Mathematical truth, axioms and theorems, propositional truth tables, quantifiers, set theory, indexed families, mathematical induction, cardinality, finite and infinite sets, denumerability, the Axiom of Choice, and the continuum hypothesis. Example syllabus.

Matrices, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, and functions of matrices. Example syllabus.

Introduction to number theory, recursively defined functions, analyzing algorithm performance, recurrence relations, generating functions, permutations and combinations, Inclusion/Exclusion, introduction to Graph Theory, and Boolean algebra. Prerequisite: 531. Example syllabus.

Three dimensional geometry, directional derivatives, gradient, divergence, curl, maximum-minimum problems, multiple integrals, parametric surfaces and curves, and line integrals. Prerequisite: 512.

Review of random variables, discrete and continuous distributions, expectation, conditional probability, and limit theorems. Introduction to the Poisson point process and Markov chains.

Review of statistical estimation and hypothesis testing. Introduction to nonparametric methods, permutation tests, the bootstrap, and Bayesian statistics. Prerequisite: 521.

Linear and nonlinear regression, logistic regression, analysis of variance, and generalized linear models. Prerequisites: 521, 522.

Principles of experimental design, randomization, blocking, factorial designs, repeated measures, and Latin squares. Prerequisites: 525.

Classes, objects, instances, messages, methods, inheritance, interfaces, polymorphism, software life cycle, variables, expressions, data objects, control structures, strings, arrays, files, searching, sorting, applets, toolkits, threads, and graphical user interfaces. Example syllabus.

Abstract data types, stacks, queues, databases, priority queues, trees, linked lists, hashing, balanced trees, self-organizing data structures, and advanced sorting. Prerequisites: 531. Example syllabus.

Computer representation and hardware, system programming, prototyping and development, memory and data organization, communications and networking, human/computer interactions, and performance analysis and improvement. Example syllabus.

Software development processes and the software life cycle, software architecture and design, emphasizing object-oriented design, user interface design, validation and verification, testing methods, systems analysis and requirements definition, software management and personnel issues. Prerequisites: 531, 535. Example syllabus.

Network technologies, protocols, and management. Programming networked applications. The effects of the Internet and World Wide Web on computing and society. Prerequisites: 522, 532.

Introduction to the mathematics of images and image processing, as well as computational methods for real data manipulation. Topics include image acquisition, image enhancement and restoration in both the spatial and frequency domains, the Fourier transform, wavelets, image compression, image segmentation, and morphological processing algorithms. Prerequisites: 512 (or MATH 315), and 518 (or MATH 215). Example syllabus.

Graph theory, graph algorithms, coloring, network flows, computational geometry, compression, randomized algorithms, parallel algorithms, and NP-completeness. Prerequisite: 532.

Option strategies, future markets, option price relations, binomial option pricing model, binomial put model, and the Black-Scholes analysis. Prerequisites: 511, 512.

Analytic functions, elementary transformations, complex integration, Cauchy theory of integration, complex power series, Laurent series, and residues. Prerequisites: 511, 512.

Finite difference methods, stability, boundary value problems, ordinary differential equations, integral equations, and partial differential equations. Prerequisites: 511, 512.

Mathematical techniques for securing data for storage or transmission in an insecure context: abstract algebra, threat modeling, information and coding theory, error-correcting codes, public and private key encryption schemes, and cryptanalytic techniques. Prerequisites: 511, 512, 516, 522.

An introduction to the background of operations including example problems and a brief history. An extensive discussion of the theory and applications of linear programming will follow. Other topics will include nonlinear programming, continuous and discrete probability models, dynamic programming, game theory and transportation and network flow models. Prerequisite: MATH 116.

Linear systems, interpolation, functional approximation, numeric differentiation and integration, and solutions to non-linear equations. Prerequisite: 531.

Building both a predictive and a classification model using data mining techniques. Software used for building models is JMP and S-Plus I-Miner. Specific topics include: the distinction between supervised and unsupervised learning; issues in data exploration (steps in data cleaning, missing data, transformations, methods of imputation, training set vs. testing set and validation sets; methods of determining model accuracy (ROC curves, lift and cumulative lift charts); cross-validation, modeling methods for discrete and continuous dependent variables; clustering algorithms; neural networks. Prerequisite: 525, 526.

Computational and statistical modeling of human cognitive processes and their implementation: modularity of mind, rule-based vs. distributed vs. prototype models, search techniques, story understanding, and statistical models of language. Prerequisites: 511, 512, 522, 532, 535. Example syllabus.

Development and application of techniques which allow computation on different components of a network: network technology and design, communications theory, applications protocols, remote procedure calls, distributed storage techniques, parallel algorithms, load balancing, scheduling, and task migration. Prerequisites: 511, 512, 525, 532, 535.

Basic tools, including statistical significance testing, overview of theory, algorithms, and applications, concept learning, reinforcement learning, clustering, advanced concept learning, neural networks, perceptrons, decision trees, general-purpose algorithmic methods, data mining, and collaborative filtering. Prerequisites: 521, 531.

BNF representation, variables, scope, binding, data types and type checking, abstract data types, control, control flow abstractions, procedural abstractions, calling mechanisms, semantic models, category theory, functional programming, lambda calculus, logic programming, functors, adjoint functors, 2-categories, and little categories. Prerequisites: 531, 535.

Cardinality of sets, uncountability of certain sets, languages, regular languages, context-free languages, the Chomsky hierarchy, Turing machines, Church-Turing thesis, problems that are not Turing computable, and an introduction to computational complexity. Prerequisites: 511, 512. Example syllabus.

Network, database, and Web security, threat models, elementary and advanced crytology, protocol analysis, covert channels, access control and trust issues, legal and ethical issues in security.

Various subjects in computational mathematics. May be repeated for credit when content changes. Prerequisite: Permission of the instructor. Example syllabus.

Directed study on a topic related to computational mathematics. May be repeated once for credit. Prerequisite: Permission of the instructor and Graduate Director.

Prerequisite: Permission of the Graduate Director.