Graduate Courses in Mathematics

*PT:Pure+Thesis, PN:Pure+Nonthesis, IT:Industrial+Thesis,
IN:Industrial+Nonthesis, TM:Teaching Mathematics, DL:Distance Learning

*R = Required course,   N = Need to pick three courses in each track,   2 = need to take twice

Courses Descriptions

MATH 5304 FOUNDATIONS OF MATHEMATICS     

     This course presents elements of mathematical logic, set theory, number theory and selected topics from Discrete Mathematics such as Combinatorial Analysis and Graph Theory. Mathematics proofs are emphasized. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5309 INTEGRATING TECHNOLOGY INTO MATHEMATICS     

     This is an introductory course related to the latest technological computer programs, especially in Mathematics. The students become familiar with a representative sample of the technology currently available for industry, and will be able to publish mathematical articles both on-line and off-line. They also will be enabled todecide how to use technology in industry. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5321 HIGHER ALGEBRA    

    The purpose of this course is to provide the necessary algebraic backgroundfor all branches of modern Mathematics that use algebraic language and methods (in particular number theory and Algebraic Geometry). Topics include basic ring theory (primes and irreducible ring elements, prime ideals and maximal ideals, integral ring extensions, Noetherian and Dedekind rings, polynomial rings over Noetherian rings (Hilbert's Basissatz)), field extensions, and basic Galois theory with the usual applications to classical problems in geometry. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5323 GROUP THEORY     

    The purpose of this course is to provide students with a concept, which arises naturally in almost every mathematical area, but also in Physics and Chemistry, the notion of a group. The course will cover at least one of the essential aspects of modern group theory, finite group theory, algebraic group theory, or combinatorial group theory. In the first case, the course will include the theorems of Jordan-Holder, Sylow, and Schur-Zassenhaus, the treatment of the generalized Fitting subgroup, a first approach to solvable as well as simple groups (including the theorems of Ph.\ Hall and Burnside). Prerequisite: MATH 5321 or consent of instructor.

MATH 5327 LIE ALGEBRAS     

    This course is an introduction to the classical theory of Lie algebras. Topics include root systems, the Weyl group, nilpotent and solvable Lie algebras, the theorems of Lie and Engel, Cartan subalgebras, Cartan's criterion for semi-simplicity, Chevalley groups and groups of Lie type.Prerequisite: MATH 5321 or consent of instructor.

MATH 5329 NUMBER THEORY    

    This course focuses on analytical or algebraic number theory. In the first case, the course covers arithmetic functions (Moebius, Euler, Dirichlet), Dirichlet series (convergence, uniqueness, multiplicative property) distribution of primes (Dirichlet, Tchebycheff, Hadamard resp. de la Vallee-Poussin), Riemann's zeta function. In the second case, the course focuses on algebraic number fields, Dedekind domains, and the class group. Prerequisite: MATH 5321 or consent of instructor.

MATH 5331 HIGHER GEOMETRY          

    This course will be on Projective, Algebraic or Convex Geometry. Projective Geometry includes basic incidence geometry, group actions on geometries, ternary rings and coordinates in projective and affine geometries, and the Fundamental Theorem of Projective Geometry. Algebraic Geometry includes basic facts on algebraic curves, the relationship between algebraic sets and radical ideals, Hilbert's Nullstellensatz. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5337 DYNAMICAL SYSTEMS     

    The main goal of this course is to understand the long term behavior of states in a system for which there is a deterministic rule for how a state evolves. The evolution of the state of the system may be very different, such as stability and instability/bifurcation/catastrophe; controllability and stabilizability; observability and detectability; isolations and attractors; oscillations and chaos. In this course we will focus on the linear control systems and preliminary nonlinear control systems. Prerequisite: MATH 5331 or consent of instructor.

MATH 5339 TOPOLOGY     

    The course treats both the general and the algebraic aspects of topology. It covers topological spaces, continuous mappings, connectedness and compactness, the fundamental group, covering spaces, the Jordan Curve Theorem and a classification of surfaces. Prerequisite: MATH 5341 or consent of instructor.

MATH 5341 HIGHER ANALYSIS     

    This course presents the system of the real numbers and the system of the complex numbers, sequences and series of real numbers, continuity and differentiability of real functions, convergence of sequences and series of functions, aspects of functions in several variables, the Riemann-Stieltjes integral and an introduction to Lebesgue theory. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5342 MEASURE AND INTEGRAL THEORY     

    The course presents the Lebesgue Theory, abstract integration, Borel measures, Lebesgue spaces, integration of differential forms. Prerequisite: MATH 5341.

MATH 5346 FUNCTIONAL ANALYSIS     

    This course introduces to topological vector spaces. It presents the theory of Hilbert spaces, Banach space techniques and their applications, and basic facts on operator theory and spectral theory. Prerequisite: MATH 5342 or consent of instructor.

MATH 5348 DIFFERENTIAL EQUATIONS     

    This course covers first order and higher order ordinary differential equations, systems of solutions of linear differential equations, the Laplace transform, and several basic concepts of partial differential equations. Prerequisite: 6 SCH of MATH 4000-courses or 3 SCH of MATH 5000-courses.

MATH 5361 MATHEMATICAL MODELING     

    In this course, we shall not deal with a specified mathematical theory. Instead, the students will learn how to develop mathematical models which reflect the real world problems. It may include modeling with difference and differential equations or with stochastic processes. The course may be project-oriented. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5362 GRAPH THEORY     

    This course provides the student with the basic ideas of graph theory as it is used in many branches of Industrial Mathematics. It contains Ramsey Theory, spanning trees, decision trees, matching theory, graphcoloring, traveling salesman problems, networks, min-max theorems, flows, Ford-Fulkerson. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5363 OPERATIONS RESEARCH    

    This course emphasizes fundamental concepts and principles as well as algorithms in Operations Research. The topics are Linear Programming (simplex and its variations), integral programming (cutting plane method, 0-1 style and assignment problems), non-linear programming (gradient, conjugate gradient, penalty functions, patterns), dynamic programming, networks, queuing theory, inventory theory, decision theory, game theory. In this course, students will be required to participate in projects. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5365 DISCRETE MATHEMATICS     

    This course is on the borderline between Mathematics and Computer Science. It contains basic graph theory (flows, min-max, Ford-Fulkerson), generating functions, (Convolutions, Dirichlet's generating function, Riemann's zeta function), design theory, basic facts on coding theory (minimal distance, Reed-Solomon Codes), combinatorial optimization, elements of asymptotics (O-notation, O-manipulation), and complexity of algorithms. Prerequisite: 6 SCH of MATH 4000.

MATH 5367 NUMERICAL ANALYSIS     

    This course deals with solutions of equations, interpolation and approximation, numerical differentiation and integration, numerical aspects of linear algebra, and with solutions of ordinary differential equations. Prerequisite: MATH 5341 or consent of instructor.

MATH 5368 CODES, CYPHERS, AND SECURITY IN COMMUNICATIONS 

    This course addresses two related problems in communications theory. The first deals with errors that occur in the transmission of information; how they can be detected and how they can be corrected. The second is concerned with the security of transmitted information. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5375 MEASURE AND PROBABILITY     

    This course is an introduction to measure-theoretic probability theory. Topics covered include sets and events, monotone sequences, algebras, sigma-algebras, probability spaces, Borel sets and Lebesgue measure; measurable functions and random variables, independence, Borel-Cantelli lemma, Kolmogorov's zero-one law; Lebesgue integral and expectation; different types of convergence, laws of large numbers, characteristic functions and the central limit theorem. Prerequisite: MATH 5341.

MATH 5379 STOCHASTIC ANALYSIS     

    The main objective of this course is to study discrete stochastic processes and their applications. The principal topics discussed include Markov process and Markov chains; transient and persistent states, irreducible, aperiodic chains, stationary distributions, convergence theorems, random walks on a lattice, stopping times, Ehrenfest chain, birth and death chains, Bernoulli-Laplace model of diffusion. Martingales, super and submartingales, reversed martingales, connection between martingales and Markov process, gambling systems, fundamental theorems of mathematical finance, trading strategies, viable markets, and market models. Prerequisite: MATH 4374 or consent of instructor.

MATH 5381 MATHEMATICAL STATISTICS     

    This is a course in inferential statistics. Topics include random sampling, distribution of means and the central limit theorem, estimation problems, tests of hypotheses, linear regression, correlation, analysis of variance. Prerequisite: MATH 4374 or consent of instructor.

MATH 5385 TIME SERIES AND ENGINEERING SYSTEMS     

    The contents of this course include the treatment of normal sequences and white noise, stationary time series, characteristic analysis of time series, the analysis of stationary time series in the time domain, linear modeling of dynamic data, linear predictions of time series, multivariate dynamic data models. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5391 SPECIAL TOPICS IN MATHEMATICS     

    The topic of this course may come from different areas of Pure and Industrial Mathematics not available in other courses. For instance, the course could be an introduction to the foundations of system engineering. This would enable the student to treat complex systems from the point of view of entire, multiple aspects, and evolution. Topics would include open and closed systems (ordered and unordered), bifurcation and catastrophe, attractors and chaos, self-organization of systems, stochastic systems. Other topics of this course could be linear optimization or non-linear optimization. The course may be repeated for credit. Prerequisite: 6 SCH of MATH 4000-courses.

MATH 5395 RESEARCH SEMINAR     

    This course is an introduction to the methods and tools of mathematical research. The participants will study (under the guidance of an instructor) a chapter of a textbook or an original research paper, and they will have to present the contents to their classmates and to faculty.

MATH 5397 THESIS     

    Supervised research. This will include the treatment of an original research problem with a written thesis, if needed with a collection and the analysis of original data, and written in a scientific style in an acceptable publication format. Prerequisite: Consent of the advisor.