Graduate Level Mathematics Courses
The department offers courses which are slashlisted so undergraduate students may take an undergraduate 4000-level course while graduate students may take a graduate 5000-level course. The lectures in a slashlisted course are the same. However, students in the 5000-level course have substantial additional requirements beyond those for students in the 4000-level course. These additional requirements are listed in the slashlisted course syllabus.
Explanation of Course Numbers
In the Department of Mathematics the second digit indicates the area within the department: 1—miscellaneous; 2—mathematics education; 3—algebra; 4—analysis; 5—foundations and logic; 6—geometry; 7—probability and statistics; 8—topology; 9—research. The third digit identifies the course within the level and area.
G4033 Applied Matrix Models. Prerequisite: 3333 and either a programming course or permission of instructor. Solution of systems of m linear equations in m unknowns; solution of m linear equations in k unknowns; linear programming; eigenvalue and vector problems; matrix models selected from various areas such as ecology, voting systems, city street sweeping, infectious diseases, population, predator prey systems, heat transfer in frozen soil, network analysis, psychology, sociology, Markov processes. (F)
G4073 Numerical Analysis I. Prerequisite: 3113 or 3413. Solution of linear and nonlinear equations, approximation of functions, numerical integration and differentiation, introduction to analysis of convergence and errors, pitfalls in automatic computation, one-step methods in the solutions of ordinary differential equations. (F)
G4083 Numerical Analysis II. Prerequisite: 3113 or 3413; 4073 or Electrical Engineering 3793; 3333 or 4373 or Biostatistics and Epidemiology 5563. Matrix inversion and related topics; numerical solution of ordinary differential equations, partial differential equations, integral equations and functional equations; numerical solution of eigenvalue problems and applications of functional analysis. (Alt. Sp)
G4103 Introduction to Functions of a Complex Variable. Prerequisite: 3113. Complex analytic functions, conformal mappings, complex integrals. Taylor and Laurent series, integration by the method of residues, complex analytic functions and potential theory. (F, Sp)
G4163 Introduction to Partial Differential Equations. Prerequisite: 3113. Physical models, classification of equations, Fourier series and boundary value problems, integral transforms, the method of characteristics. (F, Sp)
G4323 Higher Algebra I. Prerequisite: 3333 and 3513, or permission of instructor. Concepts from set theory; the system of natural numbers, extension from the natural numbers to the integers; semigroups and groups; rings, integral domain and fields. Duplicates one hour of 4383. (F, Sp, Su)
G4333 Higher Algebra II. Prerequisite: 4323. Extensions of rings and fields, elementary factorization theory; groups with operators; modules and ideals; lattices. (Sp)
4373 Abstract Linear Algebra (Slashlisted with 5373). Prerequisite: 3333. Vector spaces over arbitrary fields, bases, dimension, linear transformations and matrices, similarity and its canonical forms (rational, Jordan), spectral theorem and diagonalization of quadratic forms. No student may earn credit for 3343 and 4373 or 5373, or for both 4373 and 5373. (F, Sp, Su)
4383 Applied Modern Algebra (Slashlisted with 5383). Prerequisite: 3333. Topics from the theory of error correcting codes, including Shannon's theorem, finite fields, families of linear codes such as Hamming, Golay, BCH, and Reed-Solomon codes. Other topics such as Goppa codes, group codes, and cryptography as time permits. No student may earn credit for both 4383 and 5383. (Sp)
G4413 Intermediate Ordinary Differential Equations. Prerequisite: 3113 or 3413; 3333. Topics selected from: linear systems of equations, integral equations, stability theory, existence and uniqueness criteria, perturbation theory, dynamical systems, boundary-value problems, numerical methods. (Irreg.)
G4433 Introduction to Analysis I. Prerequisite: 3513 or permission of instructor. Differentiation of functions of one variable, the Riemann integral, uniform convergence. (F, Su)
4443 Introduction to Analysis II (Slashlisted with 5443). Prerequisite: 4433. Calculus of several variables including topics of differentiation, content, and integration; sequences and series in one and several dimensions; topology of Euclidean p-spaces; line and surface integrals and Green's theorem; other topics as time permits. No student may earn credit for both 4443 and 5443. (Sp)
4623 Convexity Theory I (Slashlisted with 5623). Prerequisite: 3333, 3513 or permission of instructor. An introduction to the theory of convex sets. Topics include basic definitions and properties, separating and supporting hyperplanes, and combinatorial theorems of Caratheodory, Radon and Helly. No student may earn credit for both 4623 and 5623. (F)
G4643 Topics in Geometry and Combinatorics. Prerequisite: 3333. May be repeated with permission of instructor; maximum credit six hours. Topics may include convexity (convex sets, combinatorial theorems in finite dimensional Euclidean space), graph theory, finite geometries, foundations of geometry. (F, Sp)
4653 Introduction to Differential Geometry I (Slashlisted with 5653). Prerequisite: 2433 and 3333, or permission of instructor. Elementary theory of curves and surfaces in three-dimensional Euclidean space, differentiable manifolds, Riemannian geometry of two dimensions, Gauss Theorem Egregium. No student may earn credit for both 4653 and 5653. (F)
4663 Introduction to Differential Geometry II (Slashlisted with 5663). Prerequisite: 4653 or 5653. Intermediate theory of surfaces, covariant differentiation, geodesics, Gauss-Bonnet Theorem. Further topics may include: rigidity theorems, minimal surfaces, the Hopf-Rinow Theorem, the Hadamard Theorem, index of vector fields. No student may earn credit for both 4663 and 5663. (Sp)
4673 Graph Theory I (Slashlisted with 5673). Prerequisite: 3513 or permission of instructor. An introduction to the theory of graphs. Topics include basic definitions, cutpoints, blocks, trees, connectivity and Menger's theorem. No student may earn credit for both 4673 and 5673. (F)
G4733 Mathematical Theory of Probability. Prerequisite: 2443 or concurrent enrollment. Probability spaces, counting techniques, random variables, moments, special distributions, limit theorems. (F)
4743 Introduction to Mathematical Statistics (Slashlisted with 5743). Prerequisite: 4733. Mathematical development of basic concepts in statistics: estimation, hypothesis testing, sampling from normal and other populations, regression, goodness-of-fit. No student may earn credit for both 4743 and 5743. (Sp)
G4753 Applied Statistical Methods. Prerequisite: 2123 or 2423 or permission of instructor. Estimation, hypothesis testing, analysis of variance, regression and correlation, goodness-of-fit, other topics as time permits. Emphasis on applications of statistical methods. (F, Sp, Su)
4773 Applied Regression Analysis (Slashlisted with 5773). Prerequisite: 3333, 4733 or 4753 or any statistical probability course at an equivalent level. The general regression problem of fitting an equation involving a single dependent variable and several independent variables, estimation and tests of regression parameters, residual analysis, selecting the “best” regression equation. No student may earn credit for both 4773 and 5773. (Alt. F)
4793 Advanced Applied Statistics (Slashlisted with 5793). Prerequisite: 4743 or 4753 or equivalent. Survey of advanced applied statistical methods other than applied regression, including exploratory data analysis, analysis of multivariate data (principal components: analysis, multiple analysis of variance, cluster analysis, etc.), and introduction to non-parametric methods. No student may earn credit for both 4793 and 5793. (Alt. F)
G4853 Introduction to Topology. Prerequisite: 2433, 3513 or permission of instructor. Metric spaces and topological spaces, continuity, connectedness, compactness and related topics. (Sp)
G5103 Mathematical Models. Prerequisite: permission of instructor or admission to the M.S. program. May be repeated with change of content; maximum credit six hours. Mathematical models are formulated for problems arising in various areas in which mathematics has been applied. In each case, techniques are developed for analyzing the resulting mathematical problem, and this analysis is used to test the validity of the model. (Sp)
G5113 Topics in Applied Mathematics (Slashlisted with 4113). Prerequisite: permission of instructor. May be repeated with change of content; maximum credit nine hours. Algebraic coding theory, linear finite state workings, numerical analysis of differential equations, asymptotic analysis, game theory or other subjects. No student may earn credit for both 4113 and 5113. (Irreg.)
G5163 Partial Differential Equations. Prerequisite: 4163 or permission of instructor. First order equations, Cauchy problem for higher order equations, second order equations with constant coefficients, linear hyperbolic equations. (Sp)
G5173 Advanced Numerical Analysis I. Prerequisite: 4433, 4443 or permission of instructor. Topics may include: error analysis of numerical methods for optimization and initial value problems, numerical approximation of aspects of control problems. (Alt. F)
G5183 Advanced Numerical Analysis II. Prerequisite: 4433, 4443 or permission of instructor. Topics may include: analysis of spline approximations as a basis of the finite element method, error analysis for finite element approximation of elliptic and parabolic boundary value problems. (Alt. Sp)
G5333 Topics in Number Theory. Prerequisite: at least one mathematics course numbered above 3000, other than 3213, 4222, or 4232. May be repeated with change of content; maximum credit nine hours. Topics may include congruencies, arithmetic functions, quadratic reciprocity, continued fractions, diophantine equations, primality testing, factorization methods, cryptography, quadratic forms and quadratic fields, computational number theory, additive number theory, coding theory, p-adic numbers. (Irreg.)
G5353 Abstract Algebra I. Prerequisite: 4323, permission of instructor. Groups, Sylow theorems, group actions, group presentations. Rings, ideals, polynomial rings, unique factorization. Fields, algebraic and transcendental extensions. (F)
G5363 Abstract Algebra II. Prerequisite: 5353. Galois theory, solvability. Modules over a principal ideal domain. Noetherian ideal theory. Group representations, semisimple rings. Classical groups. (Sp)
G5373 Abstract Linear Algebra (Slashlisted with 4373). Prerequisite: 3333. Vector spaces over arbitrary fields, bases, dimension, linear transformations and matrices, similarity and its canonical forms (rational, Jordan), spectral theorem and diagonalization of quadratic forms. No student may earn credit for 3343 and 4373 or 5373, or for both 4373 and 5373. (F, Sp, Su)
G5383 Applied Modern Algebra (Slashlisted with 4383). Prerequisite: 3333. Topics from the theory of error correcting codes, including Shannon's theorem, finite fields, families of linear codes such as Hamming, Golay, BCH, and Reed-Solomon codes. Other topics such as Goppa codes, group codes, and cryptography as time permits. No student may earn credit for both 4383 and 5383. Duplicates one hour of 4323. (Sp)
G5403 Calculus of Variations. Prerequisite: 4433 or 3423 or 4163. Linear spaces, global and local theories of optimization, necessary conditions for relative extrema of integrals. (Irreg.)
G5423 Complex Analysis I. Prerequisite: 4433. The complex numbers, topologies of the extended plane and related sphere, elementary functions, power series, properties of general holomorphic functions. The integral of a complex-valued function over an oriented rectifiable curve, the classical theorems on integrals, Taylor and Laurent expansions, analytic continuation, introduction to Riemann surfaces. (Alt. F)
G5433 Complex Analysis II. Prerequisite: 5423. Selected topics from classical and modern function theory such as geometric theory, univalent functions, Hardy spaces and Nevanlinna theory. (Alt. Sp)
G5443 Introduction to Analysis II (Slashlisted with 4443). Prerequisite: 4433. Integration of functions of a single variable. Series of real numbers. Series of functions. Differentiation of functions of more than one variable. No student may earn credit for both 4443 and 5443. (Sp)
G5453 Real Analysis I. Prerequisite: 4433 or permission of instructor. Lebesgue measure and integration theory, absolutely continuous functions, metric spaces. (F)
G5463 Real Analysis II. Prerequisite: 5453. General measure and integration theory, Banach spaces, topics from related areas. (Sp)
G5483 Wavelets. Prerequisite: 3113 and 3333. Fourier analysis on a finite cyclic group, the group of integers, and the real line. The matching pursuit algorithm. The Poisson summation formula and sampling. Multi-resolution analysis, various wavelet constructions (including those of Daubechies and Meyer) and filter banks. An introduction to the MATLAB wavelet toolbox. (F)
G5623 Convexity Theory I (Slashlisted with 4623). Prerequisite: 3333, 2513 or permission of instructor. An introduction to the theory of convex sets. Topics include basic definitions and properties, separating and supporting hyperplanes, and combinatorial theorems of Caratheodory, Radon and Helly. No student may earn credit for both 4623 and 5623. (F)
G5633 Convexity Theory II. Prerequisite: 5623 or permission of instructor. A continuation of the study of convex sets. Topics include Helly-type theorems, the Blaschke selection theorem, alternate characterizations of convex sets, convex polytopes and Eveler's formula. (Sp)
G5653 Introduction to Differential Geometry I (Slashlisted with 4653). Prerequisite: 2433 and 3333, or permission of instructor. Elementary theory of curves and surfaces in three-dimensional Euclidean space, differentiable manifolds, Riemannian geometry of two dimensions, Gauss Theorem Egregium. No student may earn credit for both 4653 and 5653. (F)
G5663 Introduction to Differential Geometry II (Slashlisted with 4663). Prerequisite: 4653 or 5653. Intermediate theory of surfaces, covariant differentiation, geodesics, Gauss-Bonnet Theorem. Further topics may include: rigidity theorems, minimal surfaces, the Hopf-Rinow Theorem, the Hadamard Theorem, index of vector fields. No student may earn credit for both 4663 and 5663. (Sp)
G5673 Graph Theory I (Slashlisted with 4673). Prerequisite: 2513 or permission of instructor. An introduction to the theory of graphs. Topics include basic definitions, cutpoints, blocks, trees, connectivity and Menger's theorem. No student may earn credit for both 4673 and 5673. (F)
G5683 Graph Theory II. Prerequisite: 5673 or permission of instructor. A continuation of the study of graphs. Topics include partitions, Eulerian and Hamiltonian graphs, planarity and colorability. (Sp)
G5693 Topics in Geometry and Combinatorics I. Prerequisite: permission of instructor. May be repeated with permission of instructor; maximum credit 12 hours. Topics may include convexity, combinatorial geometry, graph theory, or Riemannian geometry. (F, Sp, Su)
G5743 Introduction to Mathematical Statistics (Slashlisted with 4743). Prerequisite: 4733. Mathematical development of basic concepts in statistics: estimation, hypothesis testing, sampling from normal and other populations; regression, goodness of fit. No student may earn credit for both 4743 and 5743. (Sp)
G5763 Introduction to Stochastic Processes. Prerequisite: 4733 or permission of instructor. Stochastic processes in discrete time including random walks, recurrent events, Markov chains and branching processes. Processes in continuous time including linear and nonlinear birth-death processes and diffusions. Applications taken from economics, engineering, operations research. (Irreg.)
G5773 Applied Regression Analysis (Slashlisted with 4773). Prerequisite: 3333, 4733 or 4753 or any statistical probability course at an equivalent level. The general regression problem of fitting an equation involving a single dependent variable and several independent variables, estimation and tests of regression parameters, residual analysis, selecting the "best" regression equation. No student may earn credit for both 4773 and 5773. (Alt. F)
G5793 Advanced Applied Statistics (Slashlisted with 4793). Prerequisite: 4743 or 4753 or equivalent. Survey of advanced applied statistical methods other than applied regression, including exploratory data analysis, analysis of multivariate data (principal components: analysis, multiple analysis of variance, cluster analysis, etc.), and introduction to non-parametric methods. No student may earn credit for both 4793 and 5793. (Alt. F)
G5803 Topics in Mathematics. Prerequisite: permission of instructor. May be repeated with change of content; maximum credit nine hours. Topics may include any area of mathematics; these will be substantial and fundamental subjects not offered in regular courses. (F, Sp, Su)
G5853 Topology I. Prerequisite: 3513. Set theory, separation axioms, connectedness, compactness, continuity, metric spaces, nets and sequences. (F)
G5863 Topology II. Prerequisite: 5853. Metrization, product and quotient spaces, function spaces, dimension theory, Hilbert spaces, homotopy, simplicial complexes, continua. (Sp)
G5900 Graduate Mathematics Readings. 1 to 3 hours. Prerequisite: six-hour mathematics sequence at the 5000+ level. May be repeated with change of content; maximum credit 12 hours. Special background readings in advanced mathematical topics as preparation for later dissertation work. (F, Sp, Su)
G5910 Seminar—Analysis. 1 to 2 hours. Prerequisite: graduate standing. May be repeated with change of content; maximum credit 12 hours.
G5920 Seminar—Algebra and Theory of Numbers. 1 to 2 hours. May be repeated with change of content; maximum credit 12 hours. (F, Sp)
G5930 Seminar—Geometry and Topology. 1 to 2 hours. May be repeated with change of content; maximum credit 12 hours. (F, Sp)
G5940 Seminar—Applied Mathematics and Statistics. 1 to 2 hours. May be repeated with change of content; maximum credit 12 hours. (F, Sp)
G5950 Seminar—Undergraduate Mathematics Curriculum and Pedagogy. 1 to 2 hours. May be repeated with change of content; maximum credit 12 hours. This seminar will explore the current research literature on undergraduate mathematics curriculum and pedagogy. (F, Sp)
G5980 Research for Master's Thesis. Variable enrollment, two to nine hours; maximum credit applicable toward degree, four hours. (F, Sp)
G5990 Special Problems in Mathematics. 1 to 2 hours. An option for all candidates for the master's degree who do not present theses. (F, Sp, Su)
G6443 Topics in Differential Equations. Prerequisite: permission of instructor. May be repeated with change of content; maximum credit 12 hours. Topics include, but are not limited to, dynamical systems, nonlinear boundary value problems, parameter identification theory, wave theory, nonlinear functional analysis.(F, Sp)
G6473 Functional Analysis I. Prerequisite: 5463 or permission of instructor. Vector spaces with topology or norm, dual space, theorems on linear operators, spectral theory in Hilbert space, spectral decomposition of operators, convex sets and weak topologies, fixed point theorems. (Alt. F)
G6483 Functional Analysis II. Prerequisite: 6473. Banach algebras and harmonic analysis, representations of symmetric rings, unitary representations of a group, rings of operators in Hilbert space, decomposition of ring operators. Introduction to the theory of distributions. (Alt. Sp)
G6493 Topics in Analysis. Prerequisite: 5453 or permission of instructor. May be repeated with change of course content. Topics of modern research interest in analysis. (Irreg.)
G6673 Differential Geometry I. Prerequisite: 5853 or permission of instructor. Multilinear algebra, differential manifolds, exterior differential forms, affine connections, Riemannian manifolds. (F)
G6683 Differential Geometry II. Prerequisite: 6673. Riemannian manifolds, theory of connections, bundles with classical groups as structure groups, curvature and Betti numbers, complex manifolds. (Sp)
G6693 Topics in Geometry and Combinatorics II. Prerequisite: permission of instructor. May be repeated with permission of instructor; maximum credit 12 hours. Topics may include convexity, combinatorial geometry, graph theory, linear topological spaces, metric geometry, differential geometry or Riemannian geometry. (F, Sp)
G6813 Algebraic Topology I. Prerequisite: 5863. Introduction to homology theory of spaces, fundamental group and covering spaces, higher homotopy groups, CW-complexes and cellular homology, Whitehead and Hurewicz theorems, Eilenberg-Steenrod axioms. (F)
G6823 Algebraic Topology II. Prerequisite: 6813. Topics in cohomology and homology theory, universal coefficient theorems, orientation and duality on manifolds. Further topics may include: obstruction theory, cohomology operations, fibre bundles and characteristic classes, theory of sheaves, Eilenberg-MacLane spaces and Postnikov systems, spectral sequences. (Sp)
G6833 Topics in Topology I. Prerequisite: 5863. May be repeated with permission of instructor; maximum credit 12 hours. Topics may include algebraic topology, combinatorial topology, linear topological spaces, dimension theory, metrization, continua, decomposition spaces, topology of flat spaces. (F, Sp, Su)
G6843 Topics in Topology II. Prerequisite: 6833. May be repeated with permission of instructor; maximum credit 12 hours. Topics may include algebraic topology, combinatorial topology, linear topological spaces, dimension theory, metrization, continua, decomposition spaces, topology of flat spaces. (Irreg.)
G6900 Advanced Topics in Mathematics. 1 to 4 hours. May be repeated with change of content; maximum credit eight hours. A research problems course for advanced graduate students. (Irreg.)
G6910 Seminar—Analysis. 1 to 2 hours. Prerequisite: post-master's graduate standing in the department. May be repeated with change of content; maximum credit 12 hours. (F, Sp, Su)
G6920 Seminar—Algebra. 1 to 2 hours. Prerequisite: post-master's graduate standing in the department. May be repeated with change of content; maximum credit 12 hours. (F, Sp, Su)
G6930 Seminar—Geometry and Topology. 1 to 2 hours. Prerequisite: post-master's graduate standing in the department. May be repeated with change of content; maximum credit 12 hours. (F, Sp)
G6980 Research for Doctor's Dissertation. (F, Sp, Su)
Updated: August 15, 2003