Department of Mathematics (MATH)

Allan D. Mills, Interim Chairperson

DEPARTMENTAL FACULTY: Rafal F. Ablamowicz; Michael R. Allen; Andrzej Gutek; Andrew Hetzel; Richard C. Le Borne; Sabine Le Borne; Yung-Way Liu; Motoya Machida; Claude E. McHenry; Allan D. Mills; Jeffrey Norden; Brian M. O'Connor; Richard P. Savage, Jr.; Alexander Shibakov; David Smith.

DEPARTMENTAL OVERVIEW

The Department of Mathematics offers a comprehensive program leading to a Master of Science degree in Mathematics. The program of study provides suitable preparation for teaching, government, industrial work, or further study. The moderate size of the program encourages close contact between the faculty and students, thus providing a rare opportunity for students to tailor their programs of study based on individual background, interest, and goals. Graduate students compete for the Stanley Dolzycki Memorial Scholarship and the Graduate Student Mathematics Teaching Award. They participate in a weekly Graduate Seminar and develop teaching skills through participating in the Graduate Teaching seminar also held weekly. For more information, please contact the Mathematics Department at (931) 372-3441, or visit the departmental web page at http://www.math.tntech.edu.

Integrated B.S./M.S. Program

The "Accelerate to Masters" (ATM) program is designed to provide an opportunity for promising undergraduate students at TTU to begin their pursuit of an M.S. degree in Mathematics during their senior year at TTU. Although program participants in the ATM program will technically not be admitted as mathematics graduate students, participants will be permitted to enroll for up to nine (9) hours of graduate level Mathematics courses. Because these courses will not count toward their Bachelor's degree, they will be applied to their Master's degree at Tennessee Tech University or transferred to another institution as graduate credit.

Admission criteria will be similar to the traditional graduate program, except that the students will not yet have earned their Bachelor's degree. To be eligible, students must meet the following minimum criteria:

1. Have an over GPA of at least 3.00;
2. Must have satisfactorily completed at least 18 hours of upper-division (3000 or 4000 level) work in Mathematics. This total may include currently enrolled courses.
3. Must have a "B" or better in all upper division Mathematics courses.

Fulfilling the above minimum requirements does not guarantee admission to the ATM program. Students who meet the above minimum admission requirements must apply to the Mathematics Department for admission to the ATM program under the status governed by "Special Standing." The department's graduate committee will review the application and make a decision for approval.

DEPARTMENTAL ADMISSION REQUIREMENTS

Admissions decisions are made by the Department Chairperson, in consultation with the Department’s Graduate Committee, and are based upon the overall record of the applicant. In general, a strong record in undergraduate mathematics is required; however, applicants who show strong promise but lack sufficient undergraduate preparation may be admitted with a Provisional Standing and be required to complete some background courses before having their status upgraded to Full Standing. Please visit the departmental web page http://www.math.tntech.edu/graduate/gradforms.html.

DEPARTMENTAL DEGREE REQUIREMENTS

Requirements for the M.S. degree in Mathematics are:

THESIS OPTION:

1. Three semester hours of 6000-level Algebra.
2. Three semester hours of 6000-level Analysis.
3. Two one-year approved sequences totaling 12 semester hours.
4. A written thesis and six semester hours of thesis credit.
5. A total of 30 semester hours, including at least 21 at the 6000 level.

NON-THESIS OPTION:

All mathematics graduate students pursuing the non-thesis option will be required to complete three (3) credit hours of MATH 6991, Research and Independent Study.

1. Three semester hours of 6000-level Algebra.
2. Three semester hours of 6000-level Analysis.
3. Three one-year approved sequences totaling 18 semester hours.
4. A comprehensive examination on two of the three one-year approved sequences used to fulfill the 18 semester hour requirement. The selection of the two areas of examination will be left to the graduate student and to the graduate student’s advisor, subject to the approval of the student’s Graduate Advisory Committee. The exam will test both the student's knowledge of the subject areas and ability to independently solve problems and prove theorems.
5. MATH 6991--3 hours
6. A total of 33 semester hours, including at least 24 will be at the 6000 level.

List of Approved Sequences

COURSES

MATH 4050 (5050). Number Theory. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. Properties of integers, division algorithms, prime numbers, diophantine equations, congruences. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4110-20 (5110-20). Advanced Calculus I-II. Lec. 3. Rec. 1. Cr. 3.

Prerequisite: MATH 4110 (5110): C or better in MATH 3400 or consent of instructor; MATH 4120 (5120): C or better in MATH 4110 (5110). Rigorous treatment of functions of one and several variables, improper integrals, sequences, infinite series, uniform convergence and applications. Students are expected to improve their ability to work in an abstract setting using precise definitions and formal proofs and to present their work in class. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4210-20 (5210-20). Numerical Analysis I-II. Lec. 3. Cr. 3.

Prerequisite: MATH 4210 (5210): C or better in MATH 1920 (or consent of instructor for MATH 5210); MATH 4220 (5220): C or better in MATH 2120 or consent of instructor. Iterative methods for nonlinear equations, computational error analysis, convergence of iterative techniques, interpolation, numerical differentiation and integration, approximate solutions of initial-value problems, boundary-value problems, and nonlinear systems, direct and iterative methods for linear systems. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4250-60 (5250-60). Advanced Ordinary Differential Equations I-II. Lec. 3. Cr. 3.

Prerequisite: MATH 4250 (5250): C or better in MATH 2110 and MATH 2120 (or consent of instructor for MATH 5250); MATH 4260 (5260): C or better in MATH 4250 (5250). Systems of ordinary differential equations, matrix methods, approximate solutions, stability theory, basic theory of nonlinear equations and differential systems, trajectories, phase space stability, construction of Liapunov functions. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4310-20 (5310-20). Introduction to Topology I-II. Lec. 3. Cr. 3.

Prerequisite: MATH 4310 (5310): C or better in MATH 3400 (or consent of instructor for MATH 5310); MATH 4320 (5320): C or better in MATH 4310 (5310). Topological spaces, continuity, connectedness, compactness, separation axioms, function spaces, and fundamental groups. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4350 (5350). Introductory Combinatorics. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 3400 or consent of instructor. Topics to be covered include permutations, combinations, multisets, partitions, recurrence relations, generating functions, and the principle of inclusion-exclusion. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4360 (5360). Graph Theory. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 3400 or consent of instructor. Fundamental concepts of undirected and directed graphs, trees, connectivity, traversability, planarity, colorability, network flows, and matching theory. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4410 (5410). Differential Geometry. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 2010, 2110, and 3400 (or consent of instructor for MATH 5410). Geometry of curves and surfaces in three-dimensional space. Calculus on surfaces, curvature and Riemannian geometry. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4470-80 (5470-80). Probability and Statistics I-II. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 2110 or consent of instructor. Mathematical foundations of elementary statistical methods, application and theory, probability in discrete and continuous distribution, correlation and regression, sampling distributions, significance tests. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4510 (5510). Advanced Mathematics for Engineers. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 2120 and MATH 2120. Fourier series, Sturm-Liouville problems, orthogonal functions, Legendre polynomials, Bessel functions, separable partial differential equations (e.g., heat, wave, and Laplace equations), and other topics. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4530-40 (5530-40). Linear Algebra I-II. Lec. 3. Cr. 3.

Prerequisite: MATH 4530 (5530): C or better in MATH 2010 and MATH 3400; MATH 4540 (5540): C or better in MATH 4530 (5530). A theoretical study of vector spaces, bases and dimensions, subspaces, linear transformations, dual spaces, eigenvalues and eigenvectors, inner product speaces, spectral theory, duality, quadratic and bilinear forms. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4610 (5610). History of Mathematics I. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 3400 (or consent of instructor for MATH 5610). The development of mathematics and its relation to the development of civilization prior to the beginnings of calculus. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4620 (5620). History of Mathematics II. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 3400 (or consent of instructor for MATH 5620). History of mathematics from the beginnings of calculus through the modern times. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4710 (5710). Vector Analysis. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 2110. The algebra and the differential and integral calculus of vectors; applications to geometry and mechanics. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4750 (5750). Category Theory of Sets. Lec. 3. Cr. 3.

Prerequisites: C or better in MATH 3400 (or consent of instructor for MATH 5750). Abstracts sets and mappings, categories, sums, universal property, monomorphisms and parts, finite inverse limits, colimits, epimorphisms, the Axiom of Choice, mapping sets and exponentials, covariant and contravariant functionality of function spaces, Cantor's diagonal argument, powers sets, variable sets, models of additional variation, selected applications. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4850 (5850). Computational Algebraic Geometry I . Lec. 3. Cr. 3.

Prerequisites: C or better in MATH 2010, and C or better in MATH 3400 or equivalent; or consent of instructor. Additional recommended prerequisite: MATH 3510 or any other 4000/5000 level mathematics course in which proofs are required. Affine varieties and polynomial ideals. Groebner bases, elimination theory, Hilbert's Nullstellensatz, Zariski closure, decomposition into irreducible varieties.

MATH 4860 (5860). Computational Algebraic Geometry II. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 4850. Polynomial and rational functions on a variety, projective varieties, the dimension of a variety, selected applications in robotics, automatic theorem proving, and invariant theory of finite groups.

MATH 4910-20 (5910-20). Directed Readings. Cr. 1-3.

Prerequisite: Consent of instructor. These courses provide an opportunity for individual reading and study under the supervision of a qualified staff member. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 4950 (5950). Topics in Mathematics. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. A formal course in any area where there is no other course offering. May be taken more than once, provided that the topic is different. Students enrolled in the 5000-level course will be required to complete additional work as stated in the syllabus.

MATH 6010-20. Functional Analysis I-II. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 4120(5120) or consent of instructor. Metric spaces, normed and Banach spaces, inner product and Hilbert spaces. Fundamental theorems for normed and Banach spaces and their applications. Linear operators on normed and Hilbert spaces.

MATH 6070-80. Applied Linear Statistical Methods I-II. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. Regression analysis, correlation, analysis of variance, experimental designs.

MATH 6110-20. Abstract Algebra I-II. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 3520 or consent of instructor. An extensive treatment of groups, semigroups, integral domains, rings and ideals, fields, and Galois fields.

MATH 6150. Mathematical Modeling. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. Applications of mathematics to real world problems with emphasis on problem definition, research, solution, and written report presentation.

MATH 6170-80. Experimental Design I-II. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. Introduction to basic concepts of experimental design, fundamental assumptions in analysis of variance, multiple comparison tests, complete randomized design, general linear model approach to ANOVA, various experimental designs, incomplete block designs, factorial experiments, fractional factorial experiments, response surface methods, repeated measure designs.

MATH 6210-20. Topology I-II. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 4320 (5320) or consent of instructor. Topics in point-set topology, homotopy theory, triangulated spaces, homology theory, other topics in topology.

MATH 6270. Mathematical Statistics. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. Statistical hypothesis, uniform most powerful tests, sufficient statistics, completeness, Roa-Cramer Inequality, sequential probability ratio test, analysis of variance, multiple comparisons, nonparametric techniques.

MATH 6310-20. Complex Analysis I-II. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 4120 (5120) or consent of instructor. Complex numbers, calculus of complex variables, analytic function. Cauchy's Theorem and complex integration, power series including Taylor's and Laurent's, residue theory with applications, conformal mapping with physical applications.

MATH 6370-80. Probability Theory and Stochastic Processes I-II. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 4480 (5480) or consent of instructor. Probability theory of sets, random variable distribution and characteristic functions, convergence, limits and law of large numbers, convolutions, compound distribution, recurrent events, random walk models, Markov chains, homogeneous, nonhomogeneous, and queuing processes.

MATH 6410-20. Real Analysis I-II. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 4120 (5120) or consent of instructor. Theory of Lebesque measure and integration, Lp spaces. Integration in locally compact space.

MATH (CSC) 6450. Advanced Theory of Computation. Lec. 3. Cr. 3.

Prerequisite: Consent of the instructor (previous coursework involving proofs and some programming experience are needed). A rigorous treatment of the theory of computation. Topics such as: computable functions, the Church-Turing thesis, complexity theory, and P vs NP.

MATH (CSC) 6460. Computational Methods for Graphics and Modeling. Lec. 3. Cr. 3.

Prerequisite: Consent of the instructor (previous coursework involving proofs and some programming experience are needed). Mathematical methods for graphics and modeling. Topics such as: 3-D transformations, ray tracing, rendering, image processing, and compression.

MATH 6510. Finite Difference Solutions of Partial Differential Equations. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 4510 (5510) or consent of instructor. Approximate solutions of boundary and initial value problems using the finite difference method. Elliptic, parabolic, and hyperbolic PDE's. Numerical differentiation. Solution methods for linear systems.

MATH 6520. Finite Element Solutions of Partial Differential Equations. Lec. 3. Cr. 3.

Prerequisite: C or better in MATH 4510 (5510) or consent of instructor. Mathematical foundations of the finite element method. Approximate solutions of PDE's. Polynomial interpolation. Variational techniques. Numerical integration. Solution methods for linear systems. Isoparametric technique.

MATH 6530. Integral Equations and Applications. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. Voilterra and Fredholm equations. Green function, Hilbert-Schmidt and Fredholm theories. Neumann series, iterative methods.

MATH 6540. Calculus of Variations and Applications. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. Euler equation, constraints, Lagrange multipliers, Ritz method, applications.

MATH 6610. Operational Mathematics. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. Integral transforms (Laplace, Fourier) inversion and convolution theorems, applications.

MATH 6810. Partial Differential Equations. Lec. 3. Cr. 3.

Prerequisite: Consent of instructor. First and second order PDE's, wave, heat, and Laplace's equations, applications to boundary and eigen-value problems of mathematics, physics, and engineering.

MATH 6900. Mathematics Seminar. Lec. 1. Cr. 0-1.

MATH 6910-20. Special Topics in Mathematics. Cr. 1-3.

Prerequisite: Consent of instructor. Individual study of advanced mathematical topics in fields of interest under the supervision of a qualified staff member.

MATH 6990. Research and Thesis. Cr. 3,6.

MATH 6991. Research and Independent Study. Lec. 1-3. Cr. 1-3.

Prerequisite: Consent of instructor. The purpose of this course is to foster research and independent study at the graduate level in mathematics or statistics. Students will independently study a chosen area of mathematics, explore open and significant problems, draw conclusions, and, if applicable, participate in problem solving via consulting. Students will be required to give presentations on their own investigations and conclusions.

Page last updated: 5/3/08

Graduate Studies
Tennessee Tech Univesity
P.O. Box 5036
Phone: (931) 372-3233 Fax: (931) 372-3497
E-Mail: Gradstudies@tntech.edu