Mathematics

## Course Descriptions and Syllabi

**Undergraduate Course Descriptions and Syllabi**

The Mathematics Department offers a variety of entry-level courses. The prerequisite
for each is a minimum of two years of high school algebra and one year of high school
geometry. In addition, prerequisites for MATH 1730-Precalculus and MATH 1910-Calculus
I include trigonometry. For certain pairs of courses, e.g., 1710 and 1730, credit
is not given for both. The entry-level course for students planning to major in Engineering,
Mathematics, Physics, or other technical areas, but who lack the necessary preparation
for Calculus, is MATH 1730-Precalculus Mathematics. The prerequisites for this course
are 2 years of high school algebra, one year of high school geometry, and at least
12 weeks of high school trigonometry (or equivalent).**NOTE:** All mathematics prerequisite courses must be completed with a grade of "C" or better.
In addition, students cannot receive credit for a 1000-level mathematics course if
that course is a prerequisite for any mathematics course that has been completed with
a grade of "C" or better.

**NOTE:** The ** u** designation before a course listing indicates that the course meets Tennessee Tech
General Education requirements.

**NOTE:** **Graduate courses** are listed below the undergraduate courses.

**NOTE:** Please consult our** Course Offerings Page **when planning a schedule of coursework.

* MATH 1000. Transitional Algebra.* Lab 3. Credit 3.

**Syllabus for MATH 1000**

Prerequisite: ACT mathematics score greater than or equal to 19; or a Compass score of greater than or equal to 28; or completions of Learning Support Mathematics Competencies 1 thru 5, or equivalent. Exponents and roots; polynomial, rational, radical, and absolute value expressions; factoring; linear equations and inequalities; quadratic equations; graphing; functions.

* uMATH 1010. Math for General Studies.* Lec. 3. Credit 3.

**Syllabus for MATH 1010**

Mathematics as applied to real-life problems selected from such topics as preference schemes for voting, fair division and apportionment methods, routing and scheduling problems, analysis of graphs, growth and symmetry, and counting problems.

*Rec. 2. Credit 1.*

**MATH (CSC, PHYS) 1020. First-Year Connections.****Syllabus for MATH 1020**

This course is intended as a bridge course for students entering Tennessee Tech from high school. The course is designed to strengthen the student’s connection to Tennessee Tech, the College of Arts and Sciences, and the appropriate department (CSC, MATH, or PHYS) by focusing on the enhancement of skills needed for academic success. This course engages the student in meaningful academic and non-academic out-of-the-classroom activities, as learning occurs both in and out of the classroom. It emphasizes critical thinking, the formation of academic and social goals and support groups, and time-management and study skills.

*Lec. 3. Credit 3.*

**uMATH 1130. College Algebra.****Syllabus for MATH 1130**

Review of algebra and coordinate geometry; functions; polynomial, rational, exponential, and logarithmic functions; systems of equations; binomial formula; counting (multiplication principle, permutations, and combinations); and conics. Credit towards graduation will not be given for MATH 1130 and MATH 1710 or for MATH 1130 and MATH 1730.

* uMATH 1410. Number Concepts for Teachers.* Lec. 3. Credit 3.

**Syllabus for MATH 1410**

Prerequisite: Admission is restricted to students majoring in Elementary Education. Introduction to sets and operations on sets, properties and operations on whole numbers, and integers, rational and real numbers.

*Lec. 3. Credit 3.*

**MATH 1420. Geometry Concepts for Teachers.****Syllabus for MATH 1420**

Prerequisite: C or better in MATH 1410. Admission is restricted to students majoring in Elementary Education. Introduction to elements of probability and statistics and basic concepts of Euclidean Geometry including congruence, similarity, measurements, areas, and volumes.

*Lec. 3. Credit 3.*

**uMATH 1530. Introductory Statistics.****Syllabus for MATH 1530**

Descriptive statistics including measures of central location and variation, frequency distributions, histograms, and frequency polygons. Probability relating to elementary sample spaces, events, conditional probability, discrete and continuous type random variables, mathematical expectation, and the normal probability distribution. Inferential statistics relating to the confidence intervals and hypothesis tests related to the mean and proportion.

*Lec. 3. Credit 3.*

**uMATH 1630. Finite Mathematics.****Syllabus for MATH 1630**

Brief review of basic algebra; introduction to probability; matrix algebra and linear programming; and applications to business and economics.

*Lec. 3. Credit 3.*

**uMATH 1710. Pre-Calculus Algebra.****Syllabus for MATH 1710**

Review of algebra; relations and functions and their graphs, including polynomial and rational functions; conic sections; inequalities; arithmetic, and geometric sequences and series. Credit will not be given for both MATH 1710 and MATH 1130 or for MATH 1710 and MATH 1730.

*Lec. 3. Credit 3.*

**uMATH 1720. Pre-Calculus Trigonometry.****Syllabus for MATH 1720**

Circular functions and radian measure, graphs of the trigonometric functions, trigonometric identities, and equations, the inverse trigonometric functions, polar coordinates. Applications involving triangles, vectors in the plane and complex numbers. Credit will not be given for both MATH 1720 and MATH 1730.

*Lec. 5. Credit 5.*

**uMATH 1730. Pre-Calculus Mathematics.****Syllabus for MATH 1730**

Prerequisites: Two years of high school algebra, one year of high school geometry, and 12 weeks of trigonometry. Review of algebra and trigonometry; relations and functions and their graphs, including polynomial and rational functions; conic sections; inequalities; polar coordinates; complex numbers; and advanced topics in algebra. Credit will not be given for both MATH 1730 and any of MATH 1710 and MATH 1720.

*Lec. 3. Credit 3.*

**uMATH 1830. Applied Calculus.****Syllabus for MATH 1830**

Prerequisites: ACT mathematics score of 25 or above and three years of high school mathematics, including algebra and geometry; or, special permission of the Mathematics Department; or, C or better in MATH 1130 or MATH 1710 or equivalent. A survey of limits, continuity, and the differential and integral calculus with applications in business, economics and the life sciences.

* uMATH 1831. Further Topics in Applied Calculus.* Lec. 1. Credit 1.

**Syllabus for MATH 1831**

Corequisite: MATH 1830. Includes systems of linear equations, linear programming, exponential and logarithmic equations, partial differentiation, separable and linear differential equations. This course is designed to enhance students’ understanding of calculus and its applications to Economics.

* uMATH 1845. Technical Calculus.* Lec. 3. Credit 3.

**Syllabus for MATH 1845**

Prerequisites: ACT mathematics score of at least 25 and four years of high school mathematics, including algebra, geometry, trigonometry, and advanced or pre-calculus mathematics; or, special permission of the Mathematics Department; or, C or better in MATH 1730; or, C or better in MATH 1710 and 1720 or equivalent. A survey of differential and integral calculus of functions of a single variable including transcendental functions.

*Lec. 4. Credit 4.*

**uMATH 1910. Calculus I.****Syllabus for MATH 1910**

Prerequisites: ACT mathematics score of 27 or above and four years of high school mathematics, including algebra, geometry, trigonometry, and advanced or pre-calculus mathematics, or special permission of the Mathematics Department; or C or better in MATH 1730; or C or better in MATH 1720 and MATH 1710 or equivalent. Limits, continuity, derivatives and integrals of functions of one variable. Applications of differentiation and introduction to the definite integral.

*Lab. 1. Credit 0.*

**MATH 1911. Calculus I Honors Seminar.****Syllabus for MATH 1911**

Co-requisite: Concurrent enrollment in MATH 1910. An ACT score of 30 or higher is also recommended. Selected topics to add depth to the understanding of the material in MATH 1910. Honors students can receive honors credit for MATH 1910 by successfully completing both MATH 1910 and MATH 1911.

*Lec. 4. Credit 4.*

**MATH 1920. Calculus II.****Syllabus for MATH 1920**

Prerequisite: C or better in MATH 1910; or equivalent AP credit for MATH 1910. Integration techniques, applications of the definite integral, polar coordinates, parametric equations, sequences, and series.

*Lab. 1. Credit 0.*

**MATH 1921. Calculus II Honors Seminar.****Syllabus for MATH 1921**

Prerequisite: MATH 1911 or permission of the instructor. (A grade of "A" in MATH 1910 is recommended for students not taking 1911).

Co-requisite: Concurrent enrollment in MATH 1920. Selected topics to add depth to the understanding of the material in MATH 1920. Honors students can receive honors credit for MATH 1920 by successfully completing both MATH 1920 and MATH 1921.

*Lec. 3. Credit 3.*

**MATH 2010. Introduction to Linear Algebra.****Syllabus for MATH 2010**

Prerequisite: C or better in MATH 1910. Systems of linear equations, matrix algebra, inverses, matrix factorizations, determinants, vector spaces and dimension, rank, linear transformations, eigenvalues, and eigenvectors, inner product, orthogonal projections.

*Lec. 4. Credit 4.*

**MATH 2110. Calculus III.****Syllabus for MATH 2110**

Prerequisite: C or better in MATH 1920; or equivalent AP credit for MATH 1910 and MATH 1920. Analytic geometry and vectors, differential calculus of functions of several variables, multiple integration, and topics from vector calculus.

*Lec. 3. Credit 3.*

**MATH 2120. Differential Equations.****Syllabus for MATH 2120**

Prerequisite: C or better in MATH 1920. First order equations, linear equations of higher order, power series solutions (including Frobenius method), Laplace transforms, other topics. It is recommended but not required that students take MATH 2010 before taking MATH 2120.

*Lec. 3. Credit 3.*

**MATH 2610. Discrete Structures.****Syllabus for MATH 2610**

Prerequisite: C or better in MATH 1920. Topics to be chosen from algebra of sets and relations, functions, algebras, graphs and digraphs, monoids and machines, groups and subgroups, computer arithmetic, binary codes, logic, and languages.

*Lec. 1. Credit 1.*

**MATH 3000. Selected Topics in Mathematics.**Prerequisite: C or better in MATH 1920 and consent of instructor. Lectures on and discussion of topics from upper level mathematics to be selected by the instructor in a setting with less structure than in a traditional class.

*Lec. 3-3. Credit 3-3.*

**MATH 3070-3080. Statistical Methods I-II.****Syllabus for MATH 3070, Syllabus for MATH 3080**

Prerequisite: MATH 3070: Recommended C or better in MATH 1130; MATH 3080: C or better in MATH 3070. Introduction to parametric statistical methods with some non-parametric alternatives, sampling, probability, Type I and Type II error, sample size estimation, confidence interval estimation, test of hypotheses using normal, Student's t, Snedecor's F, Chi-square and the binomial distributions, linear regression, analysis of variance, and data analysis utilizing statistical software.

*Lec. 2. Rec. 2. Credit 3.*

**MATH 3400. Introduction to Concepts of Mathematics.****Syllabus for MATH 3400**

Prerequisite: C or better in MATH 1920. A rigorous treatment of elements of logic and set theory including propositional calculus (statements, connectives, conditionals, and negation), quantifiers, sets and operations on sets, mappings, equivalence relations, and mathematical induction. Students are expected to work in an abstract setting using precise definitions and formal proofs.

*Lec. 3. Credit 3.*

**MATH 3430. College Geometry.****Syllabus for MATH 3430**

Prerequisite: C or better in MATH 3400. A rigorous development of geometry from first concepts using the metric approach. Topics include constructions and hyperbolic geometry.

*Lec. 3. Credit 3.*

**MATH 3470. Introductory Probability and Statistics.****Syllabus for MATH 3470**

Prerequisite: C or better in MATH 1920. Probability, random variables, discrete and continuous distributions and their simulation, elementary sampling theory, and estimation with an overall emphasis on simulation of random processes (Not allowed for mathematics majors after having taken MATH 4480.)

*Lec. 2. Credit 2.*

**MATH 3670. Theory and Applications of Random Signals.****Syllabus for MATH 3670**

Introduction to randomization, unconditional and conditional probability, independence, and concepts of random variables. Distributions and density functions, moments and moment generating functions, univariate and multivariate random variables, random process concepts, spectral characteristics of random processes, and linear systems with random inputs.

*Lec. 3. Credit 3.*

**MATH 3810. Complex Variables.****Syllabus for MATH 3810**

Prerequisite: C or better in MATH 2110. Complex numbers, calculus of complex variables, analytic functions, Cauchy's Theorem, series, the Residue Theorem, and applications.

* MATH 3910. Independent Study.* Credit 1-3.

Prerequisite: Consent of instructor. Readings and study under the supervision of a qualified staff member.

* MATH 4010-4020 (5010-5020). Modern Algebra I-II.* Lec. 3-3. Credit 3-3.

**Syllabus for MATH 4010-4020, 5010-5020**

Prerequisite: MATH 4010 - C or better in MATH 2010 and C or better in MATH 3400; MATH 4020 - C or better in MATH 4010. Groups and subgroups including cyclic, abelian, finite; permutation groups; group homomorphisms; cosets and Lagrange's Theorem; normal subgroups and factor groups. Rings including integral domains, unique factorization domains and Euclidean domains, ideals and factor rings, ring homomorphisms, fields and their extensions, geometric constructions.

*Lec. 3. Credit 3.*

**MATH 4050 (5050). Number Theory.****Syllabus for MATH 4050-5050**

Prerequisite: C or better in MATH 3400 or consent of instructor. Properties of integers, division algorithms, prime numbers, diophantine equations, and congruences.

*Lec. 3. Credit 3.*

**MATH 4060 (5060). Topics in Cryptography.****Syllabus for MATH 4060-5060**

Prerequisite: C or better in MATH 2010 and C or better in either MATH 3400 or CSC 2700. Fundamental concepts of cryptography presented with mathematical background (including groups, fields, elements of number theory, probability and statistics). Special attention will be given to the RSA algorithm, Elliptic Curve Cryptography, the ElGamal public key cryptosystem, Diffie-Hellman key exchange and pseudo random number generators.

*Lec. 2-2. Rec. 2-2. Credit 3-3.*

**MATH 4110-4120 (5110-5120). Advanced Calculus I-II.****Syllabus for MATH 4110-4120, 5110-5120**

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.

*Lec. 3-3. Credit 3-3.*

**MATH 4210-4220 (5210-5220). Numerical Analysis I-II.****Syllabus for MATH 4210-4220, 5210-5220**

Prerequisite: MATH 4210 (5210): C or better in MATH 1920; 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, and direct and iterative methods for linear systems.

*Lec. 3-3. Credit 3-3.*

**MATH 4250-4260 (5250-5260). Advanced Ordinary Differential Equations I-II.****Syllabus for MATH 4250-4260-5250-5260**

Prerequisite: MATH 4250 (5250): C or better in MATH 2110 and MATH 2120; 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, and construction of liapunov functions.

*Lec. 3-3. Credit 3-3.*

**MATH 4310-4320 (5310-5320). Introduction to Topology I-II.****Syllabus for MATH 4310-4320, 5310-5320**

Prerequisite: MATH 4310 (5310): C or better in MATH 3400; MATH 4320 (5320): C or better in MATH 4310 (5310). Topological spaces, continuity, connectedness, compactness, separation axioms, function spaces, and fundamental groups.

*Lec. 3. Credit 3.*

**MATH 4350 (5350). Introductory Combinatorics.****Syllabus for MATH 4350-5350**

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.

*Lec. 3. Credit 3.*

**MATH 4360 (5360). Graph Theory.****Syllabus for MATH 4360-5360**

Prerequisite: C or better in MATH 3400 or consent of instructor. Fundamental concepts of undirected and directed graphs, trees, connectivity, traversability, colorability, network flows, and matching theory.

*Lec. 3. Credit 3.*

**MATH 4410 (5410). Differential Geometry.****Syllabus for MATH 4410-5410**

Prerequisites: C or better in MATH 2110, 2010 and 3400. Geometry of curves and surfaces in three dimensional space. Calculus on surfaces, curvature and Riemannian geometry.

*Lec. 3-3. Credit 3-3.*

**MATH 4470-4480 (5470-5480). Probability and Statistics I-II.****Syllabus for MATH 4470-4480, 5470-5480**

Prerequisite: MATH 4470: C or better in MATH 2110; MATH 4480: C or better in MATH 4470. Mathematical foundations of elementary statistical methods, application and theory, probability in discrete and continuous distributions, correlation and regression, sampling distributions, and significance tests.

*Lec. 3. Credit 3.*

**MATH 4510 (5510). Advanced Mathematics for Engineers.****Syllabus for MATH 4510-5510**

Prerequisites: C or better in MATH 2110 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.

*Lec. 3. Credit 3.*

**MATH 4530-4540 (5530-5540). Linear Algebra I-II.****Syllabus for MATH 4530-4540, 5530-5540**

Prerequisites: 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 dimension, subspaces, linear transformations, dual spaces, eigenvalues and eigenvectors, inner product spaces, spectral theory, duality, and quadratic and bilinear forms.

* MATH 4550/5550-4560/5560. Mathematics of Investment I-II.* Lec. 3. Cr. 3.

**Syllabus for MATH 4550/5550, 4560/5560**

MATH 4550/5550: Prerequisite: C or better in MATH 1920 or consent of instructor. Topics include examination of annuities, loans, bonds and other securities, portfolio, immunization, interest rate swaps.

MATH 4560/5560: Prerequisite: C or better in both MATH 4550/5550 and MATH 4470/5470, or consent of instructor. Topics include derivative securities, mathematical models of financial risk management, and corporate finance.

*Lec. 3. Credit 3.*

**MATH 4610 (5610). History of Mathematics I.****Syllabus for MATH 4610-5610**

Prerequisite: C or better in MATH 3400. The development of mathematics and its relation to the development of civilization prior to the beginnings of calculus.

*Lec. 3. Credit 3.*

**MATH 4620 (5620). History of Mathematics II.****Syllabus for MATH 4620-5620**

Prerequisite: C or better in MATH 3400. History of mathematics from the beginnings of calculus through the modern times.

*Lec. 3. Credit 3.*

**MATH 4710 (5710). Vector Analysis.****Syllabus for MATH 4710-5710**

Prerequisite: C or better in MATH 2110. The algebra and the differential and integral calculus of vectors, and applications to geometry and mechanics.

*Lec. 3. Credit 3.*

**MATH 4750 (5750). Category Theory of Sets.****Syllabus for MATH 4750-5750**

Prerequisite: C or better in MATH 3400 (or consent of instructor for MATH 5750). Abstract 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 functoriality of function spaces, Cantor's diagonal argument, power sets, variable sets, models of additional variation, and selected applications.

*Lec. 3. Credit 3.*

**MATH 4850 (5850). Computational Algebraic Geometry I.****Syllabus for MATH 4850-5850**

Prerequisites: C or better in MATH 2010 and C or better in MATH 3400 or equivalent (or consent of instructor for MATH 5850). 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, and decomposition into irreducible varieties.

*Lec. 3. Credit 3.*

**MATH 4860 (5860). Computational Algebraic Geometry II.****Syllabus for MATH 4860-5860**

Prerequisite: C or better in MATH 4850 (5850). 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.

*Credit 1-3.*

**MATH 4910-4920 (5910-5920). Directed Readings.**Prerequisite: Consent of instructor. These courses provide an opportunity for individual reading and study under the supervision of a qualified staff member.

*Lec. 3. Credit 3.*

**MATH 4950 (5950). Topics in Mathematics.**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.

*Lec. 1. Credit 1.*

**MATH 4970. Senior Seminar.****Syllabus for MATH 4970**

Prerequisite: Senior standing. Preparation of papers at an advanced level in mathematics to be presented both in writing and orally.

**MATH 4991, 4992, 4993. Mathematical Research. Credit 1, 2, 3.****Syllabus for MATH 4991, Syllabus for MATH 4992, Syllabus for MATH 4993**

Prerequisite: C or better in MATH 1920 and consent of instructor. This course introduces students to the process of performing research. By reading papers the students will learn how to define open and significant problems, set up a research plan and, if applicable, define relevant experiments. Students will be required to give presentations on either their own or other people's research. These courses can be taken for credit more than once.

**GRADUATE COURSE Descriptions and Syllabi**

* MATH 6001. Communicating Mathematics I. *Lec. 3. Cr. 3.

**Syllabus for MATH 6001**

This course provides mathematics graduate teaching assistants with practical training in the teaching of mathematics.

* MATH 6002. Communicating Mathematics II.* Lec. 3. Cr. 3

**Syllabus for MATH 6002**

This course provides practical training in the writing, typesetting, and oral presentation of mathematics.

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

**Syllabus for MATH 6010-6020**

Prerequisite: MATH 6010: C or better in MATH 4120(5120) or consent of instructor; MATH 6020: C or better in MATH 6010. 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.

**Syllabus for MATH 6070-6080**

Prerequisite: MATH 6070: Consent of instructor; MATH 6080: B or better in MATH 6070. These two courses cover simple linear regression, multiple regression, diagnostic measures, prediction and estimation, polynomial regression, qualitative regression, nonlinear regression, and logistic regression.

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

**Syllabus for MATH 6110-6120**

Prerequisite: MATH 6110: C or better in MATH 4010(5010) or consent of instructor; MATH 6120: C or better in MATH 4020(5020) and C or better in MATH 6110, 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.

**Syllabus for MATH 6150**

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.

**Syllabus for MATH 6170-6180**

Prerequisite: MATH 6170: Consent of instructor; MATH 6180: C or better in MATH 6170. 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.

**Syllabus for MATH 6210-6220**

Prerequisite: MATH 6210: C or better in MATH 4320 (5320) or consent of instructor; MATH 6220: C or better in MATH 6210. Topics in point-set topology, homotopy theory, triangulated spaces, homology theory, other topics in topology.

* MATH 6240-50. Representations and Characters of Groups I -II.* Lec. 3. Cr. 3.

**Syllabus for MATH 6240-6250**

Prerequisite: MATH 6240: C or better in MATH 4010/5010 while C or better in MATH 4530/5530 is recommended, or consent of instructor; MATH 6250: C or better in MATH 6240. FG-modules, reducibility, group algebras, FG-homomorphisms, Maschke’s Theorem, Schur’s Lemma, irreducible modules, characters, inner products of characters, character tables, orthogonality relations. Normal subgroups and lifted characters, tensor products, restriction to a subgroup, induced modules and characters, Frobenius reciprocity relation, applications to group theory such as real representations, groups of order pq, p-groups, characters of GL(2,q), symmetric groups, Burnside’s Theorem, and molecular vibrations

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

**Syllabus for MATH 6270**

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

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

**Syllabus for MATH 6310-6320**

Prerequisite: MATH 6310: C or better in MATH 4120 (5120) or consent of instructor; MATH 6320: C or better in MATH 6310. 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.

**Syllabus for MATH 6370-6380**

Prerequisite: MATH 6370: C or better in MATH 4480(5480) or consent of instructor; MATH 6380: C or better in MATH 6370. 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.

**Syllabus for MATH 6410-6420**

Prerequisite: MATH 6410: C or better in MATH 4120(5120) or consent of instructor; MATH 6420: C or better in MATH 6410. Theory of Lebesgue measure and integration, Lp spaces. Integration in locally compact space.

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

**Syllabus for MATH 6450**

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.

**Syllabus for MATH 6460**

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 6470. Environmental Statistics.* Lec. 3. Cr. 3.

**Syllabus for MATH 6470**

Prerequisite: MATH 6070 or MATH 6170 or their equivalents. This course covers statistical analysis used in environmental modeling. Topics include finite population parameter estimation, spatial sampling techniques, animal population size estimation, variogram estimation, kriging, logistic regression, and survival analysis. Familiarity with computers is necessary. Also necessary is a background in calculus including differentiation and integration of transcendental functions and series.

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

**Syllabus for MATH 6510**

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.

**Syllabus for MATH 6520**

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.

**Syllabus for MATH 6530**

Prerequisite: Consent of instructor. Volterra and Fredholm equations. Green's functions, Hilbert-Schmidt and Fredholm theories. Neumann series, iterative methods.

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

**Syllabus for MATH 6540**

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

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

**Syllabus for MATH 6610**

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

* MATH 6700. Graph Theory.* Lec. 3. Cr. 3.

**Syllabus for MATH 6700**

Prerequisite: C or better in Math 3400 or consent of instructor. Fundamental concepts of undirected and directed graphs, trees, connectivity, traversability, colorability, network flows, matchings and coverings, Ramsey theory, and graph minors.

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

**Syllabus for MATH 6810**

Prerequisite: Consent of instructor. First and second order PDE's, wave, heat, and Laplace's equations, applications to boundary and eigenvalue 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.* Cr. 1-3.

**Syllabus for MATH 6991**

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.