All MATH/STAT/OPRS courses offered by the Department of Mathematics are approved to satisfy requirements for the Problem Solving Goal of UNC Charlotte Education.
Undergraduate
MATH 1100. College Algebra and Probability. (3) Prerequisite: appropriate score on the Mathematics Placement Test or placement by the department. The basic mathematics course for undergraduates not majoring in mathematics, engineering, or the physical sciences. Fundamental concepts of algebra and probability. (Credit may not be given for both MATH 1100 and 1103; students who received credit for MATH 1101 between Fall 1987 and Fall 1989 may not take 1100 for credit; students who already have credit for MATH 1120 or 1241 with a grade of C or better may not take 1100 for credit.) (Fall, Spring, Summer) (Evenings)
MATH 1102. Introduction to Mathematical Thinking. (3) Prerequisite: appropriate score on the Mathematics Placement Test or placement by the department. An introduction to mathematical ideas designed primarily for non-science students. Topics are drawn from various branches of mathematics which may include algebra, geometry, number theory, probability, statistics and graph theory. Computers may be used. (Fall)
MATH 1103. Precalculus Mathematics for Science and Engineering. (3) Prerequisite: appropriate score on the Mathematics Placement Test or placement by the department. Intended for students who plan to take MATH 1241. Functions and graphs, linear and quadratic functions, polynomial and rational functions, exponential and logarithmic functions, trigonometric identities. (Credit may not be given for both MATH 1100 and 1103; students who received credit for MATH 1100 between Fall 1987 and Fall 1989 may not take MATH 1103 for credit; students who already have credit for MATH 1120 or 1241 with a grade of C or better may not take MATH 1103 for credit.) (Fall, Spring, Summer) (Evenings)
MATH 1105. Finite Mathematics. (3) Prerequisite: appropriate score on the Mathematics Placement Test or placement by the department. Review of high school algebra, elementary matrix algebra, systems of linear equations and inequalities, elementary linear programming; probability. (On demand)
MATH 1120. Calculus. (3) Prerequisite: appropriate score on the Mathematics Placement Test; MATH 1100 or 1103; or placement by the department. Intended for students majoring in fields other than engineering, mathematics or science. Elements of differential and integral calculus for polynomial, rational, exponential, logarithmic and trigonometric functions, with applications to business and the social and life sciences. (May not be taken for credit if credit has been received for MATH 1121 or 1241.) (Fall, Spring, Summer) (Evenings)
MATH 1121. Calculus (ET). (3) Prerequisite: appropriate score on the Mathematics Placement Test; MATH 1100 or 1103; or placement by the department. Intended for students majoring in engineering technology. Elements of differential and integral calculus for polynomial, rational, exponential, logarathmic and trigonometric functions, with applications to engineering. May not be taken for credit if credit has been received for MATH 1120 or 1241. (Fall, Spring, Summer) (Evenings)
MATH 1165. Introduction to Discrete Structures. (3) Prerequisite: CSCI 1100 or 1201 and its lab. Propositions and truth tables, sets, permutations and combinations, relations and functions, lattices, and trees. Credit will not be given for both MATH 1165 and 2165. (Fall, Spring, Summer) (Evenings)
MATH 1241. Calculus I. (3) Prerequisite: appropriate score on the Mathematics Placement Test; MATH 1103 with a grade of C or better, or placement by the department. Designed for students majoring in mathematics, science, or engineering. Elementary functions, derivatives and their applications, introduction to definite integrals. (May not be taken for credit if credit has been received for MATH 1141.) (Fall, Spring, Summer) (Evenings)
MATH 1242. Calculus II. (3) Prerequisite: MATH 1141 or 1241 with a grade of C or better. Methods for evaluating definite integrals, applications of integration, improper integrals, Taylor series, introduction to differential equations. (May not be taken for credit if credit has been received for MATH 1142.) (Fall, Spring, Summer) (Evenings)
MATH 2050. Topics in Mathematics. (2-3) Prerequisite: consent of the department. Topics in mathematics elected to supplement regular offerings at the 2000 level. (May or may not count for a Math core course for the CSCI major.) May be repeated for additional credit with the approval of the department. (On demand)
MATH 2101. Fundamental Concepts in Mathematics I. (3 Prerequisite: consent of the department. An intuitive development of the real number system with emphasis on the principles of teaching and learning elementary mathematics; sets and set operations; systems of numeration; arithmetic operations. Laboratory activities and practical teaching experience in an elementary classroom. May not be used to satisfy requirement for a major or minor in Mathematics. (Fall, Spring) (Evenings)
MATH 2102. Fundamental Concepts in Mathematics II. (3 Prerequisite: MATH 2101 with a grade of C or better. Mathematical systems; the study of metric and non-metric geometry; relations, functions and their graphs. Activities and applications related to elementary school mathematics goals. Practical teaching experience in an elementary classroom. May not be used to satisfy requirement for a major or minor in Mathematics. (Fall, Spring) (Evenings)
MATH 2103. Problem Solving in Mathematics Using Computers and Calculators in the Classroom. (3) Prerequisite: MATH 2101 with a grade of C or better. Calculators in the mathematics curriculum; microcomputer hardware and courseware in mathematics; LOGO; probability; data collection, analysis, and interpretation. May not be used to satisfy requirement for a major or minor in Mathematics. (Fall, Spring, Summer) (Evenings)
MATH 2164. Matrices and Linear Algebra. (3) Prerequisite: MATH 1120 or 1241 with a grade of C or better or consent of the department. Matrix algebra, systems of linear equations, vector spaces, linear transformations, determinants, inner products, eigenvalues. (Fall, Spring, Summer) (Evenings)
MATH 2171. Differential Equations. (3) Prerequisite: MATH 1242 with a grade of C or better. An introduction to ordinary differential equations including first order equations, general theory of linear equations, series solutions, special solutions, special equations such as Bessel's equation, and applications to physical and geometric problems. (Fall, Spring, Summer) (Evenings)
MATH 2241. Calculus III. (3) Prerequisite: MATH 1142 or 1242 with a grade of C or better. Functions of two or more variables, vectors in two and three dimensions, partial derivatives, optimization, double and triple integrals and their applications. (May not be taken for credit if credit has been received for MATH 2141.) (Fall, Spring, Summer) (Evenings)
MATH 2242. Calculus IV. (3) Prerequisite: MATH 2141 or 2241 with a grade of C or better. Parametric curves and surfaces, vector fields, line and surface integrals; Green's theorem, Divergence theorem, Stoke's theorem and applications. Infinite series, power series. (Fall, Spring, Summer) (Evenings)
MATH 3050. Selected Topics in Mathematics. (2-3) Prerequisite: consent of the department. Topics selected to supplement regular offerings at the 3000 level in mathematics or statistics. May be repeated for credit with the approval of the department. (On demand)
MATH 3116. Graph Theory. (3) Prerequisite: MATH 2164 or 2165 or consent of the department. Graphs as mathematical models. Planarity, colorability, connectivity, trees. Applications and algorithms for networks, matching problems and areas of computer science. (Fall) (Alternate years)
MATH 3122. Probability and Statistics I. (3) Prerequisite: MATH 2241 with a grade of C or better. Sample spaces, random variables, moment generating functions, some standard distributions, multivariate distributions, laws of large numbers, limit theorems. (Fall) (Evenings)
MATH 3123. Probability and Statistics II. (3) Prerequisite: MATH/STAT 3122. Estimation, bias, consistency, efficiency, maximum likelihood estimates, sufficient statistics, testing, the power function, chi-square test, Kolmogorov-Smirnov test. Credit for mathematics major not given for both MATH 3125 and MATH/STAT 3123. (Spring) (Evenings)
MATH 3125. Statistical Techniques. (3) Prerequisite: consent of the department. Probability models, random variables, large sample statistics, inferential statistics, analysis of variance and experimental design. Credit for mathematics major not given for both MATH 3125 and MATH/STAT 3123. (Spring) (Evenings)
MATH 3141. Advanced Calculus of One Variable. (3) Prerequisites: MATH 2241 and 2164 with grades of C or better. Topology of the real line; continuity, uniform continuity, differentiability, integration, sequences and series of functions. (Fall) (Evenings)
MATH 3142. Advanced Calculus of Several Variables. (3) Prerequisite: MATH 3141. Continuity and differentiability of functions of several variables, inverse and implicit function theorems, integration, Fubini's theorem, change of variables, the classical integral theorems of Gauss, Green and Stokes and their generalizations. (Spring) (Evenings)
MATH 3146. Introduction to Complex Analysis. (3) Prerequisite: MATH 2241 with a grade of C or better. Analytic functions, complex integration, calculus of residues, conformal mapping. (Spring) (Alternate years)
MATH 3163. Introduction to Modern Algebra. (W) (3) Prerequisite: MATH 1241 with a grade of C or better or consent of the department. Examples and elementary properties of basic algebraic structures, especially groups. The course emphasizes the writing of proofs of elementary theorems. (Fall, Spring) (Evenings)
MATH 3166. Combinatorics. (3) Prerequisites: MATH 2164. Combinatorial modeling, generating functions, recurrence relations, inclusion-exclusion principle and problems from recreational mathematics. (Spring) (Alternate years)
MATH 3171. Applied Mathematics. (3) Prerequisites: MATH 2241 and 2171 with grades of C or better. Separation of variables techniques for the classical partial differential equations of mathematical physics; Fourier series; Sturm-Liouville theory. (Fall) (Evenings)
MATH 3176. Numerical Analysis. (3) Prerequisites: CSCI 1100 or 1201 and its lab, MATH 2241 and 2171. Numerical solution of initial value and boundary value problems in ordinary differential equations, direct and iterative methods of solving systems of equations. Selected problems will be programmed for computer solution. (Spring) (Alternate years)
MATH 3181. Fundamental Concepts of Geometry. (3) Prerequisite: MATH 2164. Foundations of geometry, transformations, comparison of Euclidean and non- Euclidean geometries. (Fall, Spring) (Evenings)
MATH 3551. Mathematics Cooperative Education Experience. (0) Prerequisites: Sophomore standing, a 3.0 GPA in MATH/STAT/OPRS courses and consent of the Department of Mathematics. The student will be employed in a manner that affords him/her the opportunity of using and enhancing mathematical knowledge and skills through practical experience. After completing MATH 3551, the student must take MATH 3652. MATH 3551 may be repeated with consent of the department. (On demand)
MATH 3652. Mathematics Cooperative Education Seminar. (1) Prerequisite: MATH 3551. The student will give an exposition of his/her work experience in MATH 3551. An exposition of underlying theoretical concepts and related ideas may also be required. (On demand)
MATH 3688. Mathematics Awareness Seminar. (0) Prerequisite: sophomore standing. Visiting speakers, discussion of internships, cooperative education and job opportunities; selected topics in mathematics. (Fall)
MATH 3689. Mathematics Project Seminar. (1) Prerequisite: senior standing. Oral presentation by the student on an area of mathematics or a mathematical problem. (Fall, Spring)
MATH 3691. Seminar. (1-6) Prerequisite: consent of the department. Readings, study and discussion designed to develop the student's ability to study independently and to present results properly. (On demand)
MATH 3790. Junior Honors Seminar. (3) Prerequisite: consent of the department. May be repeated once for additional credit with approval of the department. (On demand)
MATH 3791. Senior Honors Tutorial. (3) Prerequisite: consent of the department. Individual tutorials in which the student will pursue independent study and research in any area of mathematics under the direction of one or more faculty members. The project of the student will be planned to culminate in a research paper of original or expository nature. May be repeated for additional credit with the approval of the department. (On demand)
Undergraduate/Available for Graduate Credit
MATH 4000. Topics in Foundations or History of Mathematics. (2-3) (2-3G) Prerequisite: consent of the department. Topics in the foundations or the history of mathematics selected to supplement regular course offerings in this area of mathematics. May be repeated for credit with approval of the department. Credit for the M.A. degree in mathematics requires approval of the department. (On demand)
MATH 4040. Topics in Analysis. (2-3) (2-3G) Prerequisite: consent of the department. Topics in analysis selected to supplement regular course offerings in this area of mathematics. May be repeated for credit with the approval of the department. Credit for the M.A. degree in mathematics requires approval of the department. (On demand)
MATH 4060. Topics in Algebra. (2-3) (2-3G) Prerequisite: consent of the department. Topics in algebra selected to supplement regular course offerings in this area of mathematics. May be repeated for credit with the approval of the department. Credit for the M.A. degree in mathematics requires approval of the department. (On demand)
MATH 4080. Topics in Geometry and Topology. (3) (3G) Prerequisite: consent of the department. Topics in geometry or topology selected as to supplement regular course offerings in this area of mathematics. May be repeated for credit with approval of the department. Credit for M.A. degree in mathematics requires approval of the department. (On demand)
MATH 4109. History of Mathematical Thought. (3) (3G) Prerequisite: MATH 1241 or consent of the department. A study of the development of mathematics in its historical setting from the earliest beginnings to modern times. Not approved for the M.A. in mathematics degree. (Fall) (Evenings)
MATH 4161. Number Theory. (3) (3G) Prerequisite: MATH 3163 with a grade of C or better or consent of the department. A study of the elements of classical number theory including divisibility, congruences, diophantine equations, prime numbers and their distribution, quadratic reciprocity, number-theoretic functions, and famous unsolved problems. Not approved for the M.A. in mathematics degree. (Spring) (Alternate years)
MATH 4163. Modern Algebra. (3) (3G) Prerequisite: MATH 3163 or consent of the department. Groups, rings, integral domains, fields. (Fall) (Alternate years)
MATH 4164. Abstract Linear Algebra. (3) (3G) Prerequisite: MATH 3163 and 2164 or consent of the department. Vector spaces over arbitrary fields, linear transformations, canonical forms, multilinear algebra. (Spring) (Alternate years)
MATH 4181. Introduction to Topology. (3) (3G) Prerequisite: MATH 2164. Topics from set theory and point set topology such as cardinality, order, topological spaces, metric spaces, separation axioms, compactness and connectedness. (Fall) (Alternate years)
MATH 4691. Seminar. (1-6) (1-6G) Prerequisite: consent of the department. Individual or group investigation and exposition of selected topics in mathematics. (On demand)
MATH 4692. Seminar. (1-6) (1-6G) Prerequisite: consent of the department. A continuation of MATH 4691. (On demand)
Graduate and Advanced Undergraduate
MATH 5128. Applied Probability I. (3) (3G) Prerequisite: MATH/STAT 3122 and MATH 2171 or consent of the department. Finite and countable Markov chains, Markov Decision Processes, and optimal stopping. Other topics selected from: queuing theory, inventory models, reliability theory, game theory, recurrent events, information theory, stochastic control, stochastic control with incomplete information and Kalman filtering. (Fall)(Alternate years)
MATH 5129 Applied Probability II. (3) (3G) Prerequisite: MATH 5128 or consent of the department. Continuation of MATH 5128. (Spring)(Alternate years)
MATH 5143. Analysis I. (3) (3G) Prerequisite: MATH 3141 with a grade of B or better, or consent of the department. First course of a two-semester sequence providing a rigorous treatment of continuity, differentiability and integration of functions of one and several real variables. (Fall)
MATH 5144. Analysis II. (3) (3G) Prerequisite: MATH 5143 with a grade of B or better or consent of the department. Continuation of MATH 5143. (Spring)
MATH 5165. Numerical Linear Algebra. (3) (3G) Prerequisites: CSCI 1100 or 1201 and 1201L, MATH 2164 and 2171, all with a grade of C or better, or consent of the Department. Gaussian elimination and LU decomposition methods for linear systems. Vector and matrix norms, condition numbers and accuracy of solutions. Solutions of large sparse matrix systems using skyline solvers, and Jacobi, Gauss-Seidel, and SOR iterative methods. Solution of nonlinear systems. Least squares methods using the QR factorization. Selected problems will be programmed for computer solution. (Fall) (Alternate years)
MATH 5171. Numerical Solution of Ordinary Differential Equations. (3) (3G) Prerequisites: CSCI 1100 or 1201 and 1201L, MATH 2241, 2164 and 2171, all with a grade of C or better, or consent of the Department. Numerical solution techniques for ordinary differential equations such as Runga-kutta, multistep and extrapolation methods. Stiff solvers and stability criteria. Comparative work with modern robust codes and visualization methods. (On demand)
MATH 5172. The Finite Element Method. (3) (3G) Prerequisites: CSCI 1100 or 1201 and 1201L, MATH 2241, 2164, and 2171, all with a grade of C or better, or consent of the department. Boundary value problems and their variational form. Finite element basis functions, computational techniques, isoparametric elements and curved boundaries, alternate methods, singular problems, eigenvalue problems. Some practical experience with an F.E.M. program and graphical output. (Spring) (Alternate years)
MATH 5173. Ordinary Differential Equations. (3G) Prerequisites: MATH 2171, MATH 3142, or consent of the department. Existence and uniqueness theorems for initial value problems; continuous dependence of solutions on initial values and right hand sides; linear differential equations in R2 and Rn; non-linear differential equations in R2 and Rn: phase portraits, singularities, cycles; invariant manifolds; linearization; singularities of planar systems; Lyapunov stability; examples: van der Pol oscillator, Liénard systems, Volterra-Lotka equations. (Spring)
MATH 5174. Partial Differential Equations. (3) (3G) Prerequisites: MATH 2164 and 3141, or consent of department. Classification of types of partial differential equations. Separation of variables, Sturm-Liouville problems, boundary and eigenvalue problems, fundamental solutions and Green's theorem, Fourier series and integrals, Laplace transforms. (Fall)
MATH 5176. Numerical Methods for Partial Differential Equations. (3) (3G) Prerequisite: CSCI 1100 or 1201 and 1201L, MATH 2241, 2164 and 2171 all with a grade of C or better, or consent of the department. Basic finite difference schemes for the solutions of elliptic, parabolic and hyperbolic equations. Van Neuman analysis, characteristics, boundary conditions. (Fall) (Alternate years)
Graduate Only
MATH 6004. Topics in Analysis. (3G) Prerequisite: MATH 6101 or consent of department. Topics in analysis selected so as to complement regular course offerings in this area of mathematics. May be repeated for credit with the consent of department. (On demand)
MATH 6008. Topics in Geometry and Topology. (3G) Prerequisite: consent of department. Topics selected from Euclidean geometry, non-Euclidean geometry, projective geometry, differential geometry, point-set topology, algebraic topology. May be repeated for credit with approval of department. (On demand)
MATH 6100. Foundations of Mathematics. (3G) Prerequisite: consent of department. Logic, sets and axiomatic systems. (Fall, Summer) (Alternate years)
MATH 6101. Foundations of Real Analysis. (3G) Prerequisite: MATH 6100 or consent of department. Axiomatic and historical development of the real and complex numbers; rigorous development of limits and continuity of functions, intermediate and extreme value theorems. (Fall) (Alternate years)
MATH 6102. Calculus from an Advanced Viewpoint. (3G) Prerequisite: MATH 6101 or its equivalent. A continuation of MATH 6101. A rigorous approach to differentiation and integration of functions of one real variable. (Spring) (Alternate years)
MATH 6103. Computer Techniques and Numerical Methods. (3G) Prerequisite: MATH 6101 or consent of department. Computer systems, programming, and the computer solution of numerical problems. (Summer) (Alternate years)
MATH 6105. Problem Solving in Discrete Mathematics. (3G) Prerequisite: consent of department. Propositional and predicate calculus, counting techniques, partially ordered sets, lattices, graphs and trees. (Alternate years)
MATH 6106. Modern Algebra. (3G) Prerequisite: MATH 3163 or its equivalent or consent of department. Topics chosen from group theory, rings and ideals, integral domains, fields and elementary Galois theory. (Summer) (Alternate years)
MATH 6107. Linear Algebra. (3G) Prerequisite: MATH 2164 or its equivalent or consent of department. Systems of linear equations, matrices, vector spaces, linear transformations, determinants, canonical forms of matrices, inner products. (Summer) (Alternate years)
MATH 6118. Non-Euclidean Geometry. (3G) Prerequisite: consent of department. History of Euclid's Fifth Postulate and attempts to prove it; work of Gauss, Bolyai, Lobachevsky and others; systematic development of hyperbolic geometry; relative consistency of hyperbolic geometry; relative consistency of hyperbolic and Euclidean geometries. (Alternate years)
MATH 6171. Advanced Applied Mathematics I. (3G) Prerequisites: MATH 2241 and 2171 with grades of C or better, or consent of department. Power series solutions of ordinary differential equations, vector calculus, line and surface integrals, partial differential equations and Fourier integrals. (Fall) (Evenings)
MATH 6172. Advanced Applied Mathematics II. (3G) Prerequisites: MATH 2241 and 2171 with grades of C or better or consent of department. Complex analysis; probability and statistics. (Spring) (Evenings)
MATH 6609. Seminar. (1-2-3G) Prerequisite: consent of the department. A series of regularly scheduled meetings in which each student will present one or more topics selected by the instructor. May be repeated for credit with the consent of department. (On demand)
Advanced Graduate Only
MATH 7028. Topics in Probability. (3G) Prerequisite: MATH 7120 and 7121, or consent of department. Topics of current interest in probability and advanced topics in probability. May be repeated for credit with the consent of the department. (On demand)
MATH 7050. Topics in Mathematics. (2-3G) Prerequisite: consent of department. Topics chosen from such fields as algebra, topology, analysis, applied mathematics, differential geometry, mathematical physics, graph theory, probability, statistics. May be repeated for credit as topics vary and with the approval of the department. (On demand)
MATH 7065. Topics in Applied Algebra and Algebraic Structures. (3G) Prerequisite: consent of the department. Current topics in Applied Algebra and Algebraic Structure. (On demand)
MATH 7070. Topics in Numerical Analysis. (3G) Prerequisite: consent of the department. Topics of current interest in numerical analysis. May be repeated for credit with the consent of the department. (On demand)
MATH 7071. Topics in Differential Equations. (3G) Prerequisite: consent of the department. Topics of current interest in ODE, PDE, dynamical systems, inverse problems and related subjects. May be repeated for credit with the consent of the department. (On demand)
MATH 7120. Probability Theory I. (3G) Prerequisites: MATH 7143 and MATH/STAT 3122 or consent of department. Topics include probability spaces, probability measures, sigma-algebras, characteristic functions, sequences of random variables, law of large numbers, general forms of the Central Limit Theorem. (Fall) (Alternate years)
MATH 7121. Probability Theory II. (3G) Prerequisite: MATH 7120 or consent of the department. A continuation of MATH 7120. (Spring) (Alternate years)
MATH 7125. Stochastic Processes I. (3G) Prerequisites: MATH 3122 and 7143 or consent of the department. Basic ideas in the study of stochastic processes, selected from: discrete and continuous time Markov processes, stationary and renewal processes, applications to queuing theory, reliability theory, stochastic differential equations, time-series analysis, filtering and stochastic control theory. (On demand)
MATH 7126. Stochastic Processes II. (3G) Prerequisite: MATH 7125. A continuation of MATH 7125. (On demand)
MATH 7141. Complex Analysis I. (3G) Prerequisite: MATH 5143 or consent of the department. Holomorphic functions, complex integration, residues, entire and meromorphic functions, conformal mapping, harmonic functions. (Spring) (Alternate years)
MATH 7142. Complex Analysis II. (3G) Prerequisite: MATH 7141. A continuation of MATH 7141. (On demand)
MATH 7143. Real Analysis I. (3G) Prerequisite: MATH 5144 or consent of the department. Lebesgue integration on the real line, Lp spaces, introduction to general measure and integration theory. (Fall)
MATH 7144. Real Analysis II. (3G) Prerequisite: MATH 7143 or consent of the department. A continuation of MATH 7143. (Spring)
MATH 7147. Applied Functional Analysis. (3G) Prerequisite: MATH 5144. Introduction to functional analysis and its applications to such areas as linear and non-linear differential equations, integral equations, and control theory. Topics chosen from Banach spaces, operators, the Hahn-Banach, open mapping and closed graph theorems, Sobolev spaces, spectral theory, operators in Hilbert space. (Summer) (On Demand)
MATH 7148. Functional Analysis. (3G) Prerequisite: MATH 7144 or consent of the department. Material selected from: spectral theory, spectral theory of differential operators, groups and semigroups of operators, nonlinear functional analysis, asymptotic analysis, integral equations, Fourier analysis, distributions, and Sobolev spaces. (Fall)(Alternate years)
MATH 7163. Modern Algebra I. (3G) Prerequisite: MATH 4163 and 4164 or consent of department. Topics will be selected from Galois theory, commutative algebra, modules, ring theory, homological algebra. (Fall) (Alternate years)
MATH 7164. Modern Algebra II. (3G) Prerequisite: MATH 7163. A continuation of MATH 7163. (On demand)
MATH 7172. Partial Differential Equations. (3G) Prerequisite: MATH 5174 and 7144 or consent of department. Harmonic functions, mean-value theorem, maximum principle, Green's representation for the solution of the Dirichlet problem for Laplace's equation; Poisson's equations and the Poisson formula; statement and proof of the existence theorem for general second-order elliptic operators, generalized maximum principles; Sobolev spaces. Evolution equations involving elliptic operators, such as the heat or wave equations, may also be introduced. (Spring) (Alternate years)
MATH 7173. Evolution Equations. (3G) Prerequisite: MATH 7144 and 7172 or consent of the department. Semigroups of operators and their generators, examples of semigroups. The heat equation, examples of elliptic operators that generate semigroups, Hille-Yosida theory, analytic semigroups; examples, fractional powers of operators. (On demand)
MATH 7174. Linear and Non-linear Waves. (3G) Prerequisite: MATH 5124 and 7144 or consent of the department. Hyperbolic waves, characteristics, Riemann invariants, conservation laws, weak solutions, shock structure. Burger's equation, gas dynamics, dispersive waves, group velocity, water waves, non-linear optics. (On demand)
MATH 7175. Inverse Problems. (3G) Prerequisite: MATH 7144 and MATH 5174 or consent of the department. Ill-posed problems and numerical methods for them. Applications of inverse problems to real processes. One dimensional inverse problems. Multi-dimensional inverse problems: uniqueness and numerical methods. Inverse scattering problems. (On demand)
MATH 7176. Advanced Numerical Analysis. (3G) Prerequisites: MATH 2164, 2171 and 5176 or consent of the department. A selection of topics from such areas as iterative methods of solving linear and non-linear systems of equations, approximation theory, splines, and finite element methods for partial differential equations. (Spring) (Alternate years)
MATH 7177. Applied Optimal Control. (3G) Prerequisites: MATH 5143 or consent of the department. Examples of control systems and optimization problems, optimal control of discrete-time systems, solutions of the general discrete-time optimization problem, optimal control of continuous-time systems, the calculus of variations, solution of the general continuous optimization problem, applications of the Pontryagin Maximum Principle, Dynamic programming, and Bang-bang control. Controllability and differential games may also be introduced. (Spring) (Alternate years)
MATH 7178. Computational Methods for Fluid Dynamics. (3G) Prerequisite: CSCI 1100 or 1201 and 1201L, MATH 2242, 2171, 5174 and 5176 or consent of the department. Topics on various numerical techniques for the solution of incompressible and compressible flows. Finite difference, finite element and spectral methods, and shock capturing and fitting methods. Multi-grid method and acceleration techniques. (On demand)
MATH 7179. Advanced Finite Difference Methods. (3G) Prerequisite: consent of the department. Accuracy analysis and design of high order schemes, stability theory of schemes with variable coefficients, stability theory of schemes for initial-boundary value problems, convergence theory for nonlinear cases. (On demand)
MATH 7181. Topology I. (3G) Prerequisite: consent of department. Topological spaces, continuous functions, connectedness, compactness, and metrizability, and further topics from point-set, geometric or algebraic topology. (On demand)
MATH 7182. Topology II. (3G) Prerequisite: MATH 7181. A continuation of MATH 7181. (On demand)
MATH 7184. Differential Geometry I. (3G) Prerequisite: consent of the department. Manifolds, differential structures, tangent bundles, embeddings, immersions, inverse function theorem, Morse-Sard theorem, transversality, Borsuk-Ulam theorem, vector bundles, Euler characteristics, Morse theory, Stokes theorem, Gauss-Bonnet theorem, Whitney embedding theorem. (On demand)
MATH 7185. Differential Geometry II. (3G) Prerequisite: consent of the department. Differentiable manifolds, differential forms, critical points, local and global theory of curves, local and global theory of surfaces, connections, Geodeusics, curvature, spaces of constant curvature, Lie groups and Lie algebras. (On demand)
MATH 7273. Advanced Finite Element Analysis. (3G) Prerequisite: MATH 5172 and 5174 or consent of the department. Selection of topics from such areas of finite element analysis as convergence theorems (Ciarlet), hierarchical basis functions, the h-p method, adaptive grid techniques and solution methods for nonlinear equations. (Fall)(Alternate years)
MATH 7275. Dynamical Systems I. (3G) Prerequisites: MATH 5143 and MATH 5173 or consent of the department. Cycles and separatrix cycles, Poincaré firstreturn map: diffeomorphisms, Poincaré-Bendixson Theory, flows on the two-torus; structural stability, genericity, Peixoto's theorem; singularities of planar systems. Degenerate singularities, Hopf bifurcation, saddle-node bifurcation, center bifurcation. (On demand)
MATH 7276. Dynamical Systems II. (3G) Prerequisite: MATH 7275 or consent of the department. Method of averaging, Melnikov functions, hyperbolic structure, symbolic dynamics, homocline and heteroclinic orbits, global bifurcations, infinite dimensional dynamical systems, inertial manifolds, Lyapunov exponents and dimension of attractors, codimension-two bifurcations, Duffing's equation, Lorenz equations, finite dimensional systems of dimension at least three. (On demand)
MATH 7277. Bifurcation Theory. (3G) Prerequisite: MATH 7275 or consent of the department. Implicit function theorem, manifolds and transversality, Newton polygons, Lyapunov center theorem, variational methods, Ljusternik-Schnirelman theory, mountain-pass theorem, bifurcations with one-dimensional null-spaces, Morse theory and global bifurcations, geometric theory of partial differential equations. (On demand)
MATH 7691. Research Seminar. (1-3G) Prerequisite: consent of department. A seminar in which independent study may be pursued by the student or a group of students under the direction of a professor. (On demand)
MATH 7692. Research Seminar. (1-3G) Prerequisite: consent of department. A continuation of MATH 7691. (On demand)
MATH 7893. Thesis. (0-3G) Prerequisite: consent of department. Subject to the approval of the department Graduate Committee, the thesis may be original work, work of an expository nature, or the mathematical formulation and solution of a particular industrial or business problem suggested by the career interests of the student. The thesis must be defended in an oral presentation. May be repeated for credit with the consent of department. (Fall, Spring, Summer)
MATH 7994. Doctoral Research and Reading. (0-9G) Prerequisite: consent of the department. May be repeated with consent of the department.
(On demand)