6.14
DEPARTMENT OF MATHEMATICS (MATH)
Chairperson: Professor David CruzUribe,
Office: 345 Gordon Palmer Hall
Graduate Director: Professor Vo Liem,
Office: 332C 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 twocourse
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 fulltime 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 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 nonEuclidean geometry
are also 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 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
leastsquares 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, 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 initialvalue problems for ordinary
differential equations.
MATH 512 Numerical Analysis II. Three hours.
Prerequisite: MATH 511.
Continuation of MATH 511 with emphasis on numerical methods for
solving partial differential equations. Also covers leastsquares
problems, RayleighRitz method, and numerical methods for
boundaryvalue problems.
MATH 520 Linear Optimization. Three hours.
Prerequisite: C or higher in 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 noarbitrage principle, binomial
model, time value of money, money market, risky assets such as
stocks, portfolio management, forward and future contracts and
interest rates.
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: C or higher in MATH 343.
Emphasis on boundaryvalue problems for classical partial
differential equations of physical sciences and engineering. Other
topics include boundaryvalue 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 instructor.
Introduction to complex variable methods, integral transforms,
asymptotic expansions, WKB method, matched asymptotics, and boundary
layers.
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
NeymanPearson 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 NeymanPearson lemma,
likelihood ratio tests, chisquare 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 486 or MATH 586.
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 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, continuoustime 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 486 or MATH 586 or equivalent.
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: Prerequisites: MATH 486 or MATH 586 or equivalent.
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, semidirect 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 486 (formally 380).
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 CauchyRiemann 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
586 – Introduction to Real Analysis I. Three hours.
Prerequisites: MATH 237
MATH
587 – Introduction to Real Analysis II.
Three hours.
Prerequisites: MATH 486 or MATH 586.
MATH 588 Theory of Differential Equations I. Three hours.
Prerequisites: MATH 238 and MATH 486 or MATH 586.
Topics covered include existence and uniqueness of solutions, Picard
theorem, homogenous linear equations, Floquet theory, properties of
autonomous systems, PoincareBendixson theory, stability, and
bifurcations.
MATH 591 Teaching CollegeLevel Mathematics. Three hours.
Prerequisite: Permission of the instructor or the department.
Provides a basic foundation for teaching collegelevel mathematics;
to be taken by graduate students being considered to teach
undergraduatelevel mathematics courses.
MATH 598 Research Not Related to Thesis. Three to nine hours.
MATH 599 Thesis Research. One 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 661 Algebraic Topology I. Three hours.
Prerequisite: MATH 566 or equivalent.
Indepth study of homotopy and homology. The theory of cohomology is
also introduced as are characteristic classes.
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 Ktheory, and theory of polycyclic groups.
MATH 681 Real Analysis II. Three hours.
Prerequisite: MATH 580 or permission of the
department.
Topics covered include basic theory of LP spaces, convolutions, Hahn
decomposition, the RadonNikodym 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, HahnBanach theorem,
openmapping 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 698 Research Not Related to Dissertation.
One to nine hours.
MATH 699 Dissertation Research. One to twelve hours.
