6.13
DEPARTMENT OF MATHEMATICS (MATH)
Chairperson: Professor Zhijian Wu,
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, mathematics for finance, mathematical statistics,
numerical analysis, fluid dynamics, control theory, and optimization theory.
Admission Requirements
To be admitted for a graduate degree,
students are expected to satisfy the general requirements of the
Graduate School, as stated in
the
Admission Criteria 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.
Degree Requirements
Master of arts.
Each student's program for the master's degree must be approved by the department
and the
Graduate School. Students need to follow all policies
found in the master’s degree policies section of the Graduate Catalog.
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
Plan of Study for the PhD
degree in mathematics must be approved by the department and the
Graduate School by the time the student completes 30 graduate
semester hours of UA and/or transfer course work. Students also need
to follow all policies found in the doctoral degree policies section
of the Graduate Catalog.
PhD 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 PhD 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 PhD
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.
The joint PhD 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 PhD 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.
Course Descriptions
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,
eigen values 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 522 Mathematics for Finance I. Three hours.
Prerequisites: MATH 227 and MATH 355 or with permission of the
instructor.
An introduction to financial engineering and mathematical model in
finance. This course covers basic no-arbitrage principle, binomial
model, time value of money, money market, risky assets such as
stocks, portfolio management, forward and future contracts and
interest rates.
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 and MATH 557, 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.
Designed for graduate students not majoring in mathematics. 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. Credit for this course will not be
counted toward an advanced degree in mathematics.
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/MATH 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. Credit for this course will not
be counted if it is taken after MATH 583.
MATH 588 Theory of Differential Equations I. Three hours.
Prerequisites: MATH 238 and MATH 380 or MATH 580.
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.
|