# Mathematics - MATH

### Courses

**MATH 1000/1003 COLLEGE ALGEBRA (3) **LEC. 3. Fundamental concepts of algebra, equations and inequalities, functions and graphs, polynomial and rational functions. Does not satisfy the core requirement in mathematics. Students who have previous credit in any higher-numbered math course may not receive credit.

**MATH 1100 FINITE MATH AND APPLICATIONS (3) **LEC. 3. Pr. A02 score of 22 or S02 score of 520 or MATH 1000 or MATH 1003. Mathematics Core. Overview of finite mathematics and its applications. Graph theory, matrices, finite and conditional probability; descriptive and inferential statistics, voting methods, game theory.

**MATH 1120/1123 PRE-CALCULUS ALGEBRA (3) **LEC. 3. Pr. A02 score of 22 or S02 score of 520 or MATH 1000 or MATH 1003. Mathematics Core. Preparatory course for calculus. Basic analytic and geometric rties of trigonometric functions. Complex numbers, De Moivre's Theorem, polar coordinates. No credit is given to students with higher-numbered math course. Credit for only one of MATH 1120/MATH 1123.

**MATH 1130/1133 PRE-CALCULUS TRIGONOMETRY (3) **LEC. 3. Pr. A02 score of 23 or S02 score of 540 or MATH 1120 or MATH 1123. Mathematics Core. Preparatory course for the calculus sequence. Basic analytic and geometric properties of the trigonometric functions. Complex numbers, de Moivre'e theorem, polar coordinates. Students who have previous credit in any higher-numbered math course may not receive credit. Students may receive credit for only one of MATH 1130/MATH 1133.

**MATH 1150/1153 PRE-CALCULUS ALGEBRA AND TRIGONOMETRY (4) **LEC. 4. Pr. A02 score of 23 or S02 score of 540 or MATH 1000 or MATH 1003. Mathematics Core. Algebraic functions, Exponential Logarithmic functions. Analytic and geometric properties of trigonometric functions. Students with previous credit in any higher-numbered math course may not receive credit. Students may receive credit for only one of MATH 1150/MATH 1153.

**MATH 1151 MATHEXCEL PRECALCULUS WORKSHOP (2) **LEC. 2. SU. Coreq. MATH 1150. Appropriate score on the mathematics placement exam or grade of C or better in MATH 1000. Workshop for MATH 1150. Two 2-hour sessions per week. Fall, Spring.

**MATH 1610/1613 CALCULUS I (4) **LEC. 4. Pr. A02 score of 26 or S02 score of 600 or MATH 1130 or MATH 1133 or MATH 1150 or MATH 1153. Mathematics Core. Limits, the derivative of algebraic, trigonometric, exponential, logarithmic functions. Applications of the derivative, antiderivatives, the definite integral and applications to area problems, the fundamental theorem of calculus. Students may receive credit for only one of MATH 1610/1613/1617/1710.

**MATH 1611 MATHEXCEL CALCULUS WORKSHOP I (2) **LEC. SU. Coreq. MATH 1610. Workshop for Math 1610. Two 2-hour sessions per week.

**MATH 1617 HONORS CALCULUS I (4) **LEC. 4. Pr. Honors College. and a minimum MPP score of 72. and a minimum MPQ score of 100. and a minimum MPA2 score of 72. and MATH 1130 and MATH 1150 and MATH 1133 and MATH 1153 Mathematics Core. This course covers the same material as MATH 1610 but in a greater depth appropriate for honors students. Credit will not be given for both MATH 1617 and MATH 1680. Students may receive credit for only one of MATH 1610, MATH 1617, or MATH 1710.

**MATH 1620/1623 CALCULUS II (4) **LEC. 4. Pr. MATH 1610 or MATH 1613 or MATH 1617 or MATH 1710. Techniques of integration, applications of the integral, parametric equations, polar coordinates. Vectors, lines and planes in space. Infinite sequences and series. Students may receive credit for only one of MATH 1620, MATH 1627, or MATH 1720.

**MATH 1621 MATHEXCEL CALCULUS WORKSHOP II (2) **LEC. 2. SU. Coreq. MATH 1620. Workshop for MATH 1620. Two 2-hour sessions per week.

**MATH 1627 HONORS CALCULUS II (4) **LEC. 4. Pr. Honors College. MATH 1617 The same material as MATH 1620, but in greater depth appropriate for honors students. Students may receive credit for only one of MATH 1620, MATH 1627 or MATH 1720.

**MATH 1680/1683 CALCULUS WITH BUSINESS APPLICATIONS I (4) **LEC. 3, LEC. 2. Pr. A02 score of 25 or S02 score of 580 or MATH 1120 or MATH 1123 or MATH 1130 or MATH 1133 or MATH 1150 or MATH 1153. Students in College of Business. Mathematics Core. Differentiation and integration of exponential and logarithmic functions, applications to business. Functions of several variables, partial derivatives, multiple integrals.

**MATH 1681 MATHEXCEL BUSINESS CALCULUS WORKSHOP I (2) **LEC. 2. SU. Coreq. MATH 1680. Workshop for MATH 1680. Two 2-hour sessions per week.

**MATH 1690 CALCULUS WITH BUSINESS APPS II (3) **LEC. 3. Pr. MATH 1680 or MATH 1683 or MATH 1610 or MATH 1617 or MATH 1613 or Departmental approval. Probability, random variables, probability distributions. Further topics in calculus: integration, functions of several variables, applications to probability. Applications to business and related areas. Credit will not be given to majors in Engineering or Math or Physics.

**MATH 1691 MATHEXCEL BUSINESS CALCULUS WORKSHOP II (2) **LEC. 2. SU. Coreq. MATH 1690. Workshop for MATH 1690. Two 2-hours sessions per week.

**MATH 1710 CALCULUS FOR ENGINEERING AND SCIENCE I (4) **LEC. 4. Pr. A02 score of 26 or S02 score of 600 or MATH 1130 or MATH 1133 or MATH 1150 or MATH 1153. Credit will be given for only only one of MATH 1610/1613/1617/1710. Credit will not be given for MATH 1680 and MATH 1710. Mathematics Core. Vector algebra, limits, derivatives and antiderivatives of real and vector valued functions and applications. The fundamental theorem of calculus. MATH 1710 and MATH 1720 include and re-order the material of MATH 1610 and MATH 1620, and MATH 1720.

**MATH 1720 CALCULUS FOR ENGINEERING AND SCIENCE II (4) **LEC. 4. Pr. MATH 1710. Exponents and logarithms, separation of variables, L'Hopital's rule. Techniques of integration, work and energy, line integrals, the gradient and directional derivatives, the curl. Credit will be given for only one of MATH 1620, MATH 1627, or MATH 1720.

**MATH 2630 CALCULUS III (4) **LEC. 4. Pr. MATH 1620 or MATH 1627 or MATH 1720. Multivariate calculus: vector-valued functions, partial derivatives, multiple integration, vector calculus. Credit will be given for only one of MATH 2630, MATH 2637, or MATH 2730.

**MATH 2637 HONORS CALCULUS III (4) **LEC. 4. Pr. Honors College. MATH 1627 The same material as MATH 2630, but in greater depth appropriate for honors students. Credit will be given for only one of MATH 2630, MATH 2637, or MATH 2730.

**MATH 2650 LINEAR DIFFERENTIAL EQUATIONS (3) **LEC. 3. Pr. MATH 1620 or MATH 1627 or MATH 1720 or P/C MATH 2630 or P/C MATH 2637 or P/C MATH 2730. First and second order linear differential equations including the solutions by infinite series, applications.

**MATH 2660 TOPICS IN LINEAR ALGEBRA (3) **LEC. 3. Pr. MATH 1620 or MATH 1627 or MATH 1720. Matrices, row-reduction, systems of linear equations, (finite-dimensional) vector spaces, subspaces, bases, dimension, change of basis, linear transformations, kernels, orthogonality, Gram-Schmidt.

**MATH 2730 CALCULUS FOR ENGINEERING AND SCIENCE III (4) **LEC. 4. Pr. MATH 1720. Optimization and Lagrange multipliers. Linear, spherical, cylindrical, polar transformations. The Jacobian. Surface integrals and integrals over solids. Divergence, Stokes' Theorem, Gauss' Theorem. Students may receive credit for only one of MATH 2730, MATH 2630/2637/2730.

**MATH 2790 MATHEMATICS OF INTEREST THEORY (3) **LEC. 3. Pr. MATH 1620 or MATH 1627 or MATH 1720. Mathematical foundations of the theory of interest necessary as preparation for the Society of Actuaries examination on the theory of interest.

**MATH 2850 MATHEMATICS FOR ELEMENTARY EDUCATION I (3) **LEC. 3. Departmental approval. Mathematical insights for elementary school teachers. Sets, the structure of the number system (integers, fraction, decimals). For Elementary Education majors or by

**MATH 2860 MATHEMATICS FOR ELEMENTARY EDUCATION II (3) **LEC. 3. Pr. MATH 2850. Departmental approval. Mathematical insights into measurement and geometry for elementary school teachers. Shapes in two and three dimensions. Similarities, congruences and transformations. For Elementary Education majors or by

**MATH 2870 MATHEMATICS FOR ELEMENTARY EDUCATION III (3) **LEC. 3. Departmental approval. Mathematical insights into probability, data analysis and functions for elementary school teachers. Uncertainty, probability spaces and an introduction to statistics. Relationships, functions and change. For Elementary Education majors or by

**MATH 3010 HISTORY OF MATHEMATICS (3) **LEC. 3. Pr. MATH 1620 or MATH 1627 or MATH 1720. Departmental approval. The evolution of modern mathematics from its motivational roots in the physical sciences; the lives and contributions of outstanding mathematicians; the parallel development of mathematics and western culture.

**MATH 3100 INTRODUCTION TO ADVANCED MATHEMATICS (3) **LEC. 3. Pr. MATH 2630 or MATH 2637 or MATH 2730. Teaching of the fundamental abilities necessary for the pursuance of mathematical studies. Logic and set theory, mathematical induction, basic number theory, basic analysis. Credit will not be given for both MATH 3100 and Math 3710.

**MATH 3710 DISCRETE MATHEMATICS (3) **LEC. 3. Pr. MATH 2660. Methods of proof, induction, counting, inclusion-exclusion, discrete probability, relations, partial orders, graphs, trees, languages, grammars, finite state machines, automata. Credit will not be given for both MATH 3710 and Math 3100.

**MATH 4410 ADVANCED LOGIC (3) **LEC. 3. Pr. MATH 2630 or MATH 2730. Advanced topics in logic. For example: soundness, completeness, incompleteness, set theory, proof theory, odel theory, non-standard logics. May count either MATH 4110 or PHIL 4110.

**MATH 4790 ACTUARIAL SEMINAR IN THE MATHEMATICS OF FINANCE (3) **LEC. 3. Pr. MATH 2790. Intensive seminar in the mathematical aspects of finance, and the theory of interest primarily intended as preparation for the Society of Actuaries Course 2 examination.

**MATH 4820 ACTUARIAL SEMINAR IN PROBABILITY (3) **LEC. 3. Pr. STAT 3600. or equivalent. Intensive seminar in calculus, probability, and risk theory primarily intended as preparation for the Society of Actuaries Course 1 examination.

**MATH 4930 DIRECTED STUDIES (1-3) **IND. Study of individual problems or topics of interest to students. Course may be repeated for a maximum of 3 credit hours.

**MATH 4970 SPECIAL TOPICS (1-4) **IND. Departmental approval. An individual problems course. Each student will work under the direction of a staff member on a problem of mutual interest. Course may be repeated for a maximum of 4 credit hours.

**MATH 4980 UNDERGRADUATE RESEARCH (1-3) **IND. Departmental approval. Directed research in the area of specialty under faculty supervison. Course may be repeated for a maximum of 3 credit hours.

**MATH 4997 HONORS THESIS (1-6) **IND. Pr. Honors College Course may be repeated for a maximum of 6 credit hours. Membership in Honors College.

**MATH 5000 MATH MODELING CONTINUOUS (3) **LEC. 3. Pr. MATH 2650 and MATH 2660. Introduction to mathematical models and related techniques. Includes general principles involving continuous deterministic problems and a detailed, specific term project. : Programming ability.

**MATH 5010 VECTOR CALCULUS (3) **LEC. 3. Pr. (MATH 2630 or MATH 2637 or MATH 2730) and MATH 2660. Departmental approval. Vector-valued functions, vector fields. Gradient, divergence, curl. Integral theorems: Green's Theorem, Stoke's Theorem, Gauss' Theorem. Tensors and differential forms. Applications.

**MATH 5030 COMPLEX VARIABLES WITH APPLICATIONS I (3) **LEC. 3. Pr. MATH 2650. Complex functions and their elementary mapping properties; contour integration and residues; Laurent series; applications to real integrals. MATH 5030-5040 are appropriate for students of engineering or science.

**MATH 5040 COMPLEX VARIABLES WITH APPLICATIONS II (3) **LEC. 3. Pr. MATH 5030. Linear fractional transformations; conformal mappings; harmonic functions; applications to boundary value problems; analytic continuation; entire functions. MATH 5030-5040 are appropriate for students of engineering or science.

**MATH 5050 MATRIX THEORY AND APPLICATIONS (3) **LEC. 3. Pr. MATH 2660. Canonical forms, determinants, linear equations, eigenvalue problems.

**MATH 5060 ELEMENTARY PARTIAL DIFFERENTIAL EQUATIONS (3) **LEC. 3. Pr. MATH 2650. First and second order linear partial differential equations with emphasis on the method of eigenfunction expansions.

**MATH 5120 INFORMATION THEORY (3) **LEC. 3. Pr. MATH 2630 or MATH 2637 or MATH 2730. Information and entropy, information rate optimization and channel capacity, variable-length codes, data compression (Kraft-McMillan inequality, Huffman's algorithm), maximum likelihood decoding, Shannon's Noisy Channel Theorem.

**MATH 5130 CALCULUS OF VARIATION (3) **LEC. 3. Pr. MATH 2650. Fundamental concepts of extrema of functions and functionals; first and second variations; generalizations; sufficient conditions; constrained functionals; the general Lagrange Problem; optimal control.

**MATH 5140 DATA COMPRESSION (3) **LEC. 3. Pr. MATH 1620 or MATH 1627 or MATH 1720. Lossless compression methods, including static, dynamic, and higher order Huffman and arithmetic encoding, interval and recency rank encoding, and dictionary methods; lossy transform methods (JPEG).

**MATH 5150 ALGEBRAIC CODING THEORY (3) **LEC. 3. Linear codes, Hamming and Golay codes, BCH codes, cyclic codes. Random error detection and correction. Burst-error correction. Decoding algorithms. Credit will not be given for both MATH 5150 and MATH 6150/6156.

**MATH 5160 INTRODUCTION TO APPLIED MATHEMATICS (3) **LEC. 3. Pr. MATH 2650 and MATH 2660. Fourier transforms, Z-transforms, Function spaces, eigenfunction methods. Conservation laws, diffusion, equilibrium states, wave phenomena, classical transform methods, energy integrals and conservation, maximum principles.

**MATH 5180 CRYPTOGRAPHY (3) **LEC. 3. Pr. MATH 2660. Classical cryptosystems, the Data Encryption Standard, one-way functions and relevant number theoretic problems (factoring, primality testing, discrete logarithm problem), RSA and other public key cryptosystems.

**MATH 5190 INTRODUCTION TO APPROXIMATION THEORY (3) **LEC. 3. Pr. MATH 2650. Approximation of functions by polynomials, spline functions or trigonometric functions, expansions in series.

**MATH 5200 ANALYSIS I (3) **LEC. 3. Pr. MATH 3100. or equivalent course, subject to departmental approval. Real numbers, infima and suprema; sequences and series of real numbers, convergence and limits, Cauchy sequences and completeness; topology of the real line, Bolzano-Weierstrass and Heine-Borel theorems; real-valued functions of a real variable, continuity and uniform continuity; intermediate¡value and extreme-value theorems. Emphasis on proofs.

**MATH 5210 ANALYSIS II (3) **LEC. 3. Pr. MATH 5200. Sequences and series of functions, modes of convergence, power series, elementary functions; derivatives and antiderivatives, the mean-value theorem; Riemann integration and the Fundamental Theorem of Calculus; R^n and abstract spaces, functions of several variables. Emphasis on proofs.

**MATH 5240 FOURIER ANALYSIS (3) **LEC. 3. Pr. MATH 2650. Convergence and oscillation theorems for Fourier Series. Gibbs phenomenon. Fourier transform. Fast Fourier transform.

**MATH 5280 SYSTEMS OF DIFFERENTIAL EQUATIONS AND APPLICATIONS (3) **LEC. 3. Pr. MATH 2650 and MATH 2660. Linear systems of differential equations, stability, phase portraits; non-linear systems, linearization, qualitative properties of orbits, Poincare-Bendixson Theorem; numerical methods; applications.

**MATH 5300 THEORY OF DIFFERENCE EQUATIONS (3) **LEC. 3. Pr. MATH 2660. Linear difference equations, initial value problems, Green's functions, boundary value problems, systems, periodic solutions, nonlinear difference equations, models.

**MATH 5310 INTRODUCTION TO ABSTRACT ALGEBRA I (3) **LEC. 3. Pr. MATH 3100. Departmental approval. Groups, Groups of Permutations, isomorphisms and homomorphisms; Cyclic Groups, Quotient Groups, The Fundamental Homomorphism Theorem.

**MATH 5320 INTRODUCTION TO ABSTRACT ALGEBRA II (3) **LEC. 3. Pr. MATH 5310. Theory of rings and fields, Ideals and Homomorphisms, Quotient Rings, Rings of Polynomials, Extensions of Fields, Galois Theory.

**MATH 5330 COMPUTATIONAL ALGEBRA (3) **LEC. 3. Pr. MATH 5310. Introduction to computation in multivariate polynomial rings and finite fields. Topics include Groebner bases, Buchberger's Algorithm, kinematic/robotics problems, symbolic manipulation software.

**MATH 5370 LINEAR ALGEBRA (3) **LEC. 3. Pr. MATH 2660. Introduction to the theoretical foundations of Linear Linear Algebra including vector spaces, basis, dimension, linear transformations, fundamental subspaces, matrix representations, eigenvalues, eigenspaces.

**MATH 5380 INTERMEDIATE EUCLIDEAN GEOMETRY I (3) **LEC. 3. Pr. MATH 2630 or MATH 2637 or MATH 2730. Fundamental concepts and theorems of Euclidean geometry, introduction to higher dimensions. Regular polygons and polyhedra, symmetry groups, convexity, geometric extremum problems. Geometric transformations and their invariants.

**MATH 5390 INTERMEDIATE EUCLIDEAN GEOMETRY II (3) **LEC. 3. Pr. MATH 5380. Planar graphs and Euler's theorem. The symmetry group of a set, homotheties and similitudes, path, arcs and length of curves, advanced theorems on the circle.

**MATH 5460 PERTURBATION METHODS I (3) **LEC. 3. Pr. MATH 2660. Analytical solutions of nonlinear problems, ODEs, PDEs, multiple scales, and transcendental equations in engineering, mathematics, and physics using both regular and singular perturbation methods.

**MATH 5470 DYNAMICAL SYSTEMS I (3) **LEC. 3. Pr. MATH 2650. One dimensional dynamics. The logistic equation, bifurcation theory, chaos, hyperbolicity, symbolic dynamics, Sarkovskii's Theorem, maps of the circle, homoclinic points and the theory of kneading sequences.

**MATH 5480 DYNAMICAL SYSTEMS II (3) **LEC. 3. Pr. MATH 5470. Higher dimensional and complex dynamics. Lorenz map, Henon map, toral automorphisms, stable and unstable manifolds, strange attractors, quadratic maps of the complex plane, Julia sets, Mandelbrot set.

**MATH 5500 INTRODUCTION TO TOPOLOGY (3) **LEC. 3. Pr. MATH 3100. Departmental approval. Metric spaces, topological spaces, continuity, compactness, connectedness, product and quotient spaces and local properties.

**MATH 5620 MATHEMATICAL COMPUTATION AND SCIENTIFIC VISUALIZATION (3) **LEC. 3. Pr. MATH 2650. and a programming language, or departmental approval. An introduction to the computational modeling process, numerical programming tools for large-scale scientific computation, parallel and cluster computing, and to scientific visualization techniques.

**MATH 5630 INTRODUCTION TO NUMERICAL ANALYSIS I (3) **LEC. 3. Pr. MATH 2650. Programming ability. Numerical solution of equations, polynomial approximation, numerical differentiation and integration, numerical solutions of ordinary differential equations, error analysis. Written programs using algorithms. Programming ability.

**MATH 5640 INTRODUCTION TO NUMERICAL ANALYSIS II (3) **LEC. 3. Pr. MATH 2660. Programming ability. Numerical solutions of systems of linear equations, numerical computation of eigenvalues and eigenvectors, error analysis. Written programs using the algorithms.

**MATH 5650 THEORY OF NONLINEAR OPTIMIZATION (3) **LEC. 3. Pr. MATH 2650 and MATH 2660. Kuhn-Tucker conditions, quadratic programming, search methods and gradient methods, Lagrangean and penalty function methods.

**MATH 5670 PROBABILITY AND STOCHASTIC PROCESSES I (3) **LEC. 3. Pr. MATH 2630 or MATH 2637 or MATH 2730. Random variables, discrete and absolutely continuous distributions. Poisson process, expectation and conditional expectation. Moment generating functions, limit distributions. Emphasis on probabilistic reasoning and problem solving. Credit will not be given for both MATH 5670 and STAT 5670.

**MATH 5680 PROBABILITY AND STOCHASTIC PROCESSES II (3) **LEC. 3. Pr. MATH 5670 or STAT 5670. Multivariate distributions. Central Limit Theorem, Laplace transforms, convolutions, simulation, renewal processes, Continuous-time Markov Chains, Markov renewal and semi-regenerative processes, Brownian motion and diffusion. Credit will not be given for both MATH 5680 and STAT 5680.

**MATH 5690 INTRODUCTION TO CHAOTIC AND RANDOM PHENOMENA (3) **LEC. 3. Pr. MATH 1620 or MATH 1627 or MATH 1720. Stochastic properties of random phenomena in computational complexity, data analysis, chaotic nonlinear systems. Computer simulation and experimenting within Mathematica, supported by Internet resources. Credit will not be given for both MATH 5690 and STAT 5690. Basic programming.

**MATH 5710 LINEAR OPTIMIZATION (3) **LEC. 3. Pr. MATH 2660. Theory and algorithms for standard linear optimization problems. Simplex algorithm and duality, shortest paths, network flows, min-cost flows and circulations, out-of-kilter method, assignments and matchings.

**MATH 5730 ENUMERATION (3) **LEC. 3. Pr. MATH 2630 or MATH 2637 or MATH 2730. Using generating functions and Polya theory to do sophisticated counting. Permutations and combinations, inclusion-exclusion, partitions, recurrence relations, group actions, Polya theory with applications.

**MATH 5750 GRAPH THEORY (3) **LEC. 3. Pr. MATH 2660. Algorithmic and theoretical aspects of graph theory: matchings, colorings, scheduling problems, Hamilton cycles. Euler tours, spanning trees, network reliability, connectivity, extremal graphs, planar graphs, disjoint paths.

**MATH 5770 COMBINATORIAL DESIGNS (3) **LEC. 3. Pr. MATH 1620 or MATH 1627 or MATH 1720. Latin squares, mutually orthogonal latin squares, orthogonal and perpendicular arrays, Steiner triple systems, block designs, difference sets and finite geometries.

**MATH 5800 ACTUARIAL MATHEMATICS I (3) **LEC. 3. Pr. MATH 2790 and STAT 3600. A development of the mathematical theory of life insurance and annuities. Utility functions, mortality models, life tables, insurance plans, premiums.

**MATH 5810 ACTUARIAL MATHEMATICS II (3) **LEC. 3. Pr. MATH 5800. A development of the mathematical theory of life insurance and annuities. Utility functions, mortality models, life tables, insurance plans, premiums.

**MATH 5840 FOUNDATIONS OF NUMBER THEORY FOR SECONDARY SCHOOL TEACHERS (3) **LEC. 3. Pr. MATH 2630 or MATH 2637 or MATH 2730. Divisibility, Diophantine equations, congruencies.

**MATH 5850 NUMERICAL ANALYSIS FOR SECONDARY TEACHERS (3) **LEC. 3. Pr. MATH 2630 or MATH 2637 or MATH 2730. The numerical solutions of selected problems arising in calculus and algebra along with the programming techniques. Computer familiarity.

**MATH 5860 FOUNDATIONS OF NON-EUCLIDEAN GEOMETRY FOR SECONDARY SCHOOL TEACHERS (3) **LEC. 3. Pr. MATH 2630 or MATH 2637 or MATH 2730. B.L. geometry, hyperbolic geometry, absolute geometry, parallel postulates.

**MATH 5870 FINANCIAL MATHEMATICS (3) **LEC. 3. Pr. MATH 1610 and MATH 1620 and MATH 2650 and STAT 3600. Options and spreads, pricing of such options in accordance with the Black-Scholes Equation, and the binomial pricing model.

**MATH 5970 SPECIAL TOPICS (1-3) **IND. Departmental approval. Topics may vary as needed. Course may be repeated for a maximum of 3 credit hours.

**MATH 6000 MATHEMATICAL MODELING: CONTINUOUS (3) **LEC. 3. Introduction to mathematical models and related techniques. Includes general principles involving continuous deterministic problems and a detailed, specific term-project. Programming ability.

**MATH 6010 VECTOR CALCULUS (3) **LEC. 3. Pr. (MATH 2630 or MATH 2637) and MATH 2660. Departmental approval. Vector-valued functions, vector fields. Gradient, divergence, curl. Integral theorems: Green's Theorem, Stoke's Theorem, Gauss' Theorem. Tensors and differential forms. Applications.

**MATH 6030/6036 COMPLEX VARIABLES WITH APPLICATIONS I (3) **LEC. 3. Complex functions and their elementary mapping properties; contour integration and residues; Laurent series; applications to real integrals. MATH 6030-6040 are appropriate for students of engineering or science.

**MATH 6040 COMPLEX VARIABLES WITH APPLICATIONS II (3) **LEC. 3. Pr. MATH 6030 or MATH 6036. Linear fractional transformations; conformal mappings; harmonic functions; applications to boundary value problems; analytic continuation; entire functions. MATH 6030-6040 are appropriate for students of engineering or science.

**MATH 6050 MATRIX THEORY AND APPLICATIONS (3) **LEC. 3. Canonical forms, determinants, linear equations, eigenvalue problems.

**MATH 6060 ELEMENTARY PARTIAL DIFFERENTIAL EQUATIONS (3) **LEC. 3. First and second order linear partial differential equations with emphasis on the method of eigenfunction expansions.

**MATH 6120 INFORMATION THEORY (3) **LEC. 3. Information and entropy, information rate optimization and channel capacity, variable-length codes, data compression (Kraft-McMillan inequality, Huffman's algorithm), maximum likelihood decoding, Shannon's Noisy Channel Theorem.

**MATH 6130 CALCULUS OF VARIATION (3) **LEC. 3. Pr. MATH 2650. Fundamental concepts of extrema functions and functionals; first and second generalizations; sufficient conditions; constrained functionals; the general Legrange problem; optimal control.

**MATH 6140 DATA COMPRESSION (3) **LEC. 3. Lossless compression methods, including static, dynamic, and higher order Huffman and arithmetic encoding, interval and recency rank encoding, and dictionary methods; lossy transform methods (JPEG).

**MATH 6150/6156 ALGEBRAIC CODING THEORY (3) **LEC. 3. Linear codes, Hamming and Golay codes, BCH codes, cyclic codes. Random error detection and correction. Burst-error correction. Decoding algorithms. Credit will not be given for both MATH 5150 and MATH 6150/6156.

**MATH 6160 INTRODUCTION TO APPLIED MATHEMATICS (3) **LEC. 3. Fourier transforms, Z-transforms, Function spaces, eigenfunction methods. Conservation laws, diffusion, equilibrium states, wave phenomena, classical transform methods, energy integrals and conservation, maximum principles.

**MATH 6180 CRYPTOGRAPHY (3) **LEC. 3. Classical cryptosystems, the Data Encryption Standard, one-way functions and relevant number theoretic problems (factoring, primality testing, discrete logarithm problem), RSA and other public key cryptosystems.

**MATH 6190 INTRODUCTION TO APPROXIMATION THEORY (3) **LEC. 3. Pr. MATH 2650. Approximation of functions by polynomials, spline functions or trigonometric function, expansions in series.

**MATH 6200 ANALYSIS I (3) **LEC. 3. or equivalent course, subject to departmental approval. Real numbers, infima and suprema; sequences and series of real numbers, convergence and limits, Cauchy sequences and completeness; topology of the real line, Bolzano-Weierstrass and Heine-Borel theorems; real-valued functions of a real variable, continuity and uniform continuity; intermediate¡value and extreme-value theorems. Emphasis on proofs.

**MATH 6210 ANALYSIS II (3) **LEC. 3. Pr. MATH 6200. Sequences and series of functions, modes of convergence, power series, elementary functions; derivatives and power series, elementary functions; derivatives and antiderivatives, the mean-value theorem; Riemann integration and the Fundamental Theorem of Calculus; R^n and abstract spaces, functions of several variables. Emphasis on proofs.

**MATH 6240 FOURIER ANALYSIS (3) **LEC. 3. Pr. MATH 6200. Convergence and oscillation theorems for Fourier Series. Gibbs phenomenon. Fourier transform. Fast Fourier transform.

**MATH 6280 SYSTEMS OF DIFFERENTIAL EQUATIONS AND APPLICATIONS (3) **LEC. 3. Linear systems of differential equations, stability, phase portraits; non-linear systems, linearization, qualitative properties of orbits, Poincare-Bendixson Theorem; numerical methods; applications.

**MATH 6300 THEORY OF DIFFERENCE EQUATIONS (3) **LEC. 3. Linear difference equations, initial value problems, Green's functions, boundary value problems, systems, periodic solutions, nonlinear difference equations, models.

**MATH 6310 INTRODUCTION TO ABSTRACT ALGEBRA I (3) **LEC. 3. Departmental approval. Groups, Groups of Permutations, isomorphisms and homomorphisms; Cyclic Groups, Quotient Groups, The Fundamental Homomorphism Theorem.

**MATH 6320 INTRODUCTION TO ABSTRACT ALGEBRA II (3) **LEC. 3. Pr. MATH 6310. Theory of rings and fields, Ideals and Homomorphisms, Quotient Rings, Rings of Polynomials, Extensions of Fields, and Galois Theory.

**MATH 6330 COMPUTATIONAL ALGEBRA (3) **LEC. 3. Pr. MATH 6310. Introduction to computation in multivariate polynomial rings and finite fields. Topics include Groebner bases, Buchberger's Algorithm, kinematic/robotics problems, and symbolic manipulation software.

**MATH 6370 LINEAR ALGEBRA (3) **LEC. 3. Introduction to the theoretical foundations of Linear Algebra Algebra including vector spaces, basis, dimension, linear transformations, fundamental subspaces matrix representations, eigenvalues, eigenspaces.

**MATH 6380 INTERMEDIATE EUCLIDEAN GEOMETRY I (3) **LEC. 3. Fundamental concepts and theorems of Euclidean geometry, introduction to higher dimensions. Regular polygons and polyhedra, symmetry groups, convexity, geometric extremum problems. Geometric transformations and their invariants.

**MATH 6390 INTERMEDIATE EUCLIDEAN GEOMETRY II (3) **LEC. 3. Pr. MATH 6380. Planar graphs and Euler's theorem. The symmetry group of a set, homotheties and similitudes, path, arcs and length of curves, and advanced theorems on the circle.

**MATH 6460/6466 PERTURBATION METHODS (3) **LEC. 3. Pr. MATH 2660. Departmental approval. Analytical solutions of nonlinear problems, ODES, PDEs, multiple scales, and transcendental equations in engineering, mathematics, and physics using both regular and singular perturbation methods. May count either AERO/MATH 5460 or AERO/MATH 6460.

**MATH 6470 DYNAMICAL SYSTEMS I (3) **LEC. 3. Pr. MATH 2650. One dimensional dynamics. The logistic equation, bifurcation theory, chaos, hyperbolicity, symbolic dynamics, Sarkovskii's Theorem, maps of the circle, homoclinic points and the theory of kneading sequences.

**MATH 6480 DYNAMICAL SYSTEMS II (3) **LEC. 3. Pr. MATH 6470. Higher dimensional and complex dynamics. Lorenz map, Henonmap, toral automorphisms, stable and unstable manifolds, strange attractors, quadratic maps of the complex plane, Julia sets, Mandelbrot set.

**MATH 6500 INTRODUCTION TO TOPOLOGY (3) **LEC. 3. Departmental approval. Metric spaces, topological spaces, continuity, compactness, connectedness, product and quotient spaces and local properties.

**MATH 6620 MATHEMATICAL COMPUTATION AND SCIENTIFIC VISUALIZATION (3) **LEC. 3. An introduction to the computational modeling process, numerical programming tools for large-scale scientific computation, parallel and cluster computing, and to scientific visualization techniques.

**MATH 6630/6636 INTRODUCTION TO NUMERICAL ANALYSIS I (3) **LEC. 3. Numerical solution of equations, polynomial approximation, numerical differentiation and integration, numerical solutions of ordinary differential equations, error analysis. Written programs using algorithms. Programming ability.

**MATH 6640/6646 INTRODUCTION TO NUMERICAL ANALYSIS II (3) **LEC. 3. Numerical solutions of systems of linear equations, numerical computation of eigenvalues and eigenvectors, error analysis. Written programs using the algorithms. Programming ability.

**MATH 6650 THEORY OF NONLINEAR OPTIMIZATION (3) **LEC. 3. Kuhn-Tucker conditions, quadratic programming, search methods and gradient methods, Lagrangean and penalty function methods.

**MATH 6670/6676 PROBABILITY AND STOCHASTIC PROCESSES I (3) **LEC. 3. Random variables, discrete and absolutely continuous distributions. Poisson process, expectation and conditional expectation. Moment generating functions, limit distributions. Emphasis on probabilistic reasoning and problem solving. Credit will not be given for both MATH 6670 and STAT 6670.

**MATH 6680 PROBABILITY AND STOCHASTIC PROCESSES II (3) **LEC. 3. Pr. MATH 6670 or STAT 6670. Multivariate distributions. Central Limit Theorem, Laplace transforms, convolutions, simulation, renewal processes, Continuous-time Markov Chains, Markov renewal and semi- regenerative processes, Brownian motion and diffusion. Credit will not be given for both MATH 6680 and STAT 6680.

**MATH 6690 INTRODUCTION TO CHAOTIC AND RANDOM PHENOMENA (3) **LEC. 3. Pr. MATH 1620. Stochastic properties of random phenomena in computational complexity, data analysis, chaotic nonlinear systems. Computer simulation and experimenting within Mathematica, supported by Internet resources. Credit will not be given for both MATH 6690 and STAT 6690.

**MATH 6710 LINEAR OPTIMIZATION (3) **LEC. 3. Theory and algorithms for standard linear optimization problems. Simplex algorithm and duality, shortest paths, network flows, min-cost flows and circulations, out-of-kilter method, assignments and matchings.

**MATH 6730 ENUMERATION (3) **LEC. 3. Using generating functions and Polya theory to do sophisticated counting. Permutations and combinations, inclusion-exclusion, partitions, recurrence relations, group actions, Polya theory with applications.

**MATH 6750 GRAPH THEORY (3) **LEC. 3. Algorithmic and theoretical aspects of graph theory: matchings, colorings, scheduling problems, Hamilton cycles. Euler tours, spanning trees, network reliability, connectivity, extremal graphs, planar graphs, disjoint paths.

**MATH 6770 COMBINATORIAL DESIGNS (3) **LEC. 3. Latin squares, mutually orthogonal latin squares, orthogonal and perpendicular arrays, Steiner triple systems, block designs, difference sets and finite geometries.

**MATH 6800 ACTUARIAL MATHEMATICS I (3) **LEC. 3. Departmental approval. A development of the mathematical theory of life insurance and annuities. Utility functions, mortality models, life tables, insurance plans, premiums.

**MATH 6810 ACTUARIAL MATHEMATICS II (3) **LEC. 3. Pr. MATH 6800. A development of the mathematical theory of life insurance and annuities. Utility functions, mortality models, life tables, insurance plans, premiums.

**MATH 6840 FOUNDATIONS OF NUMBER THEORY FOR SECONDARY SCHOOL TEACHERS (3) **LEC. 3. Divisibility, Diophantine equations, congruencies.

**MATH 6850 NUMERICAL ANALYSIS FOR SECONDARY TEACHERS (3) **LEC. 3. Pr. MATH 2630 or MATH 2637. The numerical solutions of selected problems arising in calculus and algebra along with the programming techniques. Computer familiarity.

**MATH 6860 FOUNDATIONS OF NON-EUCLIDEAN GEOMETRY FOR SECONDARY SCHOOL TEACHERS (3) **LEC. 3. Pr. MATH 2630 or MATH 2637. B.L. geometry, hyperbolic geometry, absolute geometry, parallel postulates.

**MATH 6870 FINANCIAL MATHEMATICS (3) **LEC. 3. Pr. MATH 1610 and MATH 1620 and MATH 2650 and STAT 3600. Options and spreads, pricing of such options in accordance with the Black-Scholes Equation, and the binomial pricing model.

**MATH 6970/6976 SPECIAL TOPICS (1-3) **IND. Departmental approval. Topics may vary as needed. Course may be repeated for a maximum of 3 credit hours.

**MATH 7000/7006 APPLIED MATHEMATICS I (3) **LEC. 3. Departmental approval. Linear spaces, matrices, eigenvalues, least squares solutions to linear systems, Hilbert spaces, orthogonal expansions, integral equations, compact operators, Green's functions for boundary value problems, eigenfunction expansions.

**MATH 7010/7016 APPLIED MATHEMATICS II (3) **LEC. 3. Pr. MATH 7000. Calculus of variations, asymptotic expansions, Spectral theory, Fourier transform, Partial differential equations, transform methods and eigenfunction expansions, vibrations, diffusion processes, equilibrium states, Green's functions, boundary layer problems.

**MATH 7040/7046 APPROXIMATION THEORY I (3) **LEC. 3. Departmental approval. Introduction and theory of some of the important methods of approximation. Includes uniform approximation, best approximation, best trigonometric approximation.

**MATH 7050/7056 APPROXIMATION THEORY II (3) **LEC. 3. Pr. MATH 7040 or MATH 7046. Least square approximation and rational approximation, and advanced topics of current interest.

**MATH 7070 INTERPOLATION I (3) **LEC. 3. Departmental approval. Techniques of approximation by interpolation, rates of convergence and methods of estimating error. Simultaneous approximation of functions and their derivatives; spline function interpolation; curve and surface fitting.

**MATH 7080 INTERPOLATION II (3) **LEC. 3. Pr. MATH 7070. Timan's Theorem. Telyakovski and Gopengauz Theorems. Brudnyi's inequality, Polynomial based discrete cosine transforms, Piecewise polynomial interpolation, B-splines, Recurrence relations, Schoenberg-Whitney Theorem, Convergence properties of spline approximations.

**MATH 7100 SPECIAL FUNCTIONS (3) **LEC. 3. Departmental approval. Special functions from classical complex analysis which play an important role in the mathematics of physics, chemistry, and engineering.

**MATH 7110 DISCRETE GEOMETRY AND CONVEXITY I (3) **LEC. 3. Departmental approval. Geometric objects and configurations with discrete symmetry groups. Regular polygons and polyhedra. Regular arrangements. Plane tilings and patterns.

**MATH 7120 DISCRETE GEOMETRY AND CONVEXITY II (3) **LEC. 3. Pr. MATH 7110. Convexity and related geometric extremum problems. Packing and covering. Arrangements of extreme density.

**MATH 7130 TENSOR ANALYSIS (3) **LEC. 3. Departmental approval. Manifolds, differential structure, vector and tensor fields, vector and tensor bundles, differential forms, chains. elements of differential geometry, advanced topics.

**MATH 7140 INTRODUCTION TO MODEL THEORY (3) **LEC. 3. Departmental approval. First-order languages, Satisfaction. Consequences. The completeness and compactness theorems, models constructed from constants. Elementary substructures and embeddings, Lowenheim-Skolem-Tarskj theorems. Ultraproducts and ultrapowers.

**MATH 7150 AXIOMATIC SET THEORY I (3) **LEC. 3. Departmental approval. Introduction to modern set theory. The axioms of ZFC, ordinals and cardinals, closed unbounded sets, the constructible universe L, Martin's Axiom.

**MATH 7160 AXIOMATIC SET THEORY II (3) **LEC. 3. Pr. MATH 7150. Introduction to forcing, independence results, iterated forcing, consistency of Martin's Axiom.

**MATH 7170 ALGORITHMS DISCRETE OPTIMIZATION (3) **LEC. 3. Pr. MATH 6750. Theory and practice of discrete algorithms: complexity class classes, reductions, approximate algorithms, greedy algorithms, search techniques, heuristics, randomized algorithms, and numeric algorithms.

**MATH 7180 ALGORITHMIC ALGEBRA I (3) **LEC. 3. Departmental approval. Fundamental aspects of algorithmic algebra. Noetherian rings. Theory of Groebner bases. Hilbert Nullstellensatz. Elimination theory. Applications to graph theory and algebraic geometry.

**MATH 7190 ALGORITHMIC ALGEBRA II (3) **LEC. 3. Pr. MATH 7180. Modules. Groebner bases for modules and syzygy computations. Improved Buchberger's algorithm. Computation of Ext. Groebner bases over rings. Primary decomposition of ideals. Dimension theory.

**MATH 7200 REAL ANALYSIS I (3) **LEC. 3. Departmental approval. Sigma algebras, measures, measurable functions, integratability, properties of Lebesgue measure, density, Lusin's theorem, Egeroff's theorem, product measures, Fubini's theorem. Limit theorems involving pointwise convergence and integration.

**MATH 7210 REAL ANALYSIS II (3) **LEC. 3. Pr. MATH 7200. L-p spaces, completeness, duals. Weak convergence, norm convergence, pointwise convergence, convergence in measure. Signed and complex measures. Absolute continuity, Lebesgue decomposition. Measure theory, Lebesgue integration, introductory functional analysis.

**MATH 7230 FUNCTIONS OF A COMPLEX VARIABLE I (3) **LEC. 3. Departmental approval. Complex numbers, analytic functions, derivatives, Cauchy integral theorem and formulae, Taylor and Laurent series, analytic continuation, residues, maximum principles, Riemann surfaces.

**MATH 7240 FUNCTIONS OF A COMPLEX VARIABLE II (3) **LEC. 3. Pr. MATH 7230. Conformal mappings, families of analytic functions and harmonic analysis.

**MATH 7280 ADVANCED THEORY OF ORDINARY DIFFERENTIAL EQUATIONS I (3) **LEC. 3. Departmental approval. Existence and continuation theorems for ordinary differential equations, continuity and differentiability with respect to initial conditions, linear systems, differential inequalities, Sturm theory.

**MATH 7290 ADVANCED THEORY OF ORDINARY DIFFERENTIAL EQUATIONS II (3) **LEC. 3. Pr. MATH 7280. Stability theory, periodic solutions, boundary value problems, disconjugacy of linear equations, Green's functions, upper and lower solutions, a priori bounds methods, current research.

**MATH 7310 ALGEBRA I (3) **LEC. 3. Departmental approval. Groups, Lagrange's Theorem, normal subgroups, factor groups, Isomorphism and Correspondence Theorems. Symmetric groups, alternating groups, free groups, torsion groups. Introduction to rings, correspondence theorems.

**MATH 7320 ALGEBRA II (3) **LEC. 3. Pr. MATH 7310. Rings, modules, vector spaces, and semi-simple modules. Commutative rings; prime and primary ideals, PIDs are UFD, factorizations in integral domains, field extensions, the Galois Correspondence Theorem.

**MATH 7330 LINEAR REPRESENTATIONS OF FINITE GROUPS (3) **LEC. 3. Pr. MATH 7320. Maschke's Theorem, characters, orthogonality relations, induced modules, Frobenius reciprocity, Clifford's Theorem, Mackey's Subgroup Theorem, Burnside's theorem on solvability.

**MATH 7340 RING THEORY (3) **LEC. 3. Pr. MATH 7320. Topics on: commutative rings (Cohen-Seidenberg theorems, Krull Intersection Theorem, Dedekind domains), or noncommutative rings (projective modules over Artinian algebras, representation type, Noether-Skolem Theorem, division algebras).

**MATH 7350 ABELIAN GROUPS (3) **LEC. 3. Pr. MATH 7320. Torsion groups: Decompositions, Ulm's theorem, uniqueness theorem for Axion 3 groups, Torsion-free groups: Completely decomposable groups, Butler groups, p-local groups, Warfield groups, splitting criteria. Homological topics.

**MATH 7360 LIE ALGEBRA (3) **LEC. 3. Pr. (MATH 5370 or MATH 6370) or (MATH 5320 or MATH 6320). General introduction of Lie algebras including their structures and classifications of semisimple Lie algebras.

**MATH 7370 MATRICES I (3) **LEC. 3. Departmental approval. Jordan form, functions of a matrix, spectral theorem, singular values, norms, quadratic forms, field of values, inertia; topics of current interest.

**MATH 7380 MATRICES II (3) **LEC. 3. Pr. MATH 7370. Matrix stability and inertia, inequalities for matrix eigenvalues and singular values, The Kronecker and Hadamard matrix products, the exponential and logarithm matrix map; topics of current interest.

**MATH 7390 MULTILINEAR ALGEBRA (3) **LEC. 3. Pr. MATH 5370 or MATH 6370. Multilinear algebra, symmetry class of tensors, induced operators, generalized matrix functions, and current research topics.

**MATH 7400 FUNCTIONAL ANALYSIS I (3) **LEC. 3. Pr. MATH 7210. Departmental approval. Bounded linear transformations and functionals on Banach and Hilbert spaces, weak topologies, linear operators, adjoints, compact operators. Banach algebras, spectral theory, Gelfand transform.

**MATH 7410 FUNCTIONAL ANALYSIS II (3) **LEC. 3. Pr. MATH 7400. C*-algebras, Hermitian, self adjoint elements, functional calculus for commutative algebras. Normal operators on Hilbert space, spectral theorem, applications, symmetric and self-adjoint operators, normal operators, the spectral theorem.

**MATH 7440 PARTIAL DIFFERENTIAL EQUATIONS I (3) **LEC. 3. Departmental approval. Second order linear elliptic and hyperbolic equations stressing non-linear and numerical problems, characteristics domains of dependence, energy integrals, finite difference schemes, Sobolev spaces, maximum principle.

**MATH 7450 PARTIAL DIFFERENTIAL EQUATIONS II (3) **LEC. 3. Pr. MATH 7440. Parabolic and hyperbolic equations, stressing numerical problems, characteristics, domains of dependence, energy integrals, reaction-diffusion problems, Navier-Stokes equations, fixed-point and Galerkin methods.

**MATH 7460/7466 ADVANCED PERTURBATION METHODS (3) **LEC. 3. Departmental approval. Solutions of nonlinear problems and integrals using WKB, Rayleigh-Janzen, Generalized Scales, Latta, van der Pol, Watson, Laplace, Adomian, homotopy, Pade, Liouville-Green and Burmann transformations. May count either AERO 7460/7466 or MATH 7460/7466.

**MATH 7470 NUMBERS AND OPERATIONS FOR ELEMENTARY TEACHERS (3) **LEC. 3. Certified Elementary Teacher or Degree in Elementary/Early Education or departmental approval. Develop a deep understanding of the base ten place value system and the operations and algorithms used with this system.

**MATH 7480 MEASUREMENT AND GEOMETRY FOR ELEMENTARY TEACHERS (3) **LEC. 3. Certified Elementary Teacher or Degree in Elementary/Early Education or departmental approval. Develop a deep understanding of topics in measurement, 2-D and 3-D geometry needed for teaching elementary school students.

**MATH 7490 DATA, STATISTICS AND FUNCTIONS FOR ELEMENTARY TEACHERS (3) **LEC. 3. Certified Elementary Teacher or Degree in Elementary/Early Education or departmental approval. Develop a deep understanding of topics in data analysis, probability, statistics and algebra needed for teaching elementary school students.

**MATH 7500 TOPOLOGY I (3) **LEC. 3. Departmental approval. Separation and countability axioms, covering properties, completeness, connectedness, metric spaces and metrizability, product and quotient spaces, function spaces.

**MATH 7510 TOPOLOGY II (3) **LEC. 3. Pr. MATH 7500. Homotopy, elementary properties of retracts, fundamental groups, covering spaces, computations of fundamental groups.

**MATH 7520 DIMENSION THEORY (3) **LEC. 3. Pr. MATH 7500 or MATH 6500. Departmental approval. Topological study of dimension in separable metric spaces. Topological invariance of dimension of Euclidean spaces. Dimension and measure.

**MATH 7530 CONTINUUM THEORY I (3) **LEC. 3. Pr. MATH 7510. Departmental approval. Topics such as inverse limits, decompositions, hyperspaces, special mappings, topological structures from the pathological (indecomposable continua), to the straightforward (Peano continua).

**MATH 7540 CONTINUUM THEORY II (3) **LEC. 3. Pr. MATH 7530. Topics in continuum theory such as confluent mappings, epsilon mappings, chains, to-the-boundary theorems, relationship to inverse limits, advanced topics.

**MATH 7550 SET THEORETIC TOPOLOGY I (3) **LEC. 3. Pr. MATH 7510. Departmental approval. Compactifications, covering properties, metrization theorems and generalized metrizable spaces, topological groups.

**MATH 7560 SET THEORETIC TOPOLOGY II (3) **LEC. 3. Pr. MATH 7550. Topological Groups, Cardinal invariants, use of set-theoretic axioms such as Martin's Axiom, independence results, advanced topics.

**MATH 7570 EUCLIDEAN TOPOLOGY I (3) **LEC. 3. Pr. MATH 7510. An introduction to concepts basic in algebraic and geometric topology through the study of simple objects such as polyhedra, manifolds, retracts, and the Brouwer fixed point theorem.

**MATH 7580 EUCLIDEAN TOPOLOGY II (3) **LEC. 3. Pr. MATH 7570. Further study of basic geometric topology. Retracts, absolute neighborhood retracts, maps into spheres, invariance of domain.

**MATH 7600/7606 ADVANCED NUMERICAL MATRIX ANALYSIS (3) **LEC. 3. Departmental approval. Topics selected from: discretization matrices, sparse matrices, QR-algorithm, symmetric eigenvalue problems, singular value decomposition, pseudo-inverses, simplex method, matrix algorithms for vector computers.

**MATH 7610/7616 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS (3) **LEC. 3. Departmental approval. The numerical solution of partial differential equations using finite difference and finite element methods.

**MATH 7620 OPTIMIZATION THEORY (3) **LEC. 3. Unconstrained problems: basic descent, conjugate gradient and quasi-Newton methods. Constrained problems: gradient projection, penalty, cutting plane and Lagrange methods. Credit will not be given for both MATH 7620 and INSY 8420. An ability to program in high- level language.

**MATH 7650 HARMONIC ANALYSIS I (3) **LEC. 3. Departmental approval. Fourier series, Fourier transforms, maximal functions, singular integral theory, introduction to function spaces.

**MATH 7660 HARMONIC ANALYSIS II (3) **LEC. 3. Pr. MATH 7650. Function spaces and interpolation, Calderon's reproducing formulas, wavelets, frames, connections to function spaces applications.

**MATH 7680/7686 ADVANCED TOPICS IN NUMERICAL ANALYSIS (3) **LEC. 3. Departmental approval. Topics include: sparse systems of equations, parallel and vector algorithms, nonlinear and singular partial differential equations, calculation of eigenvalues and eigenvectors, pseudo-random numbers, filtering techniques.

**MATH 7700 ADVANCED GRAPH THEORY (3) **LEC. 3. Algorithmioc, enmerative and theoretical aspects of graph theory: matchings and factors, colorings, Hamiltonicity, connectivity, trees, exterenal graphs, planarity. May count either MATH 6750 or MATH 7700.

**MATH 7710 COMPUTATIONAL GEOMETRY (3) **LEC. 3. Departmental approval. Design and time-complexity of computer algorithms for geometry problems studying the geometric ideas needed for computer-aided design, computer graphics and robotics.

**MATH 7720 INTRODUCTION TO CODING THEORY (3) **LEC. 3. Introduction to methods and algorithms for reliable communications through error control coding. BCH, Reed- Solomon, Reed-muller codes, convolutional codes, Berlekamp-Massey, Viterbi, and iterated decoding algorithms.

**MATH 7730 ADVANCED TOPICS IN CODING THEORY (3) **LEC. 3. Pr. MATH 7720. Departmental approval. Structure and theoretical properties of codes and related algorithms. Relations to other combinatorial and algebraic objects stressed.

**MATH 7740 ADVANCED COMBINATORIAL DESIGNS (3) **LEC. 3. Topics of current interest and research in combinatorial design theory. Areas included: latin squares, embeddings, Wilson's constructions, quadruple systems, Hadamard designs, graph designs, orthogonal arrays.

**MATH 7750 ADVANCED TOPICS IN GRAPH THEORY (3) **LEC. 3. Pr. MATH 6750. Topics of current interest and recent research in graph theory. May include edge colorings, algebraic graph theory, network flows, factor theory.

**MATH 7760 INTRODUCTION TO ALGEBRAIC TOPOLOGY I (3) **LEC. 3. Pr. MATH 7510. Departmental approval. Homology of chain complexes, the axioms of homology and their verification, computations of homology groups.

**MATH 7770 INTRODUCTION TO ALGEBRAIC TOPOLOGY II (3) **LEC. 3. Pr. MATH 7760. Homology with coefficients and universal coefficient theorem theorems, Cohomology and universal coefficient theorems, homology of products of spaces, cup and cap products, duality in manifolds.

**MATH 7780 ADVANCED ALGEBRAIC TOPOLOGY I (3) **LEC. 3. Departmental approval. Advanced topics in homology, cohomology, and duality with relations to and further study of homotopy theory Applications to and further study of manifolds and geometric topology.

**MATH 7790 ADVANCED ALGEBRAIC TOPOLOGY II (3) **LEC. 3. Pr. MATH 7780. Continuation of MATH 7780; advanced topics in homology, cohomology, and duality with relations to and further study of homotopy theory. Applications to and further study of manifolds and geometric topology.

**MATH 7800 PROBABILITY I (3) **LEC. 3. Pr., a full year of undergraduate mathematical analysis at a level commensurate with MATH 5200/5210. Measure-theoretic foundations, independence, conditioning, martingales, Markov property, stationarity, random walks, Markov chains, Poisson processes.

**MATH 7810/7816 PROBABILITY II (3) **LEC. 3. Pr. MATH 7800. Classical and modern topics in stochastic processes (Markov chains, Poisson process, Brownian motion). Applications and stochastic models (queues, stationary processes, population dynamics, finances). Credit will not be given for both MATH 7810 and STAT 7810.

**MATH 7820/7826 APPLIED STOCHASTIC PROCESSES I (3) **LEC. 3. Classical and modern topics in stochastic processes (Markov processes, Random Walks, Martingales, Brownian motion). Introduction to stochastic integrals and differential equations. Applications (queues, population dynamics, chaos, finances). Credit will not be given for both MATH 7820 and STAT 7820.

**MATH 7830 APPLIED STOCHASTIC PROCESSES II (3) **LEC. 3. Classical and modern topics in stochastic processes (Markov processes, Random Walks, Martingales, Brownian motion). Introduction to stochastic integrals and differential equations. Applications (queues, population dynamics, chaos finances).

**MATH 7870 REAL FUNCTIONS AND DESCRIPTIVE SET THEORY I (3) **LEC. 3. Pr. MATH 7210 or MATH 7500. Borel classification of sets, the Baire classification of real functions. Derivatives and approximately continuous functions. The Lebesegue density topology.

**MATH 7880 REAL FUNCTIONS AND DESCRIPTIVE SET THEORY II (3) **LEC. 3. Pr. MATH 7870. Analytic and coanalytic sets. Lebesegue measurable, universally measurable and Marczewski measurable sets and functions. Baire properties. Singular sets. Category analogs to real anaylsis.

**MATH 7950 SEMINAR (1-3) **SEM. SU. Course may be repeated for a maximum of 6 credit hours.

**MATH 7960 SPECIAL PROBLEMS (1-10) **IND. Departmental approval. Topics may vary as needed. Course may be repeated for a maximum of 10 credit hours.

**MATH 7970 SPECIAL TOPICS (1-10) **IND. Departmental approval. Topics may vary as needed. Course may be repeated with change in topics.

**MATH 7980 RESEARCH AND SPECIAL PROJECT IN APPLIED MATHEMATICS (1-10) **RES. SU. Departmental approval. For students working on the Master of Applied Mathematics degree with concentration in numerical analysis. Course may be repeated for a maximum of 10 credit hours.

**MATH 7990 RESEARCH AND THESIS (1-10) **MST. Course may be repeated with change in topics.

**MATH 8310 HOMOLOGICAL ALGEBRA I (3) **LEC. 3. Pr. MATH 7320. Departmental approval. Homology and cohomology. Hom and Tensor functors; the adjoint isomorphisms, injective/projective modules, flat modules, the classification of certain rings using homological tools.

**MATH 8320 HOMOLOGICAL ALGEBRA II (3) **LEC. 3. Pr. MATH 8310. Localizations of modules, nonsingular rings and modules, the Goldie dimension, homological classification of modules; Whitehead modules, reflexive modules, R-modules as modules over their rings of endomorphisms.

**MATH 8330 INTRODUCTION TO LIE GROUPS (3) **LEC. 3. Pr. MATH 7310 or MATH 7370. Introduce Lie groups via matrix groups. Topics include exponential map, Lie algebras, classical groups, structures and classifications, manifolds, representations.

**MATH 8400 ADVANCED FUNCTIONAL ANALYSIS I (3) **LEC. 3. Pr. MATH 7210 and MATH 7400. Topics concerning bounded and unbounded linear operators in Banach and Hilbert spaces; theory of distributions and topological vector spaces with applications, current research.

**MATH 8410 ADVANCED FUNCTIONAL ANALYSIS II (3) **LEC. 3. Pr. MATH 8400. Topics from the theory of bounded and unbounded linear opera operators in Banach and Hilbert spaces; elements of nonlinear functional analysis, topics of current research interest.

**MATH 8600 ADVANCED PROBABILITY I (3) **LEC. 3. Processes, distributions, independence, Random sequences, series, averages, characteristic functions. Classical limits theorems, conditioning. Some experience with graduate level mathematics, preferably in the areas of analysis and topology.

**MATH 8610 ADVANCED PROBABILITY II (3) **LEC. 3. Pr. MATH 8600. Martingales, Markov chains, random walks, renewal theory, Poisson processes and ergodic theory.

**MATH 8630 ADVANCED STOCHASTIC PROCESSES I (3) **LEC. 3. Pr. MATH 8610. Gaussian processes, Brownian motion, invariance principles, convergence of random processes, measures and sets, stochastic integrals and quadratic variation.

**MATH 8640 ADVANCED STOCHASTIC PROCESSES II (3) **LEC. 3. Pr. MATH 8630. Continuous martingales and Brownian motion, stochastic differential equations and martingale problems, local time, excursions, one-dimensional SDE's and diffusions.

**MATH 8700 FINITE GEOMETRY I (3) **LEC. 3. Pr. MATH 5370. Projective and affine spaces over finite fields. Inversive planes. Relationship with linear algebra over finite fields and permutation groups. Applications to combinatorial designs.

**MATH 8710 FINITE GEOMETRY II (3) **LEC. 3. Pr. MATH 8700. Projective and affine spaces over finite fields. Inversive planes. Relationship with linear algebra over finite fields and permutation groups. Applications to combinatorial designs.

**MATH 8960 SPECIAL PROBLEMS (1-10) **IND. Departmental approval. Topics may vary as needed. Course may be repeated for a maximum of 15 credit hours.

**MATH 8970 SPECIAL TOPICS (1-10) **IND. Departmental approval. Topics may vary as needed. Course may be repeated for a maximum of 15 credit hours.

**MATH 8990 RESEARCH AND DISSERTATION (1-10) **DSR. Course may be repeated with change in topics.