# Course Descriptions

## Courses for Advanced Undergraduates and Graduate Students

605. Applied Multivariable Mathematics. (3) Introduction to several topics in applied mathematics including complex numbers, probability, matrix algebra, multivariable calculus, and ordinary differential equations. May not be used toward any graduate degree offered by the department.

606. Advanced Mathematics for the Physical Sciences. (3) Advanced topics in linear algebra, special functions, integraltransforms, and partial differential equations. May not be used toward any graduate degree offered by the department. P—MTH 605.

610. Advanced Calculus. (3) A rigorous proof-oriented development of important ideas in calculus. Limits and continuity, sequences and series, pointwise and uniform convergence, derivatives and integrals. Credit not allowed for both MTH 610 and 611. May not be used toward any graduate degree offered by the department.

611, 612. Introductory Real Analysis I, II. (3, 3) Limits and continuity in metric spaces, sequences and series, differentiation and Riemann-Stieltjes integration, uniform convergence, power series and Fourier series, differentiation of vector functions, implicit and inverse function theorems. Credit not allowed for both MTH 610 and 611.

617. Complex Analysis I. (3) Analytic functions Cauchy’s theorem and its consequences, power series, and residue calculus. Credit not allowed for both MTH 603 and 617.

622. Modern Algebra II. (3) A continuation of modern abstract algebra through the study of additional properties of groups, rings, and fields.

624. Linear Algebra II. (3) A thorough treatment of vector spaces and linear transformations over an arbitrary field, canonical forms, inner product spaces, and linear groups.

626. Numerical Linear Algebra. (3) Numerical methods for solving matrix and related problems in science and engineering. Topics include systems of linear equations, least squares methods, and eigenvalue computations. Special emphasis given to parallel matrix computations. Beginning knowledge of a programming language such as Pascal, FORTRAN, or C is required. Credit not allowed for both MTH 626 and CSC 652.

631. Geometry. (3) An introduction to axiomatic geometry including a comparison of Euclidean and non-Euclidean geometries.

634. Differential Geometry. (3) Introduction to the theory of curves and surfaces in two and three dimensional space including such topics as curvature, geodesics, and minimal surfaces.

645, 646. Elementary Theory of Numbers I, II. (3, 3) Properties of integers, including congruences, primitive roots, quadratic residues, perfect numbers, Pythagorean triples, sums of squares, continued fractions, Fermat’s Last Theorem, and the Prime Number Theorem.

647. Graph Theory. (3) Paths, circuits, trees, planar graphs, spanning trees, graph coloring, perfect graphs, Ramsey theory, directed graphs, enumeration of graphs and graph theoretic algorithms.

648, 649. Combinatorial Analysis I, II. (3, 3) Enumeration techniques, generating functions, recurrence formulas, the principle of inclusion and exclusion, Polya theory, graph theory, combinatorial algorithms, partially ordered sets, designs, Ramsey theory, symmetric functions, and Schur functions.

652. Partial Differential Equations. (3) Detailed study of partial differential equations, including the heat, wave, and Laplace equations, using methods such as separation of variables, characteristics, Green’s functions, and the maximum principle.

653. Mathematical Models. (3) Development and application of probabilistic and deterministic models. Emphasis given to constructing models that represent systems in the social, behavioral, and management sciences.

654. Discrete Dynamical Systems. (3) Introduction to the theory of discrete dynamical systems as applied to disciplines such as biology and economics. Includes methods for finding explicit solutions, equilibrium and stability analysis, phase plane analysis, analysis of Markov chains and bifurcation theory.

655. Introduction to Numerical Methods. (3) Numerical computations on modern computer architectures; floating point arithmetic and round-off error. Programming in a scientific/engineering language (C or FORTRAN). Algorithms and computer techniques for the solution of problems such as roots of functions, approximation, integration, systems of linear equations and least squares methods. Credit not allowed for both MTH 655 and CSC 655.

656. Statistical Methods. (3) A project-orientated course emphasizing data analysis, with introductions to nonparametric methods, multiple and logistic regression, model selection, design, categorical data or Bayesian methods. P—MTH 109, ANT 380, BIO 380, BEM 201 or 202, HES 262 or 369, PSY 311 or 312, SOC 271, or POI.

657. Probability. (3) Prepares students for Actuarial Exam #1 and includes probability distributions, mathematical expectation, and sampling distribution. P—MTH 112 or POI.

658. Mathematical Statistics. (3) This course will cover derivation of point estimators, hypothesis testing, and confidence intervals using both maximum likelihood and Bayesian approaches. P— MTH 657 or POI.

661. Selected Topics. (1, 1.5, 2, or 3) Topics in mathematics that are not considered in regular courses. Content varies.

662. Multivariate Statistics. (3) This course will concentrate on multivariate and generalized linear methods for classification, modeling, discrimination and analysis. P—MTH 112, 121, 205, 656 or POI.

664. Computational and Nonparametric Statistics. (3) This course focuses on computationally intensive methods to fit statistical models to data. Topics include simulation, Monte Carlo integration and Markov Chain Monte Carlo, sub-sampling, and nonparametric estimation and regression. Students will make extensive use of statistic software throughout the course. P—MTH 109, 656, 657 and POI.

667. Linear Models. (3) This course focuses on theory of estimation and testing in linear models. Topics include least squares and the normal equations, the Gauss-Markov Theorem, testing general linear hypotheses, and generalized linear models. P—MTH 121, 205, 206, 656, 657 or POI. 681. Individual Study. (1 or 2) A course of independent study directed by a faculty adviser. By prearrangement.

682. Reading in Mathematics. (1, 2, or 3) Reading in mathematical topics to provide a foundational basis for more advanced study in a particular mathematical area. Topics vary and may include material from algebra, analysis, combinatorics, computational or applied mathematics, number theory, topology, or statistics. May not be used to satisfy any requirement in the mathematics MA degree with thesis. No more than three hours may be applied to the requirements for the mathematics MA degree without thesis.

711, 712. Real Analysis. (3, 3) Measure and integration theory, elementary functional analysis, selected advanced topics in analysis.

715, 716. Seminar in Analysis. (1, 1)

717. Optimization in Banach Spaces. (3) Banach and Hilbert spaces, best approximations, linear operators and adjoints, Frechet derivatives and nonlinear optimization, fixed points and iterative methods. Applications to control theory, mathematical programming, and numerical analysis.

718. Topics in Analysis. (3) Selected topics from functional analysis or analytic function theory.

721, 722. Abstract Algebra. (3, 3) Groups, rings, fields, extensions, Euclidean domains, polynomials, vector spaces, Galois theory.

723, 724. Seminar on Theory of Matrices. (1, 1)

725, 726. Seminar in Algebra. (1, 1)

728. Topics in Algebra. (3) Topics vary and may include algebraic coding theory, algebraic number theory, matrix theory, representation theory, non-commutative ring theory.

731, 732. General Topology. (3, 3) An axiomatic development of topological spaces. Includes continuity, connectedness, compactness, separation axioms, metric spaces, convergence, embedding and metrization, function and quotient spaces, and complete metric spaces.

733. Topics in Topology and Geometry. (3) Topics vary and may include knot theory, non-Euclidean geometry, combinatorial topology, differential topology, minimal surfaces and algebraic topology.

735, 736. Seminar on Topology. (1, 1)

737, 738. Seminar on Geometry. (1, 1)

744. Topics in Number Theory. (3) Topics vary and are chosen from the areas of analytic, algebraic, and elementary number theory. Topics may include Farey fractions, the theory of partitions, Waring’s problem, prime number theorem, and Dirichlet’s problem.

745, 746. Seminar on Number Theory. (1, 1)

747. Topics in Discrete Mathematics. (3) Topics vary and may include enumerative combinatorics, graph theory, algebraic combinatorics, combinatorial optimization, coding theory, experimental designs, Ramsey theory, Polya theory, representation theory, set theory and mathematical logic. 748, 749. Seminar on Combinatorial Analysis. (1, 1)

750. Dynamical Systems. (3) Introduction to modern theory of dynamical systems. Linear and nonlinear autonomous differential equations, invariant sets, closed orbits, Poincare maps, structural stability, center manifolds, normal forms, local bifurcations of equilibria, linear and non-linear maps, hyperbolic sets, attractors, symbolic representation, fractal dimensions. P—MTH 611.

752. Topics in Applied Mathematics. (3) Topics vary and may include computational methods in differential equations, optimization methods, approximation techniques, eigenvalue problems.

753. Nonlinear Optimization. (3) The problem of finding global minimums of functions is addressed in the context of problems in which many local minima exist. Numerical techniques are emphasized, including gradient descent and quasi-Newton methods. Current literature is examined and a comparison made of various techniques for both unconstrained and constrained optimization problems. Credit not allowed for both MTH 753 and CSC 753. P—MTH (or CCS) 655.

754. Numerical Methods for Partial Differential Equations. (3) Numerical techniques for solving partial differential equations (including elliptic, parabolic and hyperbolic) are studied along with applications to science and engineering. Theoretical foundations are described and emphasis is placed on algorithm design and implementation using either C, FORTRAN or MATLAB. Credit not allowed for both MTH 754 and CSC 754. P—MTH 655 or CSC 655.

758. Topics in Statistics. (3) Topics vary and may include linear models, nonparametric statistics, stochastic processes.

761. Stochastic Processes. (3) Discrete time and continuous time Markov chains, Poisson processes, general birth and death processes, renewal theory. Applications, including general queuing models. 791, 792. Thesis Research. (1-9)

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