**56:645:503-504 Theory of Functions of a Complex Variable (3,3) **

Analytic functions and the Cauchy Integral Theorem. Conformal mappings. Laplace transforms.

**56:645:505-506 Analysis (3,3) **

Infinite series, especially Fourier series. Epsilon-delta proofs of continuity and differentiability. Convergence tests. Measure theory and integration.

**56:645:507 Contemporary Issues: Teaching Methods (3)**

This course will cover some traditional as well as contemporary approaches to teaching mathematics. We will analyze and discuss strategies and methods used to successfully present concepts relative to standard middle and high school curricula. The main focus will be to investigate and connect many of the topics covered throughout 7th to 12th grades. These will include but not be limited to the following concepts: operations with sets, number systems, algebraic fundamentals, number theory, geometry in 2 and 3-D, types of reasoning, functions and the concept of patterns i.e., sequences, trigonometric concepts will be discussed if time.

56:645:508 Mathematical Reasoning (3)

This course develops two fundamental components of writing mathematics: reasoning (thinking about the proof) and writing (formulating and writing the ideas precisely using logical statements). The course begins with illustrative examples and general guidelines.

**56:645:510 Mathematical Communication and Technology (3) **

New technologies for doing and teaching mathematics and for creating mathematical documents for print and World Wide Web distribution.

56:645:511 Computer Science (3)

A survey of computer science, both theoretical and practical, for the pure mathematician. Topics could include time-complexity of algorithms, NP-completeness, Turing machines, factoring and primality testing, Strassen’s matrix reduction algorithm, and the fast Fourier transform.

**56:645:527-528 Methods of Applied Mathematics (3,3) **

Derivation of the heat and wave equations. Existence theorems for ordinary differential equations, series solutions. Bessel and Legendre equations. Sturm-Liouville Theorem. Pre- or corequisite: 56:645:549.

56:645:530 Manifolds (3)

Topological and differential manifolds. Surfaces. Fundamental groups and coverings. Differential forms and de Rham cohomology.

56:645:531 Geometry (3)

Review and reevaluation of Euclid’s geometry. Axiomatic development of Euclidean and hyperbolic geometries. The parallel postulate. The impossibility of trisecting an angle or duplicating a cube.

**56:645:532 Differential Geometry (3) **

Curves and surfaces in Euclidean space. Riemannian manifolds, connections, and curvature.

**56:645:533-534 Introduction to the Theory of Computation I,II (3,3) **

645:533: Introduction to formal languages, automata, and computability: regular languages and finite state automata; context-free grammars and languages; pushdown automata; the Church-Turing thesis; Turing machines; decidability and undecidability; Rice’s theorem.

645:534: Second course in the sequence; addresses key topics in computability and complexity theory, such as recursive and recursively enumerable sets; the Recursion Theorem; Turing reductions and completeness; Kolmogorov complexity; Space and Time complexity; NP-completeness; hierarchy theorems; probabilistic complexity classes, and interactive proof systems.

**56:645:535-536 Algebra for Computer Scientists I,II (3,3) **

Linear and abstract algebra, including group theory, with applications to image processing, data compression, error correcting codes, and encryption.

**56:645:537 Computer Algorithms (3) **

Algorithm design techniques: divide-and-conquer, greedy method, dynamic programming, backtracking, and branch-and-bound. Advanced data structures, graph algorithms, and algebraic algorithms. Complexity analysis, complexity classes, and NP-completeness. Introduction to approximation algorithms and parallel algorithms.

**56:645:538 Combinatorial Optimization (3) **

Algorithmic techniques for solving optimization problems over discrete structures, including integer and linear programming, branch-and-bound, greedy algorithms, divide-and-conquer, dynamic programming, local optimization, simulated annealing, genetic algorithms, and approximation algorithms.

**56:645:540 Computational Number Theory and Cryptography (3) **

Primes and prime number theorems and numerical applications; the Chinese remainder theorem and its applications to computers and Hashing functions; factoring numbers; cryptography; computation aspects of the topics emphasized. Students required to do some simple programming.

**56:645:541 Introduction to Computational Geometry (3) **

Algorithms and data structures for geometric problems that arise in various applications, such as computer graphics, CAD/CAM, robotics, and geographical information systems (GIS). Topics include point location, range searching, intersection, decomposition of polygons, convex hulls, and Voronoi diagrams.

**56:645:542 Parallel Supercomputing (3) **

Fundamental issues in the design and development of programs for parallel supercomputers; programming models and performance optimization techniques; application examples and programming exercises on a contemporary parallel machine; cost models and performance analysis and evaluation.

**56:645:545 Topology (3) **

Point set topology, fundamental group and coverings. Singular homology and cohomology, the Brouwer degree and fixed-point theorems, the sphere retraction theorem, invariance of domains.

**56:645:549-550 Linear Algebra and Applications (3,3) **

Finite dimensional vector spaces, matrices, and linear operators. Eigenvalues, eigenvectors, diagonalizability, and Jordan canonical form. Applications.

**56:645:551-552 Abstract Algebra (3,3) **

Introductory topics in rings, modules, groups, fields, and Galois theory. Pre- or corequisite: 56:645:549.

**56:645:554 Applied Functional Analysis (3) **

Infinite dimensional vector spaces, especially Banach and Hilbert vector spaces. Orthogonal projections and the spectral decomposition theorem. Applications to differential equations and approximation methods.

**56:645:555 Glimpses of Mathematics (3) **

The intuitive beginnings and modern applications of key ideas of mathematics, such as polyhedra and the fundamental theorem of algebra. Extensive use of computer-generated films to help visualize the methods and results.

**56:645:556 Visualizing Mathematics by Computer (3) **

Introduction to symbolic computational packages and scientific visualization through examples from calculus and geometry. Covers 2-D, 3-D, and animated computer graphics using Maple, Mathematica, and Geomview. No programming knowledge required.

**56:645:557 Signal Processing (3) **

Signal modeling: periodic, stationary, and Gaussian signals. System representation: Volterra representation, state space representation, simulation. Themes in system design: least square estimation, system identification, adaptive signal processing. Representation of discrete causal signals: role of Fourier analysis, convolutions, fast Fourier transforms. Realization of linear recurrent structures: controllability, observability and minimal realization, frequency domain analysis of signals, and the role Laplace transforms. Stability analysis: Lyapunov and linearization methods. Prediction, filtering, and identification: linear prediction, the LQR problem, Kalman filter.

**56:645:558 Theory and Computation in Probability and Queuing Theory (3) **

Basic probability structures, probability distributions, random number generations and simulations, queuing models, analysis of single queue, queuing networks, applications of queuing theory.

** 56:645:560 Industrial Mathematics (3) **This course covers various problems that can be found in industry. A problem based learning approach is used. Each problem studied motivates the need for learning the mathematical techniques necessary to solve the problem. The problems will be solved using MATLAB and C++ programs. Part of the course involves writing a report on a project and giving a presentation of the results. It is suggested that students learn to use LaTeX. Previous problems include Monte Carlo methods for a financial application, circadian rhythm analysis, atmospheric refraction correction, and the Fourier synthesis of ocean scenes. Prerequisites include a strong background in undergraduate mathematics and knowledge of C++ and MATLAB. This course is suitable for graduate students and advanced undergraduates.

**56:645:561 Optimization Theory (3) **

Linear programming: optimization, simplex algorithm, nonlinear programming, game theory.

**56:645:562 Mathematical Modeling (3) **This course introduces concepts of mathematical modeling through a hand-ons problem solving approach. Depending on course enrollment, the students are divided into groups for two projects. The first project is a competition where each group is solving the same problem. The groups will develop a design and submit a paper describing the design along with C++ code of their design. The second project is a class project where the entire class will solve a problem working in groups but in this project, each group is working on a different part of the bigger problem. The groups must coordinate their efforts and integrate the solutions to solve the main problem. Previous modeling problems include problems from various sports (football, basketball, and tennis), radar system modeling and tracking, ballistic missile modeling, and thermal expansion in a hot water heater. Prerequisites include a strong background in undergraduate mathematics and knowledge of C++ and MATLAB. This course is suitable for graduate students and advanced undergraduates.

**56:645:563 Statistical Reasoning (3) **

Random variables, uniform, Gaussian, binomial, Poisson distributions, probability theory, stationary processes, central limit theorem, Markov chains, Taguchi quality control.

**56:645:570 Special Topics in Pure Mathematics (3) **

Topics vary from semester to semester. Prerequisite: Permission of instructor. Course may be taken more than once.

**56:645:571 Computational Mathematics I (3)**Computational Mathematics I (3) This first semester of a one-year sequence covers the following numerical techniques for solving mathematical problems on a computer: the IEEE internal representation of floating point numbers, interpolation, root finding, numerical integration, numerical differentiation, and function minimization. The material is presented so that topics build on one another and applications are given to illustrate the use of the techniques. Prerequisites include calculus and knowledge of C++. The course is suitable for graduate students and advanced undergraduates who have the prerequisites. It is preferred that the first semester course 645:571 is taken before the second semester course 645:572.

**56:645:572 Computation Mathematics II (3)**Computational Mathematics II (3) This second semester of a one-year sequence covers the following numerical techniques for solving mathematical problems on a computer: numerical linear algebra (including the numerical solution of linear systems of equations and the algebraic eigenvalue problem) and the numerical solution of differential equations. The material is presented so that topics build on one another and applications are given to illustrate the use of the techniques. Prerequisites include advanced calculus, linear algebra, differential equations and knowledge of C++. The course is suitable for graduate students and advanced undergraduates who have the prerequisites

**56:645:574 Control Theory and Optimization (3) **

Controllability, observability, and stabilization for linear and nonlinear systems. Kalman and Nyquist criteria. Frequency domain methods, Liapunov functions.

**56:645:575 Qualitative Theory of Ordinary Differential Equations (3) **

Cauchy-Picard existence and uniqueness theorem. Stability of linear and nonlinear systems. Applications to equations arising in biology and engineering.

**56:645:577 Quality Engineering (3) **

Introduction to statistical tools, such as data analysis, and their use in the testing of product design and minimization of uncontrollable variation.

**56:645:578 (cross-listed with: 50:640:466) MATHEMATICAL METHODS IN SYSTEMS BIOLOGY** **(3)**

The course will provide an introduction to computational and system biology, focusing on advanced mathematical tools. In particular ordinary and partial differential equations, control theory and discrete mathematics (networks) will be used to address a wide set of biological and bio-medical applications. The latter will range from classical prey-predator populations examples to cancer immuno and drug therapies, from evolutionary math to gene networks.

**56:645:579 Celestial Mechanics**

**56:645:580 Special Topics in Applied Mathematics (3) **

Topics vary from semester to semester. Prerequisite: Permission of instructor. Course may be taken more than once.

**56:645:698 Independent Study in Pure Mathematics (3) **

Study of a particular subject independently but with frequent consultations with a faculty member.

**56:645:699 Independent Study in Applied Mathematics (3) **

Study of a particular subject independently but with frequent consultations with a faculty member.

**56:645:700 Thesis in Pure Mathematics (3) **

Expository paper written under the close guidance of a faculty member.

**56:645:701 Thesis in Applied Mathematics (3) **

Expository paper written under the close guidance of a faculty member.

**56:645:800 Matriculation Continued (0) **

Continuous registration may be accomplished by enrolling for at least 3 credits in standard course offerings, including research courses, or by enrolling in this course for 0 credits. Students actively engaged in study toward their degree who are using university facilities and faculty time are expected to enroll for the appropriate credits.

56:645:877 Teaching Assistantship (E6)