M
a t h e M a t i C s a n ds
t a t i s t i C sSTA 675 Advanced Experimental Design (3) STA 676 Sample Survey Methods (3)
STA 677 Advanced Topics in Data Analysis and Quantitative Methods (3)
STA 711 Experimental Course
Interdisciplinary Electives (3-6 hours)
Student can earn the remaining credits required for the degree either by taking any STA courses at the 500 level or above (except STA 571) or by taking a maxi- mum of six (6) hours of approved graduate courses outside of statistics. Pre-approved interdisciplinary electives are:
CSC 523/524 Numerical Analysis and Computing (3) (3) CSC 526 Bioinformatics (3)
ECO 553 Economic Forecasting (3) ECO 722 Time Series and Forecasting (1-4) ECO 723 Predictive Data Mining (1-4) ERM 669 Item Response Theory (3)
ERM 728 Factor Analysis and Multidimensional Scaling (3) ERM 729 Advanced Item Response Theory (3)
ERM 731 Structural Equation Modeling in Education (3) HEA 602 Epidemiology (3)
MAT 531 Combinatorial Analysis (3) MAT 541/542 Stochastic Processes (3) (3)
Thesis or Project (Capstone Experience)
Each candidate must elect to prepare a thesis or project. Both options require 33 hours.
Thesis (6 hours)
The candidate may prepare a thesis based on the investigation of a topic in statistics. A thesis director will be appointed by the Department Head after con- sultation with the student and the Director of Gradu- ate Study. Candidates will include 6 hours of thesis (STA 699) or 3 hours of STA 698 and 3 hours of STA 699 in the required 33 hours. An oral examination on the thesis is required.
Project (3 hours)
A candidate who does not prepare a thesis must complete a project under the direction of an advisor chosen by the Director of Graduate Study in consulta- tion with the student. Three hours of STA 698 will be included in the 33 hour program.
PURE MATHEMATICS CONCENTRATION (30-33 HOURS)
The pure mathematics concentration offers a 30-hour thesis option and a 33-hour non-thesis option. At least half the work credited towards the degree must be in 600-level courses: 15 hours for the 30-hour program, and 18 hours for the 33-hour program.
Algebra and Analysis (9 hours)
Each candidate must complete any three of the fol- lowing four courses:
MAT 591 Advanced Modern Algebra (3) MAT 592 Advanced Abstract Algebra (3) MAT 595 Advanced Mathematical Analysis (3) MAT 596 Advanced Mathematical Analysis (3)
(Note: Students who have had appropriate alge- bra or analysis courses as undergraduates may be exempted from this requirement upon approval by the Director of Graduate Study. In this case, these 3, 6, or 9 hours must be replaced by the same number of hours chosen in consultation with the Director of Graduate Study.)
Students who intend to continue in the doctoral program in computational mathematics are strongly advised to complete all four of the above courses.
Core Courses (9 hours)
At least 9 hours of course work must be chosen from the following list. At least 6 of these hours must constitute a complete year-long sequence.
MAT 631, 632 Combinatorics and Graph Theory (3) (3) MAT 647, 648 Linear Algebra and Matrix Theory (3) (3) MAT 688, 689 Mathematical Logic and Axiomatic Set
Theory (3) (3)
MAT 691, 692 Modern Abstract Algebra (3) (3) MAT 693, 694 Complex Analysis (3) (3) MAT 695, 696 Real Analysis (3) (3) MAT 697, 698 General Topology (3) (3)
Electives (6-15 hours)
With prior approval of the Director of Graduate Study, a student will select 6-15 hours of other 500-600 level mathematics courses.
Thesis or Comprehensive Examination (Capstone Experience)
Each candidate may elect to (1) prepare a thesis or (2) pass a written comprehensive examination on his/ her program of course work. The thesis option is a 30 hour program, and the non-thesis option is a 33 hour program.
Thesis (6 hours)
The candidate may prepare a thesis based on the investigation of a topic in mathematics. A thesis director will be appointed by the Department Head after consultation with the student and the Director of Graduate Study. Candidates may include up to 6 hours of thesis (MAT 699) in the required 30 hours. An oral examination on the thesis is required.
Comprehensive Examination
A candidate who does not prepare a thesis must take 33 hours of course work and pass a written comprehensive examination of his/her program. This requirement can be met by passing three of the depart- ment’s doctoral qualifying examinations. Please con- sult with the Director of Graduate Study for further details.
M
a t h e M a t i C s a n ds
t a t i s t i C sMINOR Doctoral Minor in Statistics
Students pursuing a doctorate from other depart- ments may obtain a statistics minor by completing 18 semester hours of graduate level statistics courses.
Required Courses (6 hours)
STA 661 Advanced Statistics in the Behavioral and Biological Sciences I
STA 662 Advanced Statistics in the Behavioral and Biological Sciences II
Electives (12 hours)
Four additional three-hour STA courses, excluding 571, 572, and 580.
PHD Doctor of Philosophy in Computational Mathematics
The PhD in Computational Mathematics requires a minimum of 60 semester hours, including 48 hours of course work in mathematics or related area and 12 hours of dissertation.
Application and Admission
In addition to the application materials required by The Graduate School, applicants must submit a 500-700 word Personal Statement by March 15 to be considered for Fall admission.
Degree Requirements
Course Work (48 hours)
The student selects 48 hours of course work in mathematics and related areas with the approval of the Director of Graduate Study. With the approval of the Director of Graduate Study, up to 18 of the 48 hours may be accepted from UNCG’s MA in mathematics program or from a comparable master’s program.
Qualifying Examinations
Qualifying examinations, covering a student’s chosen field of research and related advanced course work, must be taken after the student has removed any provisions or special conditions attached to ad- mission and should be taken prior to the beginning of the fifth semester. These examinations each cover the material of two courses. The student must pass exami- nations in three of the following five areas.
Algebra—MAT 591 Advanced Modern Algebra, MAT 592 Ad- vanced Abstract Algebra
Analysis—MAT 595 Advanced Mathematical Analysis, MAT 596 – Advanced Mathematical Analysis
Topology—MAT 697 General Topology, MAT 698 General Topol- ogy
Combinatorics—MAT 631 Combinatorics, MAT 632 Graph Theory
Numerical Mathematics—MAT 623 Numerical Mathematics, MAT 624 Numerical Mathematics
Programming Project
The student must complete a programming project of such quality that it can become part of a computer algebra system, could be distributed as a package for a computer algebra system, or yields new mathematical data.
Other Reviews and Examinations
When a student has passed the qualifying exami- nation, and completed the programming project successfully, the student submits a dissertation research proposal, which is defended before the dissertation advisor and the advisory committee in an oral examination. After passing this examination, the student may then make a formal application to the Graduate School for admission for candidacy for a doctoral degree.
The advisory committee shall examine the disser- tation, and no dissertation shall be accepted unless it secures unanimous approval of the advisory com- mittee. The doctoral candidate who has successfully completed all other requirements for the degree will be scheduled by the chair of the advisory committee to take a final oral examination.
Schedule for Examinations and Projects
Semester Examination or Project
1-5 3 written qualifying examinations
2-6 Programming project
3-7 Dissertation proposal (oral examination) 6-14 Dissertation defense (oral examination)
Dissertation (12 hours)
M
a t h e M a t i C s a n ds
t a t i s t i C s503 Problem Solving in Mathematics (3:3)
Pr. grade of at least C in 191 and 303 or permission of instructor
Investigates the nature of problem solving, covers procedures involved in problem solving, develops individual problem solving skills, and collects a set of appropriate problems. Required for middle grades mathematics concentration. This course can not be applied toward the requirements for the MA degree in mathematics.
504 Foundations of Geometry (3:3)
Pr. grade of at least C in 292 or permission of instructor
Primarily for students seeking teacher certification. Includes log- ic and axiom systems, history, plane and solid Euclidean geom- etry, proof strategies, introduction to non-Euclidean geometries, and transformational geometry. This course can not be applied toward the requirements for the MA degree in mathematics.
505 Foundations of Mathematics (3:3)
Pr. grade of at least C in 292 or 303 or permission of instructor
Primarily for students seeking teacher certification. Includes properties and algebra of real numbers; analytic geometry; polynomial, rational, exponential, logarithmic, and trigonomet- ric functions; complex numbers; concept of limits of functions. This course can not be applied toward the requirements for the MA degree in mathematics.
513 Historical Development of Mathematics (3:3)
Pr. grade of at least C in 292
Study of the historical development of mathematics—not a his- tory of the persons involved in this development. This course can not be applied toward the requirements for the MA degree in mathematics.
514 Theory of Numbers (3:3)
Pr. grade of at least C in 311
Introduction to multiplicative and additive number theory. Divisibility, prime numbers, congruences, linear and non-linear Diophantine equations (including Pell’s equation), quadratic residues, number-theoretic functions, and other topics.
515 Mathematical Logic (3:3)
Pr. grade of at least C in 311 or 353
Formal languages, recursion, compactness, and effectiveness. First-order languages, truth, and models. Soundness and com- pleteness theorems. Models of theories.
516 Intermediate Abstract Algebra (3:3)
Pr. grade of at least C in 311 or permission of instructor
Rings, integral domains, fields, division algorithm, factoriza- tion theorems, zeros of polynomials, greatest common divisor, formal derivatives, prime polynomials, Euclidean rings, the Fundamental Theorem of Algebra.
517 Theory of Groups (3:3)
Pr. grade of at least C in 311 or permission of instructor
Elementary properties of groups and homomorphisms, quo- tients and products of groups, the Sylow theorems, structure theory for finitely generated Abelian groups.
518 Set Theory and Transfinite Arithmetic (3:3)
Pr. grade of at least C in 311 or 395 or permission of instructor
The axioms of set theory, operations on sets, relations and func- tions, ordinal and cardinal numbers.
519 Topology (3:3)
Pr. grade of at least C in 302
Countability and separation axioms, compact spaces, covering properties, metric spaces, completeness, connectedness, dimen- sion, fundamental group, covering spaces.
520 Non-Euclidean Geometry (3:3)
Pr. grade of at least C in 311 or 395 or permission of instructor
The fifth postulate, hyperbolic geometries, elliptic geometries, the consistency of the non-Euclidean geometries, models for Euclidean and non-Euclidean geometries, elements of inversion.
521 Projective Geometry (3:3)
Pr. permission of instructor
Transformation groups and projective, affine, and metric geom- etries of the line, plane, and space. Homogeneous coordinates, principles of duality, involutions, cross-ratio, collineations, fixed points, conics, ideal and imaginary elements, models, and Euclidean specifications.
522 Introductory Functional Analysis (3:3)
Pr. grade of at least C in 395
Basic concepts in Banach spaces, Hilbert spaces, linear opera- tors and their applications.
525 Intermediate Mathematical Analysis (3:3)
Pr. grade of at least C in 395
Integration, infintie series, sequences and series of functions.
531 Combinatorial Analysis (3:3)
Pr. grade of at least C in 253 or 295 or 311 or 395, or permission of instructor
The pigeon-hole principle, permutations, combinations, gener- ating functions, principle of inclusion and exclusion, distribu- tions, partitions, recurrence relations.
532 Introductory Graph Theory (3:3)
Pr. grade of at least C in 310 and any one of the following courses: 253, 295, 311, 395, 531
Basic concepts, graph coloring, trees, planar graphs, networks.
540 Introductory Complex Analysis (3:3)
Pr. 394, grade of at least C in 395 for mathematics majors
The complex number system, holomorphic functions, power series, complex integration, representation theorems, the calcu- lus of residues.
541, 542 Stochastic Processes (3:3), (3:3)
Pr. grade of at least C in 394 and either 353 or STA 351
Markov processes, Markov reward processes, queuing, deci- sion making, graphs and networks. Applications to perfor- mance, reliability, and availability modeling.
545 Differential Equations and Orthogonal Systems (3:3)
Pr. grade of at least C in 293 and 390 or permission of instructor
An introduction to Fourier series and orthogonal sets of func- tions, with applications to boundary value problems.
546 Partial Differential Equations with Applications (3:3)
Pr. grade of at least C in 545
Fourier integrals, Bessel functions, Legendre polynomials and their applications. Existence and uniqueness of solutions to boundary value problems.
549 Topics in Applied Mathematics (3:3)
Pr. grade of at least C in 293 and 390 or permission of instructor
Selected topics of current interest in applied mathematics. May be repeated for credit with approval of department head.
556 Topics in Discrete Mathematics (3:3)
Pr. grade of at least C in 353 or permission of instructor
Selected topics of current interest in discrete mathematics.
589 Experimental Course
This number reserved for experimental courses. Refer to the Course Schedule for current offerings.