# Mathematical Science

MATH 5301 Elementary Analysis I (3 semester credit hours) Real numbers, differentiation, integration, metric spaces, basic point set topology, power series, analytic functions, Cauchy's theorem. Prerequisite: MATH 2451 and MATH 3310 or equivalent. (3-0) Y

MATH 5302 Elementary Analysis II (3 semester credit hours) Continuation of MATH 5301. Prerequisite: MATH 5301. (3-0) Y

MATH 5304 Applied Mathematical Analysis for Non-Majors (3 semester credit hours) Techniques of mathematical analysis applicable to the social, behavioral and management sciences. Differential and integral calculus of one and many variables. May not be used to fulfill degree requirements. Prerequisite: MATH 1314. (3-1) S

MATH 5305 Higher Geometry for Teachers (3 semester credit hours) Topics in modern Euclidean geometry including distinguished points of a triangle, circles including the nine-point circle, cross ratio, transformations; introduction to projective geometry. May not be used to fulfill degree requirements for mathematical sciences majors except those in the Master of Arts in Teaching (MAT) program. Prerequisite: Junior-level mathematics course. (3-0) T

MATH 5306 Non-Euclidean Geometry for Teachers (3 semester credit hours) The relations among elliptic, Euclidean and hyperbolic geometries, Euclidean models of elliptic and hyperbolic geometries. May not be used to fulfill degree requirements for mathematical sciences majors except those in the Master of Arts in Teaching (MAT) program. Prerequisite: Junior-level mathematics course. (3-0) T

MATH 5313 Modern Algebra for Teachers (3 semester credit hours) Study of modern algebra involving groups, rings, fields and Galois Theory. May not be used to fulfill degree requirements for mathematical sciences majors except those in the Master of Arts in Teaching (MAT) program. Prerequisite: Junior-level mathematics course. (3-0) R

MATH 5390 Topics in Mathematics - Level 5 (3 semester credit hours) May be repeated for credit as topics vary (9 semester credit hours maximum). Instructor consent required. (3-0) R

MATH 6301 Real Analysis (3 semester credit hours) Lebesgue measure in finite- dimensional spaces, Abstract measures, measurable functions, convergence a.e., Egorov's Theorem, convergence in measure, Lebesgue integral, Lebesgue's bounded convergence theorem, Levi's monotone convergence theorem, Fatou's Lemma, Fubini's theorem, Lp-spaces. Prerequisite: MATH 5302. (3-0) Y

MATH 6302 Functional Analysis I (3 semester credit hours) Banach and Hilbert spaces, classical theorems of functional analysis, compact operators, Fredholm operators, elements of spectral theory, introduction to unbounded operators. Prerequisite: MATH 6301. (3-0) Y

MATH 6303 Theory of Complex Functions I (3 semester credit hours) Complex integration, Cauchy's theorem, calculus of residues, power series, entire functions, Riemann mapping theorems. Riemann surfaces, conformal mapping with applications. Prerequisite: MATH 4301 and MATH 4302. (3-0) Y

MATH 6304 Theory of Complex Functions II (3 semester credit hours) Riemann surfaces, meromorphic and holomorphic functions and differentials, the normalization theorem, the Riemann-Roch theorem, Abel theorem, applications to nonlinear equations. Prerequisite: MATH 6303. (3-0) T

MATH 6305 Mathematics of Signal Processing (3 semester credit hours) The course is devoted to a mathematical foundation of some of the key topics in signal processing: discrete and continuous signal transforms, least squares methods and adaptive filtering, compressed sensing and related topics. Prerequisites: (MATH 2418 and MATH 2451) or instructor consent required. (3-0) T

MATH 6307 Wavelets and Their Applications (3 semester credit hours) An introduction to windowed Fourier and continuous wavelet transforms, generalized frames, discrete wavelet frames, multiresolution analysis, Daubechies' orthogonal wavelet bases, and their applications in partial differential equations and signal processing. Prerequisite: (MATH 2418 and MATH 2420) or equivalent. (3-0) T

MATH 6308 Inverse Problems and Applications (3 semester credit hours) Exact and approximate methods of nondestructive inference, such as tomography and inverse scattering theory in one and several dimensions, with applications in physical and biomedical sciences and engineering. Prerequisite: (MATH 2418 and MATH 2420) or equivalent. (3-0) T

MATH 6309 Differential Geometry (3 semester credit hours) Smooth manifolds, tangent bundles, smooth partitions of unity, submanifolds, Sard's theorem, transversality, embeddings, Whitney theorem, differential forms, Frobenius Theorem, de Rham cohomology, degree theory on manifolds, Riemannian metric, Gauss-Bonnet theorem. Prerequisite: MATH 4301 or MATH 5301. (3-0) T

MATH 6310 Topology (3 semester credit hours) Metric spaces, introduction to topology, elements of homotopy theory, covering spaces, fundamental group, homotopy groups, fibrations, simplicial complexes and CW-complexes, degree theory. Prerequisite: MATH 4301 or MATH 5301. (3-0) Y

MATH 6311 Abstract Algebra I (3 semester credit hours) Basic properties of groups, rings, fields, and modules. Prerequisite: MATH 3311 or equivalent. (3-0) Y

MATH 6313 Numerical Analysis (3 semester credit hours) A study of numerical methods including the numerical solution of non-linear equations, linear systems of equations, interpolation, iterative methods and approximation by polynomials. Prerequisites: Knowledge of a high-level programming language and MATH 2418 and MATH 2451. (3-0) Y

MATH 6315 Ordinary Differential Equations (3 semester credit hours) The study of ordinary differential equations with emphasis on existence, uniqueness, linear systems, boundary value problems, and stability. Prerequisites: [MATH 2418 or equivalent and MATH 2420 and (MATH 4301 and MATH 4302)] or (MATH 5301 and MATH 5302). (3-0) Y

MATH 6316 Differential Equations (3 semester credit hours) Continuation of MATH 6315 and an introduction to partial differential equations. Prerequisite: MATH 6315. (3-0) T

MATH 6318 Numerical Analysis of Differential Equations (3 semester credit hours) Practical and theoretical aspects of numerical methods for both ordinary and partial differential equations are discussed. Topics selected from: initial value problems for ordinary differential equations, two-point boundary value problems, projection methods, finite difference, finite element and boundary element approximations for partial differential equations. Application of methods will be illustrated using Matlab. Prerequisite: MATH 6313 or equivalent. (3-0) T

MATH 6319 Principles and Techniques in Applied Mathematics I (3 semester credit hours) Mathematical methods usually used in applied sciences and engineering. Topics chosen from advanced linear algebra; Hilbert spaces; positivity; quaternions; integral equations; Fourier analysis; distributions; convexity; asymptotic methods; special functions. Prerequisites: MATH 2418 and MATH 2420 or equivalent. (3-0) T

MATH 6320 Principles and Techniques in Applied Mathematics II (3 semester credit hours) Continuation of Math 6319. Prerequisite: MATH 6319. (3-0) T

MATH 6321 Optimization (3 semester credit hours) Introduction to theoretical and practical concepts of optimization in finite and infinite dimensional setting, least-squares estimation, optimization of functionals, local and global theory of constrained optimization, iterative methods. Prerequisites: MATH 2420 and MATH 2418. (3-0) T

MATH 6331 Linear Systems and Signals (3 semester credit hours) Basic principles of systems and control theory: state space representations, stability, observableness, controllability, realization theory, transfer functions, feedback. Prerequisites: (MATH 2418 and MATH 2451) or instructor consent required. (3-0) T

MATH 6332 Advanced Control (3 semester credit hours) Theoretical and practical aspects of modern control methodologies in state space and frequency domain, in particular LQG and H-infinity control: coprime factorizations, internal stability, Kalman filter, optimal regulator, robust control, sensitivity minimization, loop shaping, model reduction. Prerequisite: MATH 6331. (3-0) T

MATH 6336 Nonlinear Control Systems (3 semester credit hours) Differential geometric tools, input-output maps, feedback linearization, nonlinear observers, input-output linearization, output tracking, and regulation, passivity based control, control systems on Lie groups. Prerequisites: (MATH 6315 and MATH 6331) or instructor consent required. (3-0) T

MATH 6339 Control of Distributed Parameter Systems (3 semester credit hours) Theoretical and technical issues for control of distributed parameter systems in the context of linear infinite dimensional dynamical systems: Evolution equations and control on Euclidean space, elements of functional analysis, semigroups of linear operators, abstract evolution equations, control of linear infinite dimensional dynamical systems, approximation techniques. Prerequisites: MATH 4362 and MATH 4301. (3-0) T

MATH 6340 Numerical Linear Algebra (3 semester credit hours) Topics include direct and iterative methods for solving linear systems; vector and matrix norms; condition numbers; least squares problems; orthogonalization, singular value decomposition; computation of eigenvalues and eigenvectors; conjugate gradients; preconditioners for linear systems; computational cost of algorithms. Topics will be supplemented with programming assignments. Recommended Prerequisites: MATH 4334 or equivalent and a prior programming course. Prerequisite: MATH 2418. (3-0) Y

MATH 6341 Bioinformatics (3 semester credit hours) Fundamental mathematical and algorithmic theory behind current bioinformatics techniques are covered and implemented. They include hidden Markov models, dynamic programming, genetic algorithms, simulated annealing, neural networks, cluster analysis, and information theory. Prerequisites: Knowledge of Unix and a high level programming language. (3-0) T

MATH 6342 Scientific Computing (3 semester credit hours) Introduction to scientific computing through projects in computational science and engineering. Topics include mathematical modeling; theoretical analysis of such models; numerical and symbolic computation; verification and validation; computational simulation. Representative projects will include applications of dynamical systems, Monte Carlo simulations, numerical optimization, and linear and nonlinear partial differential equations. The course includes an introduction to symbolic computation and to programming in MATLAB, Python, and/or C. Some prior programming experience is recommended. Prerequisites: MATH 4334 and MATH 4362 and MATH 6315 or instructor consent required. (3-0) T

MATH 6343 Computational Biology (3 semester credit hours) Mathematical and computation methods and techniques to analyze and understand problems in molecular biology are covered. Topics include sequence homology and alignment, genetic mapping, protein folding, and DNA computing. Prerequisite: MATH 2418 or equivalent. (3-0) T

MATH 6345 Mathematical Methods in Medicine and Biology (3 semester credit hours) Introduction to the use of mathematical techniques in solving biologically important problems. Some examples of topics that might be covered are biochemical reactions, ion channels, cellular signaling mechanisms, kidney function, and nerve impulse propagation. Recommended Prerequisite: MATH 2420. Prerequisites: MATH 2417 and MATH 2419. (3-0) T

MATH 6350 Quantum Computation and Information (3 semester credit hours) Quantum states, channels, measurements; entropy, subadditivity; entanglement measures, discord; teleportation, dense coding, quantum key distribution; Shor's algorithm, Grover's search algorithm, hidden subgroup algorithms. Prerequisite: MATH 2418 and instructor consent required. (3-0) T

MATH 6364 Stochastic Calculus in Finance (3 semester credit hours) Brownian Motion, Ito Calculus, Feynman-Kac formula and an outline of Stochastic Control, Black Scholes Analysis, Transaction Costs, Optimal Portfolio Investment. Prerequisites: STAT 4351 or equivalent and MATH 2451 or equivalent. (3-0) T

MATH 6390 Topics in Mathematics - Level 6 (3 semester credit hours) May be repeated for credit as topics vary (9 semester credit hours maximum). Instructor consent required. (3-0) R

MATH 6V81 Special Topics in Mathematics - Level 6 (1-9 semester credit hours) May be repeated for credit as topics vary. Instructor consent required. ([1-9]-0) S

MATH 6V98 Masters Thesis (3-9 semester credit hours) Pass/Fail only. May be repeated for credit. Instructor consent required. ([3-9]-0) S

MATH 7313 Partial Differential Equations I (3 semester credit hours) Classical and modern solution techniques for initial and boundary value problems for parabolic, elliptic, and hyperbolic linear partial differential equations. Existence, uniqueness, well-posedness, fundamental solutions, and Green's functions. First-order nonlinear equations, scalar conservation laws, and the method of characteristics. An introduction to weak solutions and the theory of Sobolev spaces. Prerequisite: MATH 6301 and Math 6315 or equivalent. (3-0) T

MATH 7314 Partial Differential Equations II (3 semester credit hours) Continuation of MATH 7313. Prerequisite: MATH 7313. (3-0) T

MATH 7316 Wave Propagation with Applications (3 semester credit hours) Study of the wave equation in one, two and three dimensions, the Helmholtz equation, associated Green's functions, asymptotic techniques for solving the propagation problems with applications in physical and biomedical sciences and engineering. Prerequisites: MATH 6303 and MATH 6318. (3-0) T

MATH 7319 Functional Analysis II (3 semester credit hours) Topological vector spaces, locally convex spaces, Frechet spaces, test function spaces and tempered distributions, Fourier transforms and applications to differential equations. Recommended Prerequisite: MATH 6303. Prerequisites: MATH 6301 and MATH 6302. (3-0) T

MATH 7390 Topics in Mathematics - Level 7 (3 semester credit hours) May be repeated for credit as topics vary (9 semester credit hours maximum). Instructor consent required. (3-0) R

MATH 8V02 Individual Instruction in Mathematics (1-6 semester credit hours) Pass/Fail only. May be repeated for credit as topics vary. Instructor consent required. ([1-6]-0) S

MATH 8V04 Topics in Mathematics - Level 8 (1-6 semester credit hours) Pass/Fail only. May be repeated for credit as topics vary. Instructor consent required. ([1-6]-0) R

MATH 8V07 Research (1-9 semester credit hours) Open to students with advanced standing subject to approval of the Graduate Advisor. Pass/Fail only. May be repeated for credit. Instructor consent required. ([1-9]-0) S

MATH 8V99 Dissertation (1-9 semester credit hours) Pass/Fail only. May be repeated for credit. Instructor consent required. Prerequisite: Open to PhD students only. ([1-9]-0) S