Department of Mathematics
Graduate Course Descriptions
400-level courses may be taken by either undergraduate or graduate students. 500-level courses are for graduate students.
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400-3 History of Mathematics
An introduction to the development of major mathematics concepts. Particular attention given to the evolution of the abstract concept of space, to the evolution of abstract algebra, to the evolution of the function concept, and to the changes in the concept of rigor in mathematics from 600 B.C. Does not count toward a mathematics major in the College of Liberal Arts or in the College of Science. Prerequisite: 319 and 352 or consent of instructor.
405-3 Intermediate Differential Equations
This course features the study of several sets of differential equations with the aid of computers. The equations are actual applications taken from the areas of biology, chemistry, economics, engineering, finance, medicine, and physics; where possible, problems will be chosen to match student's interests. Students from these areas are particularly welcome. Basic theory of differential equations is cited, particularly as it is needed or encountered in the problems. The prerequisite is Math 305, but highly motivated students with a good calculus background and an interest in learning to use mathematical software may enroll with permission of the instructor.
406-3 Linear Analysis
An elementary introduction to function spaces and operators as used in quantum mechanics, partial differential equations, etc. Topics include: discrete and continuous models for the vibrating string; separation of variables and eigenfunction analysis; inner product spaces; operators on inner product spaces; the spectral theorem for Hermitian operators on finite dimensional spaces with selected applications; the Courant-Fisher max-min characterization of eigenvalues; the spectral theorem for compact Hermitian operators with selected applications to Sturm-Liouville boundary value problems and Fredholm integral equations. Prerequisite: 221 and 305.
407-3 Introduction to Partial Differential Equations
The purpose of this course is to teach the student how to solve linear partial differential equations that arise in engineering and the sciences. Topics studied will include: the heat equation, the wave equation, Laplace's equation, separation of variables, boundary and initial value problems, uniqueness via the energy method, the maximum principle, and characteristics. Solutions to the vibrating string and dissipation of heat in a bar will be discussed. Prerequisite: 251 and 305.
409-3 Fourier Analysis
A practical modern introduction to the theory, techniques, and applications of elementary Fourier analysis. Topics include: the Fourier synthesis and analysis equations for periodic and aperiodic functions on the reals and the integers; convolution; the calculus for finding Fourier transforms, Fourier series, and DFT's; operators and their Fourier transforms; the FFT and related algorithms; generalized functions, such as Dirac's delta, the comb, and "1/x"; and selected applications of Fourier analysis to sampling theory, partial differential equations, probability, the synthesis of musical tones, diffraction, and wavelets. Prerequisite: 221 and 305.
411-1 to 6 (1 to 3, 1 to 3) Mathematical Topics for Teachers
Variety of short courses in mathematical ideas useful in curriculum enrichment in elementary and secondary mathematics. May be repeated as topics vary. Does not count toward a mathematics major.
412-3 Problem Solving Approaches to Basic Mathematical Skills
Content of basic skills at all levels of education and the development of these skills from elementary school through college; emphasis on problem solving and problem solving techniques; determination of student skills and proficiency level. Credit may not be applied toward degree requirements in mathematics. Prerequisite: 314 or equivalent.
417-3 Applied Matrix Theory
Selected applications of matrices to physics, chemistry and economics. This material is also useful for engineering and computer science. Topics will include matrix representation of symmetry groups, non-negative matrices and the subsidy problem, location of eigenvalues. Prerequisite: 221.
418-3 Computer Algebra Systems
This course presents modern computer algebra systems (CAS) as a research tool in mathematics. The use of a CAS in the preparation of reports, theses and dissertations will also be covered. Topics will include: Solving differential equations with a CAS; Plotting techniques with a CAS; Symbolic packages for such areas as abstract algebra, number theory; and combinatorics: Programming with a CAS; Exporting results to TeX or word processing software; The AMS-LaTeX package. Prerequisite: Graduate standing and consent of instructor.
419-3 Introduction to Abstract Algebra II
A detailed study of polynomial equations in one variable. Solvable groups and the Galois theory of field extensions are developed and applied to extensions of the quadratic formula, proving the impossibility of trisecting an angle with only a straight-edge and a compass, and to the basic facts about finite fields as needed in coding theory and computer science. Prerequisite: 319 or consent of instructor.
421-3 Linear Algebra
The extension of basic linear algebra to arbitrary scalars. The theory and computation of Jordan forms of matrices (as needed, e.g., for certain diffusion equations). Inner products, quadratic forms and Sylvester's Law of Inertia. Prerequisite: 221.
425-3 Introduction to Number Theory
Properties of integers, primes, divisibility, congruences, quadratic forms, Diophantine equations, and other topics in number theory. Prerequisite: 319 or consent of department.
430-3 Introduction to Topology
Study of the real line and the plane, metric spaces, topological spaces, compactness, connectedness, continuity, products, quotients and fixed point theorems. This course will be particularly useful to students who intend to study analysis or applied mathematics. Prerequisite: 302 or 352 or consent of instructor.
435-3 Elementary Differential Geometry
An introduction to modern differential geometry through the study of curves and surfaces in R^3. Local curve theory with emphasis on the Serret-Frenet formulas; global curve theory including Fenchel's theorem; local surface theory motivated by curve theory; global surface theory including the Gauss-Bonnet theorem. Prerequisite: 221 and 251.
447-3 Introduction to Graph Theory (Same as Computer Science 447)
Graph theory is an area of mathematics which is fundamental to future problems such as computer security, parallel processing, the structure of the World Wide Web, traffic flow, and scheduling problems. It is also playing an increasingly important role within computer science. Topics covered include: trees, coverings, planarity, colorability, digraphs, depth-first and breadth-first searches. Prerequisite: 349 or consent of instructor.
449-3 Introduction to Combinatorics (Same as Computer Science 449)
This course will introduce the student to various basic topics in combinatorics that are widely used throughout applicable mathematics. Possible topics include: elementary counting techniques, pigeonhole principle, multinomial principle, inclusion and exclusion, recurrence relations, generating functions, partitions, designs, graphs, finite geometry, codes and cryptography. Prerequisite: 349 or consent of instructor.
450-3 Methods of Advanced Calculus
This course presents multivariable calculus, an area that is fundamental to fields such as continuum mechanics, differential geometry, electromagnetism, relativity, and thermodynamics. Topics will include: parametric curves and surfaces, the inverse and implicit function theorems, contraction mapping and fixed point theorems, differentials, convergence of multivariate integrals, coordinate systems in space, Jacobians, surfaces, volumes, and Green's, Gauss', and Stokes' theorems. The emphasis in this course will be on explicit computations. Prerequisite: 251.
452-3 Introduction to Analysis
This course develops the basic mathematical tools that are necessary for the understanding of all other advanced courses in analysis. Its principal content is a rigorous development of one-variable calculus. Topics will include: sets, axioms for the real numbers, continuity and limits, differentiation, the Riemann integral, and infinite sequences and series of functions. If time allows, additional topics may be chosen from areas such as Riemann-Stieltjes integration or the analysis of functions of several variables. Prerequisite: 250.
455-3 Complex Analysis with Applications
This course introduces the mathematical techniques that are commonly used to analyze those problems in the sciences and engineering that are inherently two dimensional in nature. Its content is the analysis of differentiable functions of a single complex variable. Topics will include: the complex plane, analytic functions, the Cauchy-Riemann equations, line integrals, the Cauchy integral formula, Taylor and Laurent series, the residue theorem, and conformal mappings. Applications will be made to topics selected from fluids, electrostatics, and control theory. Prerequisite: 251.
458-3 Statistical Methods in Business and Industry
The course gives an introduction to statistical techniques using a limited calculus background. Topics covered include probability; random variables; standard distributions such as binomial, Poisson, normal, and exponential; estimation including the method of moments and of maximum likelihood; tests of hypotheses; simple linear regression. Applications to business and engineering problems will be emphasized. The course does not count toward a mathematics major or a mathematics minor. Prerequisite: 140 or equivalent.
460-3 Transformation Geometry
Geometry viewed as the study of properties invariant under the action of a group. Topics include collineations, isometries, Frieze groups, Leonardo's Theorem, the classification of isometries of Euclidean and hyperbolic geometries. Recommended elective for secondary education majors in mathematics. Prerequisite: 221 and 319.
471-3 Optimization Techniques (Same as Computer Science 471)
An elementary introduction to algorithms for finding extreme values of nonlinear functions of several variables with and without constraints. Topics include: convex sets and functions; the arithmetic-geometric mean inequality; Taylor's theorem for functions of several variables; positive definite, negative definite, and indefinite matrices; iterative methods for unconstrained optimization such as the method of steepest descent; the Kuhn-Tucker algorithm; unconstrained and constrained geometric programming; Lagrange multipliers, and penalty function methods. Students will use a computer to study the numerical properties of these algorithms. Prerequisite: 221, 250.
472-3 Linear Programming (Same as Computer Science 472)
An introduction to the theory for finding extreme values of linear functionals subject to linear constraints. Topics include: recognition, formulation, and solution of real problems via the simplex algorithm; development of the simplex algorithm; artificial variables; the dual problem and the duality theorem; complementary slackness; sensitivity analysis; and applications of linear programming to integer programming, cutting plane algorithms, the distribution problem, the transportation problem, and the assignment problem. Students will use a computer to study the numerical performance of these algorithms. Prerequisite: 221.
473-3 Reliability and Survival Models
The course provides an introduction to the statistical analysis of data on lifetimes. Topics covered include hazard functions and failure distributions; multicomponent systems; estimation and hypothesis testing in life testing experiments with complete as well as censored data. Engineering applications include standby redundancy; repairable systems; preventive maintenance. Biomedical and actuarial applications will also be discussed. Prerequisite: 458 or 483 or 480 or consent of instructor.
475-6 (3, 3) Numerical Analysis (Same as Computer Science 475)
A practical introduction to the theory and techniques for computation with digital computers. Topics include: the solution of nonlinear equations; interpolation and approximation; solution of systems of linear equations; numerical integration, solution of ordinary differential equations; computation of eigenvalues and eigenvectors; and solution of partial differential equations. Students will use MATLAB to study the numerical performance of the algorithms introduced in the course. Prerequisite: (a) 221 and 250 (b) 305 and 475a.
480-3 Probability, Stochastic Processes, and Applications I
An introduction to the central topics of modern probability including some elementary stochastic processes. A student taking this course will learn about random variables and their properties, including sum of independent random variables and the Central Limit Theorem. In addition, random walks and discrete-time finite state Markov chains will be introduced. Applications to random number generators and image and signal processing will be discussed. Principal topics studied, in addition to those already listed, include generating functions, conditional probability and independence, expectation and moments, covariance and correlation, and characteristic functions. Prerequisite: 251.
481-3 Probability, Stochastic Processes, and Applications II
A continuation of part I with additional emphasis on stochastic processes and their applications. Students will see a thorough introduction to Markov processes and Martingales. Principal topics include the laws of large numbers, classification of states, recurrence, and convergence to the stationary distribution in Markov chains, birth processes and Poisson processes, stopping times, and the Martingale convergence theorem. Additional topics may include the renewal equation, stationary processes and the ergodic theorem and their applications, diffusion, and Kalman filtering with applications to signal processing and estimation. Prerequisite: 480.
483-4 Mathematical Statistics in Engineering and the Sciences
The course develops the basic statistical techniques used in applied fields like engineering, and the physical and natural sciences. Principal topics include probability; random variables; expectations; moment generating functions; transformations of random variables; point and interval estimation; tests of hypotheses. Applications include one-way classification data and chi-square tests for cross classified data. Prerequisite: 250.
484-3 Applied Regression Analysis and Experimental Design
The course provides an introduction to linear models and design of experiments used extensively in applied statistical work. Principal topics include linear models; analysis of variance; analysis of residuals; regression diagnostics; randomized blocks; Latin squares; factorial designs. Applications include response surface methodology and model building. Computations are an integral part of the course and will require the use of a statistical package such as SAS. Prerequisite: 483 and 221 or consent of instructor.
485-3 Applied Statistical Methods
The course gives an introduction to sampling methods and categorical data analysis which are widely used in applied areas such as social and biomedical sciences, and business. In sampling methods: topics covered include simple random and stratified sampling; ratio and regression estimators. In categorical data analysis: topics covered include: contingency tables; loglinear models; logistic regression; model selection; use of a computer package. Prerequisite: 483 or consent of instructor.
495-1 to 6 Special Topics in Mathematics
Individual study or small group discussions in special areas of interest under the direction of a member of the faculty. Prerequisite: consent of chair and instructor.
501-3 Measure and Integration
This course is an introduction to measure theory and the Lebesgue integral. Its purpose is to develop many of the advanced mathematical tools that are necessary for the understanding of all other advanced courses in analysis. Topics will include: measures and measurable functions, Egoroff's theorem, the Lebesgue integral, Fatou's lemma, the monotone and dominated convergence theorems, functions of bounded variation and absolutely continuous functions, Lp-spaces, the Radon-Nikodým theorem, product measures, and Tonelli's and Fubini's theorems. Prerequisite: 452.
502-3 Linear Analysis
This course is an introduction to analysis in linear infinite-dimensional spaces. Its purpose is to introduce function spaces that are used in the formulation of modern mathematical models in economics, the sciences, and engineering involving topics such as control theory, partial differential equations, and probability. Topics will include: Banach spaces, the Hahn-Banach Theorem, the uniform boundedness principle, the closed-graph theorem, the open-mapping theorem, weak convergence, reflexive and separable spaces, adjoint operators, Hilbert spaces, and the Riesz representation theorem. Prerequisite: 501.
505-3 Ordinary Differential Equations
Existence and uniqueness theorems; general properties of solutions; linear systems; geometric theory of nonlinear equations; stability; self-adjoint boundary value problems; oscillation theorems. Theory will be illustrated with computer simulation of several real-world problems. Prerequisite: 452 and 421 or consent of instructor.
506-1 to 12 Advanced Topics in Ordinary Differential Equations
Selected advanced topics in ordinary differential equations chosen from such areas as: stability, oscillation, functional differential equations, perturbations, boundary value problems. Prerequisite: consent of instructor.
507-3 Partial Differential Equations
This course introduces the student to the mathematical techniques that are used to analyze qualitative properties of solutions to partial differential equations that arise in engineering and the sciences. Topics studied will include: function spaces including Sobolev spaces; weak derivatives; the Sobolev and Poincaré inequalities; existence, uniqueness, and continuous dependence for model equations. Prerequisite: 407 and 501.
508-3 Integral Equations
Origins of integral equations. Volterra equations of the first and second kind. Fredholm equations of the first and second kind. Fredholm's alternative theorem. The resolvent equation. Orthonormal eigensystems of a symmetric Fredholm operator. The Hilbert-Schmidt expansion theorem and its applications to Sturm-Liouville problems. Exact and approximation methods of solution. Prerequisite: 452 and 406 or 421.
511-3 Advanced Topics in the Teaching of Mathematics (Same as Curriculum and Instruction 529.)
Selected advanced topics in the teaching of mathematics chosen from such areas as: pedagogical theories; instructional strategies; applications of mathematics; problem solving. This course is counted by the Mathematics department only as part of an approved minor. Prerequisite: consent of instructor.
512-1 to 21 Topics in Mathematics for Teachers of Elementary, Middle School and Junior High Mathematics
(a) Abstract Algebra. (b) Geometry. (c) Probability and Statistics. (d) Sets, Logic and Number Systems. (e) Applications of Mathematics. (f) Algebra. (g) History of Mathematics. This course is counted by the Mathematics department only as part of an approved minor.
513-1 to 27 Topics in Mathematics for Teachers of Secondary Mathematics
(a) Abstract Algebra. (b) Geometry. (c) Probability and Statistics. (d) Sets, Logic and Number Systems. (e) Applications of Mathematics. (f) Analysis. (g) Discrete Mathematics. (h) Topology. (I) Computer Simulation. This course is counted by the Mathematics department only as part of an approved minor.
516-8 (4,4) Statistical Analysis in the Social Sciences
(a) Descriptive statistics; graphic display of data; concepts of probability; statistical estimation, and hypothesis testing. Applications to social science data. (b) Matrix algebra; general linear model; multivariate statistics, ordinal and nominal measures of associations and causal modeling. Applications to social science data. This course does not give credit toward a mathematics major. Prerequisite: one year of high school algebra or equivalent.
519-3 Algebraic Structures I
Introduction to the basic techniques in the classification of finite groups, including homomorphism theorems, classification of finitely generated abelian groups, Sylow's theorems and classification of small groups, divisibility theory in rings, especially polynomial rings. Prerequisite: 419 or consent of instructor.
520-3 Algebraic Structures II
Algebraic field extensions; splitting fields, algebraic closure, separable and inseparable extensions; finite fields; norms and traces, the fundamental theorem of Galois theory. Free modules, torsion modules, tensor products of modules, finitely generated modules over principal ideal domains, application of abelian groups. Prerequisite: 519.
522-1 to 12 Advanced Topics in Algebra and Number Theory
Selected topics in modern algebra and number theory chosen from such areas as: group theory, commutative algebra, non-commutative algebra, field theory, representation theory, analytical number theory, algebraic number theory, additive number theory, Diophantine approximations, Dirichlet series and automorphic form. Prerequisite: consent of instructor.
525-3 Number Theory
Introduction to modern analytic and algebraic techniques used in the study of quadratic forms, the distribution of prime numbers, Diophantine approximations and other topics of classical number theory. Prerequisite: 425.
530-3 Geometry and Topology I
First part of a sequence that provides students with foundational material useful for research in dynamical systems, classical mechanics, relativity as well as other areas of mathematics. Topics include a review of point set topology, an introduction to differentiable manifolds, and the fundamental group. Prerequisite: 430 or consent of instructor.
531-3 Geometry and Topology II
Second part of a sequence that provides students with foundational material useful for research in dynamical systems, classical mechanics, relativity as well as other areas of mathematics. Topics include homology and cohomology with differential forms. Prerequisite: 530 or consent of instructor.
532-1 to 12 Topics in Geometry and Topology
Topics may include dynamical systems, topological groups, knot theory, complexity theory, uniform spaces and frames, differential and Riemannian geometry, voting theory and mathematical physics. Prerequisite: consent of instructor.
540-3 Convex Analysis
The course develops the basic results on convex sets and functions which are extensively used in several areas of applied mathematics and in business and engineering. Both finite and infinite dimensional spaces will be discussed. Topics covered include separation theorems, extreme points and the Krein-Milman Theorem. For infinite dimensional spaces elementary aspects of locally convex spaces will be covered. Applications include inequalities, constrained optimization and minimax theory. Prerequisite: 452 or consent of instructor.
549-3 Combinatorial Theory
This course will introduce the student to various advanced topics in combinatorial theory that are basic to modern methods in applicable mathematics. Possible topics include: enumeration, Polya-Burnside theory, DeBruijn sequences, graph theory, Cayley's Theorem, Ramsey's Theorem, Hall's Theorem, design theory, distinct representatives, Latin squares, and Finite geometries. Prerequisite: 449 or consent of instructor.
551-3 Functional Analysis
This course will introduce the student to various advanced topics in functional analysis that are basic to modern methods in differential equations, mathematical physics, probability theory, and quantum theory. Possible topics include: Banach algebras, distributions, locally convex spaces, quantum probability, self-adjoint operators, the spectral theory of operators, and topological vector spaces. Prerequisite: 502.
553-1 to 12 Advanced Topics in Analysis and Functional Analysis
Advanced topics in analysis and functional analysis from such areas as: harmonic analysis, approximation theory, integration theory, advanced complex variables, topological vector spaces, operator theory, Banach algebras, distribution theory. Prerequisite: consent of instructor.
559-1 to 12 Advanced Topics in Combinatorics
Selected advance topics in combinatorics chosen from such areas as: graph theory; combinatorial designs; enumeration; random graphs; finite geometry; coding theory; cryptography; combinatorial algorithms. Prerequisite: consent of instructor.
566-3 Continuum Mechanics
This course will provide a rigorous development of the mechanics of solids and fluids. Topics will include: elements of tensor analysis; kinematics; balance of mass, linear momentum, and angular momentum; the concept of stress; constitutive equations for fluid and solid bodies; and invariance of constitutive equations under a change in observer. Applications of continuum mechanics to the solution of problems in materials science will be included as time permits. Prerequisite: 450 or 452.
569-1 to 12 Advanced Topics in Applied Mathematics
Selected advanced topics in applied mathematics chosen from such areas as: continuum mechanics; electromagnetic theory, control theory; mathematical physics. Prerequisite: consent of instructor.
570-1 to 12 Advanced Topics in Optimization.
Selected advanced topics in optimization and operations research chosen from such areas as: calculus of variations, optimal control theory, nonlinear programming, convex analysis, nonsmooth analysis, new flows, advanced computer simulation, large scale linear programming. Prerequisite: consent of instructor.
572-1 to 12 Advanced Topics in Numerical Analysis (Same as Computer Science 572)
Selected advanced topics in numerical analysis chosen from such areas as: approximation theory; spline theory; special functions; wavelets; numerical solution of initial value problems; numerical solution of boundary value problems; numerical linear algebra; numerical methods of optimization; and functional analytic methods. Prerequisite: consent of instructor.
574-3 Approximation Theory.
A study of techniques for approximating functions by polynomials, trigonometric polynomials, polynomial splines, wavelets, etc. Topics include: existence, uniqueness, and characterization of best approximations in normed linear spaces; projection methods for good approximation; the Weierstrass, Muntz-Szasz, and Stone-Weierstrass theorems; degree of approximation and the Jackson theorems; construction of optimal min-max and least squares approximation using rational functions, splines, wavelets. Students will use MATLAB to study the quality of various approximations developed in the course. Prerequisite: 452, 475a, and one of 406, 421.
575-3 Matrix Computations
A practical introduction to modern numerical linear algebra. Topics include: vector and matrix norms; Householder, Givens, and Gauss transforms; factorization methods for solving systems of linear equations with roundoff error analysis; QR and SVD methods for solving linear least squares problems; the QR algorithm for computing the eigenvalues of a matrix. Students will use MATLAB to study the algorithms developed in the course. Prerequisite: 475a and one of 406, 421.
580-3 Statistical Theory
The course gives a rigorous introduction to statistical inference. Topics covered include statistical models; sufficiency and completeness; Cramér-Rao bound; Rao-Blackwell theorem; best estimators; most powerful tests; likelihood ratio tests; elements of Bayes and minimax procedures. Prerequisite: 480 or 483.
581-3 Probability
A rigorous, measure-theoretic introduction to probability theory. Principal topics include general probability spaces, product spaces and product measures, random variables as measurable functions, distribution functions, conditional expectation, types of convergence, characteristic functions and the central limit theorem, tail events and 0-1 laws, the Borel-Cantelli lemma, and the weak and strong law of large numbers. Prerequisite: Concurrent course in real variables, 501.
582-1 to 6 Advanced Topics in Probability
Selected advanced topics in probability chosen from such areas as: martingales, Markov processes, Brownian motion, infinitely divisible laws. Prerequisite: consent of instructor.
583-1 to 6 Advanced Topics in Statistics
Selected advanced topics in statistics chosen from such areas as: advanced linear models, advanced experimental design, multivariate statistical analysis, decision theory, advanced nonparametric theory. Prerequisite: consent of instructor.
585-1 to 2 Statistical Consulting
Consulting with university researchers under the supervision of a member of the statistics faculty. A write up of each consultation will be required. Prerequisite: 484 or 485 and consent of instructor.
590-1 to 6 Contemporary Mathematics Research
Lectures on various mathematical topics of current research interest by members of the department and by distinguished visitors. Prerequisite: consent of the graduate advisor.
595-1 to 12 per topic Special Project
An individual project, including a written report. (a) Algebra. (b) Geometry. (c) Analysis. (d) Probability and Statistics. (e) Mathematics Education. (f) Logic and Foundations. (g) Topology. (h) Applied mathematics. (i) Differential Equations. (j) Number Theory. Graded S / U only. Prerequisite: consent of instructor.
599-1 to 6 Thesis
Minimum of three hours to be counted toward the Master of Arts degree.
600-1 to 30 (1 to 16 per semester) Dissertation
Minimum of 24 hours to be earned for the Doctor of Philosophy degree.
601-1 per semester Continuing Enrollment
For those graduate students who have not finished their degree programs and who are in the process of working on their dissertation, thesis, or research paper. The student must have completed a minimum of 24 hours of dissertation research, or the minimum thesis, or research hours before being eligible to register for this course. Concurrent enrollment in any other course is not permitted. Graded S/U or DEF only.
Department of Mathematics home page: http://www.math.siu.edu/