# 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

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.

617. Complex Analysis I. (3) Analytic functions Cauchy’s theorem and its consequences, power series, and residue calculus.

622. Modern Algebra II. (3) A continuation of modern abstract algebra through the study ofadditional 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) An introduction to numerical methods for solving matrix and related problems in science and engineering using a high-level matrix-oriented language such as MATLAB. Topics include systems of linear equations, least squares methods, and eigenvalue computations. Special emphasis is given to applications.

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. Elementary Number Theory. (3) Course topics include properties of integers, congruences, and prime numbers, wit h additional topics chosen from arithmetic functions, primitive roots, quadratic residues,Pythagorean triples, and sums of squares.

646. Modern Number Theory (3) Course topics include a selection of number-theory topics of recent Interest. Some examples include elliptic curves, partitions, modular forms, the Riemann zeta function, and algebrai number theory.

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. Probability Models. (3) Course topics include an introduction to probability models, Markov chains, Poisson process and Markov decision processes. Applications will emphasize problems in business and management science.

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) An introduction to numerical computations on modern computer architectures; floating point arithmetic and round-off error including programming in a scientific/engineering language such as MATlAB, Cor Fortran. Topics include algorithms and computer techniques for the solution of problems such as roots of functions, approximations, integration, systems of linear equations and least squares methods. Also listed as 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. May not be used toward any graduate degree offered by the department.

657. Probability. (3) Course topics include probability distributions, mathematical expectation, andsampling distributions. MTH 657 covers much of the material on the syllabus for the first actuarial exam.

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

662. Multivariate Statistics. (3) This course will concentrate on multivariate and generalized linear methods for classification, modeling, discrimination and analysis.

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. P—657 or 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—657 or POI

669. Advanced Topics in Statistics. (1, 2, or 3) Topics in statistics not considered in regular courses or which continue study begun in regular courses. Content varies.

681. Individual Study. (1 or 2) A course of independent study directed by a faculty adviser. By prearrangement. May be repeated for credit.

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. May be repeated for credit for a total of 3 hours.

683. Advanced Topics in Mathematics. (1, 2 or 3) Topics in mathematics that are not considered in regular courses. Content varies.

669. Advanced Topics in Statistics. (1, 2 or 3) Topics in statistics not considered in regular courses or which continue study begun in regular courses. Content varies.

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 metrication, 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. May be repeated for credit.

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 655 or CSC 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. May be repeated for credit.

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). May be repeated for credit. Satisfactory / Unsatisfactory