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Professor Zhijian Wu, Chairperson
Office: 345 Gordon Palmer Hall
The department offers programs leading to the master of arts and the doctor of philosophy degrees. The department offers courses in the following areas: algebra, analysis, topology, differential equations, mathematical methods for engineering, numerical analysis, mathematical statistics, fluid dynamics, control theory, and optimization theory.
To be admitted for a graduate degree, students are expected to satisfy the general requirements of the Graduate School, as stated in the "Academic Policies" section of this catalog. In support of the application, each applicant must submit scores on the general test of the Graduate Record Examination; the advanced portion is desirable but not required.
Master of arts. Each student's program for the master's degree must be approved by the department. A total of 30 graduate hours is required to obtain a master's degree in mathematics.
Candidates for the master's degree may choose either of two plans. One plan (Plan I) requires successful completion of 24 semester hours of coursework, plus a thesis. The other plan (Plan II) requires no thesis, but requires successful completion of 27 semester hours of coursework plus 3 hours of work devoted to a project supervised by a member of the graduate faculty in mathematics. At least 21 of the course hours must be taken in mathematics; courses in related areas, such as physics, finance, or computer science, may be taken with the approval of the graduate advisory committee. An oral examination is required for completion of the degree. Candidates for the master's degree must complete three of the following four core courses: MATH 510 Numerical Linear Algebra, MATH 532 Graph Theory and Applications, MATH 580 Real Analysis I, and MATH 585 Introduction to Complex Calculus.
Doctor of philosophy. The student's program for the Ph.D. degree in mathematics must be approved by the department. Ph.D. students in mathematics normally take three two-course sequences in mathematics/applied mathematics. A total of at least 48 hours of coursework is required. Dissertations for the Ph.D. degree in mathematics may be written in any one of several areas approved by the department. A total of at least 24 hours of dissertation research must be taken. Before officially becoming a Ph.D. candidate, the student must pass qualifying examinations in two areas within three years of becoming a full-time graduate student. One of the passes obtained should normally be in the area of the dissertation.
Joint Ph.D. program in applied mathematics is a program with the UA System campuses in Birmingham and Huntsville. Admission to the program is obtained by passing the joint program examination in linear algebra, numerical linear algebra, and real analysis. Each program of study requires a minimum of 54 semester hours of coursework approved by the student's joint graduate study supervisory committee. Those hours must include a major area concentration consisting of at least six courses in addition to the courses needed to prepare for the joint program examination, and an application minor consisting of at least four related graduate courses in some area outside the department. Before officially becoming a Ph.D. candidate in this program, a student must pass the comprehensive qualifying examination that covers the entire program of study. Neither the joint program examination nor the comprehensive qualifying examination can be taken more than twice.
MATH 500 Mathematical Methods of Physics I. Three hours.
Prerequisite: MATH 238.
Vector calculus, tensors and matrices, functions of a complex variable, and special functions.
MATH 501 Mathematical Methods of Physics II. Three hours.
Prerequisite: MATH 500.
Special functions, Fourier series and integral transforms, Green's functions, and group theory.
MATH 502 History of Mathematics. Three hours.
Prerequisite: Permission of the department.
Designed to increase awareness of the historical roots of the subject and its universal applications in a variety of settings, showing how mathematics has played a critical role in the evolution of cultures over both time and space.
MATH 504 Topics in Modern Mathematics for Teachers. Three hours.
Prerequisite: Permission of the department.
Diverse mathematical topics designed to enhance skills and broaden knowledge in mathematics for secondary mathematics teachers.
MATH 505 Geometry for Teachers. Three hours.
Prerequisite: MATH 125 or permission of the department.
A survey of the main features of Euclidean geometry, including the axiomatic structure of geometry and the historical development of the subject. Some elements of projective and non-Euclidean geometry are also discussed.
MATH 506 Curriculum in Secondary Mathematics. Three hours.
Prerequisite: Admission to Teacher Education Program in secondary mathematics or permission of the department.
MATH 507 Theory of Numbers. Three hours.
Prerequisite: Permission of the department.
Designed for graduate students not majoring in mathematics. Familiarity with the types of arguments to prove facts about divisibility of integers, prime numbers, and modular congruencies. Other topics, such as the Fermat theorem, Euler's theorem, and the law of quadratic reciprocity, will be discussed.
MATH 508 Topics in Algebra. Three hours.
Prerequisite: Permission of the department.
Content changes from semester to semester to meet the needs of students. Designed for graduate students not majoring in mathematics.
MATH 509 Discrete Mathematics for Computer Science. Three hours.
Prerequisite: Graduate standing.
Develops students' abilities to think abstractly and logically by applying techniques of discrete mathematics to computer science problems.
MATH 510 Numerical Linear Algebra. Three hours.
Prerequisites: MATH 237 (or MATH 257) or equivalent.
Direct solution of linear algebraic systems, analysis of errors in numerical methods for solutions of linear systems, linear least-squares problems, orthogonal and unitary transformations, eigenvalues and eigenvectors, and singular value decomposition.
MATH 511 Numerical Analysis I. Three hours.
Prerequisites: MATH 237, MATH 238 or MATH 257, and CS 226; or equivalent.
Numerical methods for solving nonlinear equations; iterative methods for solving linear systems of equations; approximations and interpolations; numerical differentiation and integration; and numerical methods for solving initial-value problems for ordinary differential equations.
MATH 512 Numerical Analysis II. Three hours.
Prerequisite: MATH 411, MATH 511, or equivalent.
Continuation of MATH 511 with emphasis on numerical methods for solving partial differential equations. Also covers least-squares problems, Rayleigh-Ritz method, and numerical methods for boundary-value problems.
MATH 513 Finite-Element Methods. Three hours.
Prerequisites: MATH 343 and MATH 382.
Corequisite: MATH 510.
Quadratic functional on finite dimensional vector spaces, variational formulation of boundary value problems, the Ritz-Galerkin method, the finite-element method, and direct and iterative methods for solving finite-element equations.
MATH 520 Linear Optimization. Three hours.
Prerequisite: MATH 237.
Topics include formulation of linear programs, simplex methods and duality, sensitivity analysis, transportation and networks, and various geometric concepts.
MATH 521 Optimization Theory II. Three hours.
Prerequisite: MATH 321 or MATH 520.
Corequisite: MATH 510 or permission of the instructor.
Emphasis on traditional constrained and unconstrained nonlinear programming methods, with an introduction to modern search algorithms.
MATH 523 Convex Analysis I. Three hours.
Prerequisites: MATH 257 and MATH 380, or permission of the department.
Introduction to convex analysis. Topics include basic concepts, topological properties, duality correspondences, and representation and inequalities of convex sets and functions.
MATH 528 Introduction to Optimal Control. Three hours.
Prerequisite: MATH 238.
Corequisite: MATH 510 or permission of the department.
Introduction to the theory and applications of deterministic systems and their controls. Major topics include calculus of variations, the Pontryagin maximum principle, dynamic programming, stability, controllability, and numerical aspects of control problems.
MATH 532 Graph Theory and Applications. Three hours.
Prerequisites: MATH 237 or MATH 257, and MATH 382 or permission of the instructor.
Survey of several of the main ideas of general graph theory with applications to network theory. Topics include oriented and nonoriented linear graphs, spanning trees, branchings and connectivity, accessibility, planar graphs, networks and flows, matchings, and applications.
MATH 537 Special Topics in Applied Mathematics I. Three hours.
Prerequisite: Permission of the department.
MATH 538 Special Topics in Applied Mathematics II. Three hours.
Prerequisite: Permission of the department.
MATH 541 Boundary Value Problems. Three hours.
Prerequisites: MATH 343 and MATH 382, or permission of the department.
Emphasis on boundary-value problems for classical partial differential equations of physical sciences and engineering. Other topics include boundary-value problems for ordinary differential equations and for systems of partial differential equations.
MATH 542 Integral Transforms and Asymptotics. Three hours.
Prerequisite: MATH 441, MATH 541, or permission of the department.
Introduction to complex variable methods, integral transforms, asymptotic expansions, WKB method, matched asymptotics, and boundary layers.
MATH 544 Singular Perturbations. Three hours.
Prerequisites: MATH 382 and MATH 441 (or MATH 541), or permission of the department.
Topics include regular perturbation methods for solving partial differential equations, matched asymptotic methods for differential equations, and the methods of strained coordinates and multiple scales, with applications to problems in combustion theory, fluid dynamics, and biology.
MATH 545 Theoretical Foundations of Fluid Dynamics I. Three hours.
Prerequisite: MATH 343, AEM 264 or equivalent, or permission of the department.
Introduction to continuum mechanics and tensors. Local fluid motion. Equations governing fluid flow and boundary conditions. Some exact solutions of the Navier-Stokes equations. Vortex motion. Potential flow and aerofoil theory.
MATH 546 Theoretical Foundations of Fluid Dynamics II. Three hours.
Prerequisite: MATH 545 or equivalent, or permission of the department.
Introduction to asymptotic methods and other approximate methods applied to classical problems in boundary-layer theory, low Reynolds number flows, surface gravity waves, shallow-water theory, and hydrodynamic stability.
MATH 551 Mathematical Statistics with Applications I. Three hours.
Prerequisites: MATH 237 and MATH 355.
Introduction to mathematical statistics. Topics include bivariate and multivariate probability distributions; functions of random variables; sampling distributions and the central limit theorem; concepts and properties of point estimators; various methods of point estimation; interval estimation; tests of hypotheses; and Neyman-Pearson lemma with some applications. Credit for this course will not be counted toward an advanced degree in mathematics.
MATH 552 Mathematical Statistics with Applications II. Three hours.
Prerequisite: MATH 551.
Considers further applications of the Neyman-Pearson lemma, likelihood ratio tests, chi-square test for goodness of fit, estimation and test of hypothesis for linear statistical models, the analysis of variance, analysis of enumerative data, and some topics in nonparametric statistics. Credit for this course will not be counted toward an advanced degree in mathematics.
MATH 554 Mathematical Statistics I (equivalent to ST 554). Three hours.
Prerequisites: MATH 237 and MATH 382.
Distributions of random variables, moments of random variables, probability distributions, joint distributions, and change of variable techniques.
MATH 555 Mathematical Statistics II (equivalent to ST 555). Three hours.
Prerequisite: MATH 554.
Order statistics, asymptotic distributions, point estimation, interval estimation, and hypothesis testing.
MATH 556 Mathematical Statistics III (equivalent to ST 610). Three hours.
Prerequisite: MATH 555.
Generalized inverse matrices; distribution of quadratic forms; regression analysis when the model is of full rank; regression using dummy variables and analysis of variance models; and regression analysis when the model is not of full rank.
MATH 557 Stochastic Processes with Applications I. Three hours.
Prerequisite: MATH 554 or ST 554.
Introduction to the basic concepts and applications of stochastic processes. Markov chains, continuous-time Markov processes, Poisson and renewal processes, and Brownian motion. Applications of stochastic processes including queueing theory and probabilistic analysis of computational algorithms.
MATH 559 Stochastic Processes with Applications II. Three hours.
Prerequisite: MATH 355 or permission of the department.
Continuation of MATH 557. Advanced topics of stochastic processes including Martingales, Brownian motion and diffusion processes, advanced queueing theory, stochastic simulation, and probabilistic search algorithms (simulated annealing).
MATH 560 Introduction to Differential Geometry. Three hours.
Prerequisites: MATH 380 or MATH 382, and permission of the department.
Introduction to basic classical notions in differential geometry: curvature, torsion, geodesic curves, geodesic parallelism, differential manifold, tangent space, vector field, Lie derivative, Lie algebra, Lie group, exponential map, and representation of a Lie group.
MATH 565 Introduction to General Topology. Three hours.
Prerequisite: MATH 380.
Basic notions in topology that can be used in other disciplines in mathematics. Topics include topological spaces, open sets, closed sets, basis for a topology, continuous functions, separation axioms, compactness, connectedness, product spaces, quotient spaces, and metric spaces.
MATH 566 Introduction to Algebraic Topology. Three hours.
Prerequisites: MATH 565 and a course in abstract algebra.
Homotopy, fundamental groups, covering spaces, covering maps, and basic homology theory, including the Eilenberg Steenrod axioms.
MATH 570 Principles of Modern Algebra I. Three hours.
Prerequisite: MATH 257.
A first course in abstract algebra. Topics include groups, permutations groups, Cayley's theorem, finite Abelian groups, isomorphism theorems, rings, polynomial rings, ideals, integral domains, and unique factorization domains.
MATH 571 Principles of Modern Algebra II. Three hours.
Prerequisite: MATH 470 or equivalent.
The basic principles of Galois theory are introduced in this course. Topics covered are rings, polynomial rings, fields, algebraic extensions, normal extensions, and the fundamental theorem of Galois theory.
MATH 573 Abstract Algebra I. Three hours.
Prerequisite: MATH 470 or equivalent.
Fundamental aspects of group theory are covered. Topics include Sylow theorems, semi-direct products, free groups, composition series, nilpotent and solvable groups, and infinite groups.
MATH 574 Cryptography. Three hours.
Prerequisite: MATH 307, MATH 470/570, or permission of department.
Introduction to a rapidly growing area of cryptography, an application of algebra, especially number theory.
MATH 580 Real Analysis I. Three hours.
Prerequisites: MATH 380 and permission of the department.
Topics covered include measure theory, Lebesgue integration, convergence theorems, Fubini's theorem, and LP spaces.
MATH 583 Complex Analysis I. Three hours.
Prerequisites: MATH 380 and permission of the department.
The basic principles of complex variable theory are discussed. Topics include Cauchy-Riemann equations, Cauchy's integral formula, Goursat's theorem, the theory of residues, the maximum principle, and Schwarz's lemma.
MATH 585 Introduction to Complex Calculus. Three hours.
Prerequisite: MATH 227.
Some basic notions in complex analysis. Topics include analytic functions, complex integration, infinite series, contour integration, and conformal mappings.
MATH 588 Theory of Differential Equations I. Three hours.
Prerequisites: MATH 238 and MATH 380.
Topics covered include existence and uniqueness of solutions, Picard theorem, homogenous linear equations, Floquet theory, properties of autonomous systems, Poincare-Bendixson theory, stability, and bifurcations.
MATH 589 Theory of Differential Equations II. Three hours.
Prerequisite: MATH 588.
Typical topics covered include principal Lyapunov stability and instability theorems; invariance theory; perturbation of linear systems including stable and unstable manifolds; periodic solutions of systems; Hopf bifurcations; and degree theory.
MATH 591 Teaching College-Level Mathematics. Three hours.
Prerequisite: Permission of the instructor or the department.
Provides a basic foundation for teaching college-level mathematics; to be taken by graduate students being considered to teach undergraduate-level mathematics courses.
MATH 592 Introduction to Graduate Mathematics. Three hours.
Prerequisite: MATH 237, MATH 257, or permission of the department.
Should prepare beginning graduate students for graduate-level mathematics. Dependent on students' backgrounds, analysis and linear algebra topics will be covered. Proofs and examples will form major course components.
MATH 598 Research Not Related to Thesis. Three to nine hours.
MATH 599 Thesis Research. Three to six hours.
MATH 610 Iterative Methods for Linear Systems. Three hours.
Prerequisite: MATH 511.
Corequisite: MATH 512.
Describes some of the best iterative techniques for solving large sparse linear systems.
MATH 623 Convex Analysis II. Three hours.
Prerequisite: MATH 523 or permission of the department.
Second course in convex analysis. Topics include differential theory, constrained extremum problems, and saddle functions and minimax theory for convex functions.
MATH 625 Optimization by Vector Space Methods. Three hours.
Prerequisite: MATH 580 or permission of the department.
Involves applications of geometric principles of linear vector space theory to complex infinite-dimensional optimization problems. Topics include linear spaces, Hilbert spaces, least-square estimation, dual spaces, linear operators and adjoints, optimization of functionals, global and local theory of optimization, and interactive methods of optimization.
MATH 639 Seminar: Topics in Applied Mathematics. One to three hours.
MATH 640 Waves in Fluids. Three hours.
Prerequisites: MATH 545 (or AEM 500) and MATH 542, or permission of the department.
Analysis of various wave motions and development of fundamental ideas of general application to waves in fluids. Sound waves, water waves, and internal waves.
MATH 642 Viscous Flows. Three hours.
Prerequisites: MATH 545 (or AEM 500) and MATH 541, or permission of the department.
Review of equations of fluid motion, tensors, and the Navier-Stokes equation. The role of viscosity in creeping flows and boundary layers.
MATH 644 Hydrodynamic Stability. Three hours.
Prerequisite: MATH 545 or AEM 500.
Fundamental ideas, methods, results, and applications of hydrodynamic stability. Introduction to some current research topics.
MATH 659 Seminar: Probability Models. One to three hours.
MATH 661 Algebraic Topology I. Three hours.
Prerequisite: MATH 566 or equivalent.
In-depth study of homotopy and homology. The theory of cohomology is also introduced as are characteristic classes.
MATH 663 General Topology I. Three hours.
Prerequisite: MATH 565 or permission of the department.
Typical topics covered in this course include countable and uncountable sets; axiom of choice; well-ordered sets; connectedness and compactness; countability and separation axioms; Tychonoff's theorem; fundamental group; and covering spaces.
MATH 664 General Topology II. Three hours.
Prerequisite: MATH 663.
Topics of interest to the instructor will be introduced.
MATH 665 Topological Structures I. Three hours.
Prerequisite: Permission of the department.
Topics covered in previous courses include selected works of Pontryagin.
MATH 669 Seminar: Topics in Topology. One to three hours.
MATH 674 Abstract Algebra II. Three hours.
Prerequisite: MATH 573 or equivalent.
Fundamental aspects of ring theory are covered. Topics include Artinian rings, Wedderburn's theorem, idempotents, polynomial rings, matrix rings, Noetherian rings, free and projective modules, and invariant basis number.
MATH 677 Topics in Algebra I. Three hours.
Prerequisite: Permission of the department.
Content decided by instructor. Recent topics covered include linear groups, representation theory, commutative algebra and algebraic geometry, algebraic K-theory, and theory of polycyclic groups.
MATH 679 Seminar: Topics in Algebra. One to three hours.
MATH 681 Real Analysis II. Three hours.
Prerequisite: MATH 580.
Topics covered include basic theory of LP spaces, convolutions, Hahn decomposition, the Radon-Nikodym theorem, Riesz representation theorem, and introduction to Banach spaces.
MATH 684 Complex Analysis II. Three hours.
Prerequisite: MATH 583 or permission of the department.
Typical topics covered include analytic functions, the Riemann mapping theorem, harmonic and subharmonic functions, the Dirichlet problem, Bloch's theorem, Schottley's theorem, and Picard's theorems.
MATH 686 Functional Analysis I. Three hours.
Prerequisites: MATH 681 and a course in complex analysis.
Topics covered in recent courses include Hilbert spaces, Riesz theorem, orthonormal bases, Banach spaces, Hahn-Banach theorem, open-mapping theorem, bounded operators, and locally convex spaces.
MATH 687 Functional Analysis II. Three hours.
Prerequisite: MATH 686.
Topics covered in recent courses include spectral theory, Banach algebras, C* algebras, nest algebras, Sobolev spaces, linear p.d.e.'s, interpolation theory, and approximation theory.
MATH 688 Seminar: Topics in Analysis. One to three hours.
MATH 689 Seminar: Topics in Functional Analysis. One to three hours.
MATH 698 Research Not Related to Dissertation. Three to nine hours.
MATH 699 Dissertation Research. Three to twelve hours.
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