * Prereq: 307 or 317. * Divisibility, integer representations, primes and divisors, linear diophantine equations, congruences, and multiplicative functions. Applications to cryptography. Nonmajor graduate credit.

** Math 365. Complex Variables with Applications. ** (3-0) Cr. 3. F.S. * Prereq: 265. * Functions of a complex variable, including differentiation, integration and series expansions, residues, evaluation of integrals, conformal mapping. Nonmajor graduate credit.

** Math 373. Introduction to Scientific Computation. ** (3-0) Cr. 3. S. * Prereq: 265. * Vector, matrix and graphics programming in MATLAB for scientific applications. Algorithms for interpolation, systems of linear equations, least squares, nonlinear equations and optimization in one and several variables. Additional topics may include ordinary differential equations, symbolic calculation and the Fast Fourier Transform. Emphasis on effective use of mathematical software, and understanding of its strengths and limitations.

** Math 385. Introduction to Partial Differential Equations. ** (3-0) Cr. 3. F.S. * Prereq: 265 and one of 266, 267. * Separation of variables methods for elliptic, parabolic, and hyperbolic partial differential equations. Fourier series, Sturm-Liouville theory, Bessel functions, and spherical harmonics. Nonmajor graduate credit.

** Math 398. Cooperative Education. ** Cr. R. Repeatable. F.S.SS. * Prereq: Permission of the department cooperative education coordinator; junior classification. * Required of all cooperative education students. Students must register for this course prior to commencing each work period.

** Math 414. Analysis I. ** (3-0) Cr. 3. F.S.SS. * Prereq: 201; 265; and 307 or 317. * A careful development of calculus of functions of a real variable: limits, continuity, differentiation, integration, series. Nonmajor graduate credit.

** Math 415. Analysis II. ** (3-0) Cr. 3. S. * Prereq: 414. * Sequences and series of functions of a real variable, uniform convergence, power series and Taylor series, Fourier series, topology of n-dimensional space, implicit function theorem, calculus of the plane and 3-dimensional space. Additional topics may include metric spaces or Stietjes or Lebesgue integration. Nonmajor graduate credit.

** Math 421. Logic for Mathematics and Computer Science. ** (Cross-listed with Com S). (3-0) Cr. 3. S. * Prereq: Math 301 or 307 or 317 or Com S 330. * Propositional and predicate logic. Topics selected from Horn logic, equational logic, resolution and unification, foundations of logic programming, reasoning about programs, program specification and verification, model checking and binary decision diagrams. Nonmajor graduate credit.

** Math 426. Mathematical Methods for the Physical Sciences. ** (3-0) Cr. 3. F. * Prereq: 266 or 267. * A fast-paced course primarily for first-year graduate students in physics and chemistry. Emphasis on techniques needed for quantum mechanics and electrodynamics. Functions of a complex variable and contour integration, integral transforms and applications, series methods for ordinary differential equations, Green's functions, Sturm-Liouville problems and orthogonal functions, boundary-value problems for partial differential equations. Nonmajor graduate credit.

** Math 435. Geometry I. ** (3-0) Cr. 3. F. * Prereq: 307 or 317. * Euclidean geometry. Points, lines, circles, triangles, congruence, similarity, properties invariant under rigid motions. Synthetic, analytic, and axiomatic methods. Nonmajor graduate credit.

** Math 436. Geometry II. ** (3-0) Cr. 3. S. * Prereq: 435. * Continuation of Euclidean geometry with topics from elliptic, projective, or hyperbolic geometry. Emphasis on analytic methods. Nonmajor graduate credit.

** Math 439. Mathematics of Fractals and Chaos. ** (3-0) Cr. 3. Alt. S., offered 2008. * Prereq: 265. * Topology of metric spaces; iterated function systems; algorithms for generation of fractals; fractal dimension; Julia sets and the Mandelbrot set; applications to chaotic systems. Nonmajor graduate credit.

** Math 465. Advanced Calculus for Applied Mathematics. ** (4-0) Cr. 4. S.SS. * Prereq: 265. * Frequently applied concepts from multivariable calculus, presented with enough theory to promote understanding of applications. Topics may include derivative matrices, Taylor polynomials, curvilinear coordinates, Green's theorem, divergence theorem, Stokes's theorem, uniform convergence, operations on series and integrals, improper integrals. Nonmajor graduate credit.

** Math 471. Computational Linear Algebra and Fixed Point Iteration. ** (Cross-listed with Com S). (3-0) Cr. 3. F.S. * Prereq: Math 265 and either Math 266, or 267; knowledge of a programming language. * Computational error, solutions of linear systems, least squares, similarity methods for eigenvalues, solution of nonlinear equations in one and several variables. Nonmajor graduate credit.

** Math 481. Numerical Solution of Differential Equations and Interpolation. ** (Cross-listed with Com S). (3-0) Cr. 3. S.SS. * Prereq: Math 265 and either Math 266 or 267; knowledge of a programming language. * Polynomial and spline interpolation, orthogonal polynomials, least squares, numerical differentiation and integration, numerical solution of ordinary differential equations. Nonmajor graduate credit.

** Math 489. History of Mathematics. ** (3-0) Cr. 3. S. * Prereq: 6 credits in mathematics at the 300 level or above. Recommended credit or enrollment in 301 or 414. * History of mathematical ideas found in the undergraduate curriculum. It includes a discussion of the historical and cultural settings in which these ideas arose, and the influence of the culture on the type of mathematical ideas that developed. Some of the particular cultures and their mathematics that are studied include: Babylonian and Ancient Egyptian. Ancient Greek, Arabic, Indian, Western European and Chinese. Nonmajor graduate credit.

** Math 490. Independent Study. ** Cr. 1-3. Repeatable. * Prereq: 301 or 317; 6 credits in mathematics. * No more than 9 credits of Math 490 may be counted toward graduation.

H. Honors

** Math 491. Undergraduate Thesis. ** Cr. 2-3. Writing a formal mathematics paper. Upon approval by the department, the paper will satisfy the departmental advanced English requirement.

** Math 492. Undergraduate Seminar. ** (2-0) Cr. 2. S. * Prereq: Consent of instructor. * Introduction to mathematics research, a participating seminar on advanced topics in mathematics. Mathematical literature search, reading a mathematical article with the guidance of the instructor, mathematical presentation. Seminar content varies.

** Math 497. Teaching Secondary School Mathematics. ** (Cross-listed with C I). (3-0) Cr. 3. F. * Prereq: 15 credits in college mathematics; if in a teacher licensure program, concurrent enrollment in C I 426 or 526. * Theory and methods for teaching mathematics in grades 7-12. Includes critical examination of instructional strategies, curriculum materials, learning tools, assessment methods, National Standards in Mathematics Education, and equity issues.

** Math 498. Cooperative Education. ** Cr. R. Repeatable. F.S.SS. * Prereq: Permission of the department cooperative education coordinator; senior classification. * Required of all cooperative education students. Students must register for this course prior to commencing each work period.

**Courses primarily for graduate students, open to qualified undergraduate students**

** Math 501. Introduction to Real Analysis. ** (3-0) Cr. 3. F. * Prereq: 265 and 307 or 317. * A development of the real numbers. Study of metric spaces, completeness, compactness, sequences, and continuity of functions. Differentiation and integration of real-valued functions, sequences of functions, limits and convergence, equicontinuity.

** Math 502. Numerical Analysis I. ** (3-0) Cr. 3. F. * Prereq: 414 or 501. * Numerical linear algebra including eigenvalue problems; numerical solution of nonlinear equations and optimization problems.

** Math 503. Numerical Analysis II. ** (3-0) Cr. 3. S. * Prereq: 414 or 501. * Approximation theory, including polynomial and spline interpolation and best approximation; numerical differentiation and integration; numerical methods for ordinary differential equations.

** Math 504. Abstract Algebra I. ** (3-0) Cr. 3. F. * Prereq: 302. * Algebraic systems and their morphisms, including groups, rings, modules, and fields.

** Math 505. Abstract Algebra II. ** (3-0) Cr. 3. S. * Prereq: 504. * Continuation of Math 504.

** Math 510. Linear Algebra. ** (3-0) Cr. 3. F. * Prereq: 307 or 317. * Advanced topics in linear algebra including canonical forms; unitary, normal, Hermitian and positive-definite matrices; variational characterizations of eigenvalues, and applications to other branches of mathematics.

** Math 511. Functions of a Single Complex Variable. ** (3-0) Cr. 3. S. * Prereq: 414 or 465 or 501. * Theory of analytic functions, integration, topology of the extended complex plane, singularities and residue theory, maximum principle.

** Math 515. Real Analysis I. ** (3-0) Cr. 3. F. * Prereq: 414 or 501. * Measure and integration with special emphasis on Lebesgue measure, modes of convergence of sequences of functions, decomposition of measures, differentiation, metric spaces, Lp spaces.

** Math 516. Real Analysis II. ** (3-0) Cr. 3. S. * Prereq: 515. * Continuation of Math 515. Hilbert and Banach spaces, product integration, Fubini's theorem, other topics at the discretion of the instructor.

** Math 517. Finite Difference Methods. ** (3-0) Cr. 3. F. * Prereq: 481 or 503. * Finite difference methods for partial differential equations, with emphasis on parabolic and hyperbolic equations, hyperbolic conservation laws, and other partial differential equations from application areas. Topics include convergence, stability and implementation issues.

** Math 519. Methods of Applied Mathematics I. ** (3-0) Cr. 3. F. * Prereq: 414 or 465 or 501. * Techniques of classical and functional analysis with applications to partial differential equations, integral equations, and boundary value problems for ordinary differential equations. Vector spaces, metric spaces, Hilbert and Banach spaces, important function spaces, contraction mapping theorem, distributions, Fourier series and Fourier transform, linear operators, spectral theory of differential and integral operators, Green s functions and boundary value problems, weak solutions of partial differential equations and variational methods, calculus in Banach spaces and applications.

** Math 520. Methods of Applied Mathematics II. ** (3-0) Cr. 3. S. * Prereq: 519. * Continuation of Math 519.

** Math 525. Numerical Analysis of High Performance Computing. ** (Cross-listed with Com S, Cpr E). (3-0) Cr. 3. S. * Prereq: Cpr E 308, or one of Math 471, 481; experience in scientific programming; knowledge of FORTRAN or C. * Development, analysis, and testing of efficient numerical methods for use on current state-of-the-art high performance computers. Applications of the methods to the students' areas of research.

** Math 533. Cryptography. ** (Cross-listed with Cpr E, InfAs). (3-0) Cr. 3. S. * Prereq: Math 301 or Cpr E 310 or Com S 330. * Basic concepts of secure communication, DES and AES, public-key cryptosystems, elliptic curves, hash algorithms, digital signatures, applications. Relevant material on number theory and finite fields.

** Math 535. Steganography and Watermarking. ** (Cross-listed with Cpr E, InfAs). (3-0) Cr. 3. S. * Prereq: Cpr E 531 or E E 524 or Math 533/Cpr E 533/InfAs 533. * Basic principles of steganography and watermarking. Algorithms based on spatial domain approaches, transformations of data, statistical approaches. Techniques for images, video, and audio data. Communications models for data hiding. Analysis, detection and recovery of hidden data. Military, commercial and e-commerce applications. Known theoretical results. Software packages for data hiding. Social and legal issues, case studies, and digital rights management issues that affect technological development of steganography and watermarking. Current developments in the area.

** Math 540. Seminar in Mathematics Education. ** (3-0) Cr. 3. * Prereq: Enrollment in the master of school mathematics program or professional studies in education. * Offered on a 3-year cycle, offered SS. 2008. Research studies in mathematics learning and teaching, exemplary practices in mathematics education, and current state and national trends in the mathematics curriculum in grades K-12.

** Math 542. Investigating the Teaching and Learning of Secondary Mathematics. ** (1-0) Cr. 1. Repeatable. Alt. F., offered 2008. * Prereq: Enrollment in master of school mathematics program, professional studies in education or by permission for secondary mathematics education majors. * Research, discussion and evaluation of efforts to improve instruction in the mathematics classroom.

** Math 543. Topics in Mathematics Education. ** (1-0) Cr. 1. F. * Prereq: Teaching a mathematics course. * Selected topics in collegiate mathematics education including cooperative learning, instructional use of technology, writing in mathematics, and cognitive learning theories. Research studies, exemplar practices, and trends in mathematics education.

** Math 545. Intermediate Calculus. ** (4-0) Cr. 4. * Prereq: 3 semesters of calculus and enrollment in the master of school mathematics program. * Offered on a 3-year cycle, offered SS. 2010. Further development of the fundamental concepts of calculus and their applications with an emphasis on a constructivist approach to learning, cooperative groups, problem solving, the use of technology.

** Math 546. Algorithms in Analysis and Their Computer Implementation. ** (2-2) Cr. 3. * Prereq: 3 semesters in calculus or concurrent enrollment in 545 and enrollment in the master of school mathematics program. * Offered on a 3- year cycle, offered SS. 2010. The use of technology in secondary mathematics with an emphasis on the exploration and implementation of algorithms.

** Math 547. Discrete Mathematics and Applications. ** (4-0) Cr. 4. * Prereq: Enrollment in the master of school mathematics program. * Offered on a 3-year cycle, offered SS. 2009. Applications of graph theory, game theory, linear programming, recursion, combinatorics and algebraic structures. Issues in integrating discrete topics into the secondary curriculum. Use of the computer to explore discrete mathematics.

** Math 549. Intermediate Geometry. ** (3-0) Cr. 3. * Prereq: 435 or equivalent and enrollment in the master of school mathematics program. * Offered on a 3-year cycle, offered SS. 2009. A study of geometry with emphasis on metrics, the group of isometries, the group of similarities, and the affine group. Specific spaces studied normally include the Euclidean plane, the 2-sphere, and projective 2-space. Emphasis on analytical methods.

** Math 554. Introduction to Stochastic Processes. ** (Cross-listed with Stat). (3-0) Cr. 3. F. * Prereq: Stat 542. * Markov chains on discrete spaces in discrete and continuous time (random walks, Poisson processes, birth and death processes) and their long-term behavior. Optional topics may include branching processes, renewal theory, introduction to Brownian motion.

** Math 557. Ordinary Differential Equations I. ** (3-0) Cr. 3. F. * Prereq: 415 or 465 or 501. * The initial-value problem, existence and uniqueness theorems, continuous dependence on parameters, linear systems, stability and asymptotic behavior of solutions, linearization, topics from dynamical systems and two-point boundary-value problems.

** Math 569. Bioinformatics III (Structural Genome Informatics). ** (Cross-listed with BCB, Com S, BBMB, Cpr E). (3-0) Cr. 3. F. * Prereq: BCB 567, Gen 411, Stat 401, Stat 432. * Algorithmic and statistical approaches in structural genomics including protein, DNA and RNA structure. Structure determination, refinement, representation, comparison, visualization, and modeling. Analysis and prediction of protein secondary and tertiary structure, disorder, protein cores and surfaces, protein-protein and protein-nucleic acid interactions, protein localization and function.

** Math 573. Random Signal Analysis and Kalman Filtering. ** (Cross-listed with Aer E, E E, M E). (3-0) Cr. 3. F. * Prereq: E E 324 or Aer E 331 or M E 370 or M E 411 or Math 341 or 395. * Elementary notions of probability. Random processes. Autocorrelation and spectral functions. Estimation of spectrum from finite data. Response of linear systems to random inputs. Discrete and continuous Kalman filter theory and applications. Smoothing and prediction. Linearization of nonlinear dynamics.

** Math 574. Optimal Control. ** (Cross-listed with Aer E, E E, M E). (3-0) Cr. 3. S. * Prereq: E E 577. * The optimal control problem. Variational approach. Pontryagin's principle. Hamilton-Jacobi equation. Dynamic programming. Time-optimal, minimum fuel, minimum energy control systems. The regulator problem. Structures and properties of optimal controls.

** Math 575. Introduction to Robust Control. ** (Cross-listed with E E, M E, Aer E). (3-0) Cr. 3. * Prereq: E E 577. * Introduction to modern robust control. Model and signal uncertainty in control systems. Uncertainty description. Stability and performance robustness to uncertainty. Solutions to the H2, Hoo, and l1 control problems. Tools for robustness analysis and synthesis.

** Math 576. Digital Feedback Control Systems. ** (Cross-listed with Aer E, E E, M E). (3-0) Cr. 3. F. * Prereq: E E 475 or Aer E 432 or M E 411 or 414 or Math 415; and Math 267. * Sampled-data, discrete data, and the z-transform. Design of digital control systems using transform methods: root locus, frequency response and direct design methods. Design using state-space methods. Controllability, observability, pole placement, state estimators. Digital filters in control systems. Microcomputer implementation of digital filters. Finite wordlength effects. Linear quadratic optimal control in digital control systems. Simulation of digital control systems.

** Math 577. Linear Systems. ** (Cross-listed with Aer E, E E, M E). (3-0) Cr. 3. F. * Prereq: E E 324 or Aer E 331 or M E 414 or Math 415; and Math 307. * State variable and input-output descriptions of linear continuous-time and discrete-time systems. Solution of linear dynamical equations. Controllability and observability of linear dynamical systems. Canonical descriptions of linear equations. Irreducible realizations of rational transfer function matrices. Canonical form dynamical equations. State feedback. State estimators. Decoupling by state feedback. Design of feedback systems. Stability of linear dynamical systems.

** Math 578. Nonlinear Systems. ** (Cross-listed with Aer E, E E, M E). (3-0) Cr. 3. S. * Prereq: E E 577. * Classification of nonlinear control systems. Existence and uniqueness of solutions. Approximate analysis methods. Periodic orbits. Concept of stability and Lyapunove stability theory. Absolute stability of feedback systems. Input-and output stability. Passivity.

** Math 590. Special Topics. ** Cr. arr. Repeatable.

** Math 597. Introductory Computational Structural Biology. ** (Cross-listed with BCB). (3-0) Cr. 3. S. * Prereq: Permission of instructor. * Mathematical and computational approaches to protein structure prediction and determination. Topics include molecular distance geometry, potential energy minimization, and molecular dynamics simulation.

** Math 599. Creative Component. ** Cr. arr.

**Courses primarily for graduate students**

** Math 601. Mathematical Logic I. ** (3-0) Cr. 3. Alt. F., offered 2008. * Prereq: 504. * First semester of full-year course. Completeness and compactness of propositional and predicate logic, incompleteness and undecidability of set theory and arithmetic, Goedel's theorems, recursive functions, computability, models, ultraproducts, and ultralimits.

** Math 602. Mathematical Logic II. ** (3-0) Cr. 3. Alt. S., offered 2009. * Prereq: 601. * Continuation of Math 601.

** Math 605. Design Theory and Association Schemes. ** (3-0) Cr. 3. Alt. F., offered 2008. * Prereq: 504. * Combinatorial designs and Latin squares. Construction methods including finite fields. Error-correcting codes. Adjacency matrices and algebraic combinatorics.

** Math 606. Enumerative Combinatorics and Ordered Sets. ** (3-0) Cr. 3. Alt. S., offered 2009. * Prereq: 504 or permission of instructor. * Ordered sets and lattices. Generating functions. Moebius inversion and other enumeration methods.

** Math 607. Modern (Structural) Graph Theory. ** (3-0) Cr. 3. Alt. F., offered 2007. * Prereq: 504 or permission of instructor. * Structural and extremal theory of graphs. Topics include basic structures (trees, paths and cycles), networks, colorings, connectivity, topological graph theory, Ramsey theory, forbidden graphs and minors, introduction to random graphs, applications.

** Math 610. Seminar. ** Cr. arr.

** Math 615. General Theory of Algebraic Structures I. ** (3-0) Cr. 3. F. * Prereq: 504. * First semester of full-year course. Subalgebras, homomorphisms, congruence relations, and direct products. Lattices and closure operators. Varieties and quasivarieties of algebras, free algebras, Birkhoff's theorems, clones, Mal'cev conditions. Advanced topics.

** Math 616. General Theory of Algebraic Structures II. ** (3-0) Cr. 3. Alt. S., offered 2008. * Prereq: 615. * Continuation of Math 615.

** Math 617. Category Theory. ** (3-0) Cr. 3. Alt. F., offered 2008. * Prereq: 504. * Categories and functors and their applications.

** Math 618. Boolean Algebras. ** (3-0) Cr. 3. Alt. S., offered 2009. * Prereq: 504. * Structure of Boolean algebras and their representations. Stone spaces and duality. Atomicity, completeness, distributivity, operators, extensions of homomorphisms. Examples and applications from mathematical logic and topology.

** Math 621. Topology. ** (3-0) Cr. 3. Alt. F., offered 2008. * Prereq: Permission of instructor. * Introduction to general topology. Topological spaces, continuous functions, connectedness, compactness. Topics selected from countability and separation axioms, metrization, and complete metric spaces.

** Math 622. Algebraic Topology. ** (3-0) Cr. 3. Alt. S., offered 2009. * Prereq: 504. * Foundations of algebraic topology. The fundamental group, simplicial homology groups, and singular homology groups.

** Math 624. Manifolds, Tensors and Differential Geometry. ** (3-0) Cr. 3. Alt. S., offered 2008. * Prereq: 501 or 515. * Topics selected from: Geometry of curves and surfaces. Manifolds, coordinate systems. Tensors, differential forms, Riemannian metrics. Connections, covariant differentiation, curvature tensors.

** Math 633. Functional Analysis I. ** (3-0) Cr. 3. Alt. F., offered 2007. * Prereq: Permission of instructor. * Fundamental theory of normed linear spaces and algebras emphasizing aspects that provide a framework for the study of boundary-value problems, eigenvalue problems, harmonic analysis, analytic function theory, and modern operator theory.

** Math 634. Functional Analysis II. ** (3-0) Cr. 3. Alt. S., offered 2008. * Prereq: 633. * Continuation of Math 633.

** Math 642. Advanced Probability Theory. ** (Cross-listed with Stat). (4-0) Cr. 4. F. * Prereq: Stat 542. * Measure spaces, extension theorem and construction of Lebesgue-Stieljes measures on Euclidean spaces, Lebesgue integration and the basic convergence theorems, Lp-spaces, absolute continuity of measures and the Radon Nikodym theorem, absolute continuity of functions on R and the fundamental theorem of Lebesgue integration, product spaces and Fubini-Tonelli Theorems, convolutions. Fourier series and transforms, probability spaces; Kolmogorov's existence theorem for stochastic processes; expectation; Jensen's inequality and applications, independence, Borel-Cantelli lemmas; weak and strong laws of large numbers and applications, renewal theory.

** Math 645. Advanced Stochastic Processes. ** (Cross-listed with Stat). (3-0) Cr. 3. S. * Prereq: Permission of instructor. * Weak convergence. Random walks and Brownian motion. Martingales. Stochastic integration and Ito's Formula. Stochastic differential equations and applications.

** Math 646. Mathematical Modeling of Complex Physical Systems. ** (3-0) Cr. 3. S. * Prereq: Permission of instructor. * Modeling of the dynamics of complex systems on multiple scales: Classical and dissipative molecular dynamics, stochastic modeling and Monte-Carlo simulation; macroscale non-linear dynamics and pattern formation.

** Math 655. Partial Differential Equations I. ** (3-0) Cr. 3. F. * Prereq: 515 or 519. * First order equations and systems, conservation laws, general theory of linear partial differential equations of elliptic, parabolic and hyperbolic types, maximum principles, fundamental solutions, Sobolev spaces, variational and Hilbert space methods.

** Math 656. Partial Differential Equations II. ** (3-0) Cr. 3. S. * Prereq: 655. * Continuation of Math 655.

** Math 658. Dynamical Systems. ** (3-0) Cr. 3. Alt. S., offered 2009. * Prereq: 501 or 515. * Smooth mappings and flows. Fixed points, stable, unstable and center manifolds, normal forms. Structural stability, bifurcations. Horseshoe maps, introduction to chaotic behavior.

** Math 666. Finite Element Methods. ** (3-0) Cr. 3. S. * Prereq: 503 or 516 or 520 or 656. * Elements of functional analysis; Sobolev spaces; variational principles and weak formulations; approximation theory in finite element spaces; analysis of finite element methods; implementation issues; applications.

** Math 690. Advanced Topics. ** Cr. 3. Repeatable.

A. Algebra

B. Functional Analysis

C. Control Theory

D. Approximation Theory

E. Linear Algebra

G. Number Theory

H. Harmonic Analysis

I. Combinatorics and Graph Theory

J. Mathematical Biology and Bioinformatics

K. Mathematics Education

L. Logic and Foundations

M. Complex Analysis

N. Numerical Analysis

O. Ordinary Differential Equations

P. Partial Differential Equations

Q. Group Theory

R. Applied Mathematics

S. Set Theory

T. Topology

U. Automata Theory

V. Optimization Theory

W. Probability and Stochastic Processes

Y. Special Functions

Z. Ring Theory

** Math 699. Research. ** Cr. arr. Repeatable.