Mathematics

This course presents fundamentals of college algebra and trigonometry, with an introduction to concepts in 2D geometry and linear algebra. Topics include: polynomial, rational, trigonometric, exponential and logarithmic functions as well as their inverses; analytic trigonometry, trigonometric identities, the unit circle, and trigonometric functions of a real variable; introduction to linear systems, basics of linear transformations in 2D; vectors, parametric lines, dot product, and projections in 2D.

This course presents fundamentals of probability and statistics without calculus. Topics include: data representation, population mean, variance, and standard deviation, finite probabilities, events, conditional and marginal probability, discrete random variables, binomial distribution, normal distribution, sampling distributions for mean and variance, estimation of means, confidence intervals, hypothesis testing, inference, and chi-square tests.

This course presents the mathematics needed for calculus including: function domain and codomain, composition of functions, inverse functions, polynomials, exponential and logarithmic functions, trigonometric functions, trigonometric identities, graphs of trigonometric functions, and applications of trigonometry. Additional topics may include an introduction to vectors and matrices.

This course explores the mathematical foundations of music and sound. Topics include scale systems, just and tempered intervals, oscillations and trigonometry, sound waves, and basic discrete mathematics.

This course explores further topics in the mathematical foundations of music and sound, with emphasis on digital signal processing. Topics include digital signals and sampling, spectral analysis and synthesis, convolution, filtering, sound synthesis, and physical modeling.

Topics in vector geometry include vector arithmetic, dot products, cross products, numerical representations of lines and planes, distances, angles, and intersections. Topics in linear algebra include matrices, systems of linear equations, linear transformations, and affine transformations. Additional topics may include barycentric coordinates and perspective projections.

This course introduces the calculus of functions of a single real variable. Topics include limits, continuity, differentiation, techniques of differentiation, optimization, integration, Riemann sums, the fundamental theorem of calculus, and u-substitution.

This course introduces the calculus of functions of a single real variable, and differential calculus of multivariate functions. Topics include: limits; continuity; differentiation; techniques of differentiation; optimization; integration; Riemann sums; the Fundamental Theorem of Calculus; curves in space; partial derivatives; gradient, divergence, and curl; and extrema of multivariate functions.

Topics include an introduction to differential equations, applications of integrals, techniques of integration, sequences and series of real numbers, power series, and Taylor series.

This course explores further topics in the mathematical foundations of music and sound, with emphasis on digital signal processing. Topics include: Digital signals and sampling, spectral analysis and synthesis, discrete fourier transforms, FFT, convolution, filtering, wave equation, Bessel functions, sound synthesis and physical modeling.

This course extends the basic ideas of calculus to the context of functions of several variables and vector-valued functions. Topics include partial derivatives, tangent planes, and Lagrange multipliers. The study of curves in two- and three space focuses on curvature, torsion, and the TNB-frame. Topics in vector analysis include multiple integrals, vector fields, Green'92s Theorem, the Divergence Theorem and Stokes'92 Theorem. Additionally, the course may cover the basics of differential equations.

This course studies sequences and series, and integral calculus of single and multivariate functions. Topics include: an introduction to differential equations; applications of integrals; techniques of integration; sequences and series of real numbers; power series; Taylor series; double, triple, line, and surface integrals; and the theorems of Green, Gauss and Stokes.

This course presents the mathematical foundations of linear algebra, which includes a review of basic matrix algebra and linear systems of equations as well as basics of linear transformations in Euclidean spaces, determinants, and the Gauss-Jordan Algorithm. The more substantial part of the course begins with abstract vector spaces and the study of linear independence and bases. Further topics may include orthogonality, change of basis, general theory of linear transformations, and eigenvalues and eigenvectors. Other topics may include applications to least-squares approximations and Fourier transforms, differential equations, and computer graphics.

This course introduces the basic theory and applications of first and second-order linear differential equations. The course emphasizes specific techniques such as the solutions to exact and separable equations, power series solutions, special functions and the Laplace transform. Applications include RLC circuits and elementary dynamical systems, and the physics of the second order harmonic oscillator equation.

This course gives an introduction to several mathematical topics of foundational importance in the mathematical and computer sciences. Typically starting with propositional and first order logic, the course considers applications to methods of mathematical proof and reasoning. Further topics include basic set theory, number theory, enumeration, recurrence relations, mathematical induction, generating functions, and basic probability. Other topics may include graph theory, asymptotic analysis, and finite automata.

This course is an introduction to parameterized polynomial curves and surfaces with a view toward applications in computer graphics. It discusses both the algebraic and constructive aspects of these topics. Algebraic aspects include vector spaces of functions, special polynomial and piecewise polynomial bases, polynomial interpolation, and polar forms. Constructive aspects include the de Casteljau algorithm and the de Boor algorithm. Other topics may include an introduction to parametric surfaces and multivariate splines.

This course explores the mathematical foundations of digital signal processing, with applications to digital audio programming. Topics include: digital signals, sampling and quantization, complex numbers and phasors, complex functions, feedforward filters, feedback filters, frequency response and transfer functions, periodic signals and Fourier series, discrete Fourier transform and fast Fourier transform, comb and string filters, Z-transform and convolution.

This course continues to explore the mathematical foundations of digital signal processing, with applications to digital audio programming. Topics include: Review of digital signals, Z-transforms and convolution, filter types, applications of fast Fourier transform, switching signals on and off, windowing, spectrograms, aliasing, digital to analog conversion, Nyquist Theorem, filter design, Butterworth filters, reverb, and the phase vocoder.

This course is an introduction to basic probability and statistics with an eye toward computer science and artificial intelligence. Basic topics from probability theory include sample spaces, random variables, continuous and discrete probability density functions, mean and variance, expectation, and conditional probability. Basic topics from statistics include binomial, Poisson, chi-square, and normal distributions; confidence intervals; and the Central Limit Theorem. Further topics may include fuzzy sets and fuzzy logic.

This course presents a variety of computational tools for modeling and understanding complex data. Topics include manipulating data, exploratory data analysis, statistical inference, spam filters and naive Bayes, neural networks, and machine learning algorithms such as linear regression, k-nearest neighbors, and k-means. The course will focus on both understanding the mathematics underlying the computational methods and gaining hands-on experience in the application of these techniques to real datasets.

This course focuses on the conceptual understanding of a core set of practical and effective statistical methods for modeling and analyzing complex data, and applies them to solve real world problems. Topics include linear and logistic regression, linear models for classification, deep learning and neural networks, support vector machines and kernel methods, unsupervised methods, classification trees, boosting, and random forests.

This course is a continuation of MAT 300 with topics taken from the theory and applications of curves and surfaces. The course treats some of the material from MAT 300 in more detail, like the mathematical foundations for non-uniform rational B-spline (NURBS) curves and surfaces, knot insertion, and subdivision. Other topics may include basic differential geometry of curves and surfaces, tensor product surfaces, and multivariate splines.

This course gives an introduction to several mathematical topics of foundational importance to abstract algebra, and in particular the algebra of quaternions. Topics covered may include: operations, groups, rings, fields, vector spaces, algebras, complex numbers, quaternions, curves over the quaternionic space, interpolation techniques, splines, octonions, and Clifford algebras.

This course presents the foundations of wavelets as a method of representing and approximating functions. It discusses background material in complex linear algebra and Fourier analysis. Basic material on the discrete and continuous wavelet transforms forms the core subject matter. This includes the Haar transform, and multi-resolution analysis. Other topics may include subdivision curves and surfaces, and B-spline wavelets. Applications to computer graphics may include image editing, compression, surface reconstruction from contours, and fast methods of solving 3D simulation problems.

This course presents an introduction to differential geometry, with emphasis on curves and surfaces in three-space. It includes background material on the differentiability of multivariable functions. Topics covered include parameterized curves and surfaces in three-space and their associated first and second fundamental forms, Gaussian curvature, the Gauss map, and an introduction to the intrinsic geometry of surfaces. Other topics may include an introduction to differentiable manifolds, Riemannian geometry, and the curvature tensor.

Topics covered in this course include convex hulls, triangulations, Art Gallery theorems, Voronoi diagrams, Delaunay graphs, Minkowski sums, path finding, arrangements, duality, and possibly randomized algorithms, time permitting. Throughout the course, students explore various data structures and algorithms. The analysis of these algorithms, focusing specifically on the mathematics that arises in their development and analysis is discussed. Although CS 330 is not a prerequisite, it is recommended.

This course provides an introduction to the basic theorems and algorithms of graph theory. Topics include graph isomorphism, connectedness, Euler tours, Hamiltonian cycles, and matrix representation. Further topics may include spanning trees, coloring algorithms, planarity algorithms, and search algorithms. Applications may include network flows, graphical enumeration, and embedding of graphs in surfaces.

This course covers the advanced theory and applications of ordinary differential equations. The first course in differential equations focused on basic prototypes, such as exact and separable equations and the second-degree harmonic oscillator equation. This course builds upon these ideas with a greater degree of generality and theory. Topics include qualitative theory, dynamical systems, calculus of variations, and applications to classical mechanics. Further topics may include chaotic systems and cellular automata. With this overview, students will be prepared to study the specific applications of differential equations to the modeling of problems in physics, engineering, and computer science.

This course covers both the theoretical and practical study of numerical methods used in many areas of computer science, applied mathematics, science and engineering. Topics include: solutions of non-linear equations, interpolation, approximation of functions, quadrature rules, numerical solutions of ordinary differential equations, and numerical methods in linear algebra. Further topics may include Fourier series, wavelets, and stability theory.

This course introduces computational algebra as a tool to study the geometry of curves and surfaces in affine and projective space. The central objects of study are affine varieties and polynomial ideals, and the algebra-geometry dictionary captures relations between these two objects. The precise methods of studying polynomial ideals make use of monomial orderings, Grobner bases, and the Buchberger algorithm. Students have opportunities to program parts of these algorithms and to use software packages to illustrate key concepts. Further topics may include resultants, Zariski closure of algebraic sets, intersections of curves and surfaces, and multivariate polynomial splines.

This course is an introduction to elementary number theory and cryptography. Among the essential tools of number theory that are covered, are divisibility and congruence, Euler'92s function, Fermat'92s little theorem, Euler'92s formula, the Chinese remainder theorem, powers modulo m, kth roots modulo m, primitive roots and indices, and quadratic reciprocity. These tools are then used in cryptography, where the course discusses encryption schemes, the role of prime numbers, security and factorization, the DES algorithm, public key encryption, and various other topics, as time allows.

This course introduces the basic theory of fuzzy sets and fuzzy logic and explores some of their applications. Topics covered include classical sets and their operations, fuzzy sets and their operations, membership functions, fuzzy relations, fuzzification/ defuzzification, classical logic, multi-valued logic, fuzzy logic, fuzzy reasoning, fuzzy arithmetic, classical groups, and fuzz groups. Students will also explore a number of applications, including approximate reasoning, fuzzy control, fuzzy behavior, and interaction in computer games.

This course explores partial differential equations (PDEs) and fluid dynamics. Topics covered in this class include Fourier series, Fourier transforms, classification of PDEs, Poisson'92s equation, heat equation, wave equation, and introductory topics of fluid dynamics. Solution methods of initial and boundary value problems of various types will be investigated. Numerical methods, such as finite difference, finite volume, and finite element will be studied.

Combinatorial Game Theory studies finite, two-player games in which there are no ties. Techniques from logic combinatorics and set theory are used to prove various properties of such games. Typical games include Domineering, Hackenbush, and Nim. The analysis of such games can also be used to study other more complex games like Dots and Boxes, and Go. Topics covered in this course include Conway'92s theory of numbers as games, impartial and partizan games, winning strategies, outcome classes and algebra of games.

This course introduces topology and its applications. n Topics covered include topological spaces, quotient and product spaces, metric and normed spaces, connectedness, compactness, and separation axioms. Further topics may include basic algebraic topology, fixed point theorems, theory of knots, and applications to kinematics, game theory, and computer graphics.

This course covers the fundamental techniques and algorithms of counting. Topics include combinations, permutations, lists and strings, distributions, Stirling numbers, partitions, rearrangements and derangements, the principle of inclusion and exclusion, generating functions, and recursion. The course may include further topics such as the Polya-Redfield method, partially ordered sets, enumeration problems from graph theory, Ramsey'92s Theorem, block designs, codes, difference sets, finite geometries, Latin squares and Hadamard matrices.

This course introduces the basic theory of fuzzy sets and fuzzy logic, fuzzy systems, neural networks and neuro-fuzzy systems. Topics in Fuzzy Systems include: fuzzy sets and their operations, membership functions, fuzzy systems of various types, fuzzy control, and fuzzy clustering. Topics in Artificial Neural Networks include: artificial neural networks, the backpropagation algorithm, deep learning, adaptive neuro-fuzzy inference systems. Additional topics may include parameter selection and regularization for neural networks, and convolutional neural networks.

The content of this course may change each time it is offered. n It is for the purpose of offering a new or specialized course of interest to the faculty and students that is not covered by the courses in the current catalog.

This course introduces the foundations of real analysis by means of a rigorous reexamination of the topics covered in elementary calculus. The course starts with the topology of the real line and proceeds to a formal examination of limits, continuity, and differentiability. The course also covers the convergence of sequences and series of real numbers and the uniform convergence of sequences of real valued functions.

A continuation of MAT 400, this course emphasizes the formal treatment of the theory of integration of functions of a real variable. It reexamines the Riemann integral and the Fundamental theorem of calculus as well as the theory of the Stieltjes and Lebesgue integral and their applications in probability and Fourier analysis. The course concludes with a discussion of the topology of R^n, and the differentiability and integrability of functions of several variables, including the theorems of Green and Stokes and the divergence theorem.

This course provides an introduction to the foundations of abstract algebra. The fundamental objects of study are groups, rings, and fields. The student builds on previous courses in algebra, particularly linear algebra, with an even greater emphasis here on proofs. The study of groups is an ideal starting point, with few axioms but a rich landscape of examples and theorems, including matrix groups, homomorphism theorems, group actions, symmetry, and quotient groups. This course extends these ideas to the study of rings and fields. Topics in ring theory include polynomial rings and ideals in rings. The course also covers fields, their construction from rings, finite fields, basic theory of equations, and Galois theory.

This course builds on the foundations established in MAT 450. n It extends the fundamental objects of groups, rings, and fields to include modules over rings and algebras. The course gives the basic ideas of linear algebra a more rigorous treatment and extends scalars to elements in a commutative ring. In this context, students study the general theory of vector spaces and similarity of transformations. The curriculum also discusses non-commutative algebras and rings, emphasizing examples, such as quaternion algebras. Further topics may include non-associative rings and algebras, Galois theory, exact sequences, and homology.

This course is an introduction to parameterized polynomial curves and surfaces with a view toward applications in computer graphics. It discusses both the algebraic and constructive aspects of these topics. Algebraic aspects include vector spaces of functions, special polynomial and piecewise polynomial bases, polynomial interpolation, and polar forms. Constructive aspects include the de Casteljau algorithm and the de Boor algorithm. Other topics may include an introduction to parametric surfaces and multivariate splines.

This course is a continuation of MAT 300 with topics taken from the theory and applications of curves and surfaces. The course treats some of the material from MAT 300 in more detail, like the mathematical foundations for non-uniform rational B-spline (NURBS) curves and surfaces, knot insertion, and subdivision. Other topics may include basic differential geometry of curves and surfaces, tensor product surfaces, and multivariate splines.

This course gives an introduction to several mathematical topics of foundational importance to abstract algebra, and in particular the algebra of quaternions. Topics covered may include: operations, groups, rings, fields, vector spaces, algebras, complex numbers, quaternions, curves over the quaternionic space, interpolation techniques, splines, octonions, and Clifford algebras.

This course presents the foundations of wavelets as a method of representing and approximating functions. It discusses background material in complex linear algebra and Fourier analysis. Basic material on the discrete and continuous wavelet transforms forms the core subject matter. This includes the Haar transform, and multi-resolution analysis. Other topics may include subdivision curves and surfaces, and B-spline wavelets. Applications to computer graphics may include image editing, compression, surface reconstruction from contours, and fast methods of solving 3D simulation problems.

This course presents an introduction to differential geometry, with emphasis on curves and surfaces in three-space. It includes background material on the differentiability of multivariable functions. Topics covered include parameterized curves and surfaces in three-space and their associated first and second fundamental forms, Gaussian curvature, the Gauss map, and an introduction to the intrinsic geometry of surfaces. Other topics may include an introduction to differentiable manifolds, Riemannian geometry, and the curvature tensor.

Topics covered in this course include convex hulls, triangulations, Art Gallery theorems, Voronoi diagrams, Delaunay graphs, Minkowski sums, path finding, arrangements, duality, and possibly randomized algorithms, time permitting. Throughout the course, students explore various data structures and algorithms. The analysis of these algorithms, focusing specifically on the mathematics that arises in their development and analysis is discussed. Although CS 330 is not a prerequisite, it is recommended.

This course provides an introduction to the basic theorems and algorithms of graph theory. Topics include graph isomorphism, connectedness, Euler tours, Hamiltonian cycles, and matrix representation. Further topics may include spanning trees, coloring algorithms, planarity algorithms, and search algorithms. Applications may include network flows, graphical enumeration, and embedding of graphs in surfaces.

This course covers the advanced theory and applications of ordinary differential equations. The first course in differential equations focused on basic prototypes, such as exact and separable equations and the second-degree harmonic oscillator equation. This course builds upon these ideas with a greater degree of generality and theory. Topics include qualitative theory, dynamical systems, calculus of variations, and applications to classical mechanics. Further topics may include chaotic systems and cellular automata. With this overview, students will be prepared to study the specific applications of differential equations to the modeling of problems in physics, engineering, and computer science.

This course covers both the theoretical and practical study of numerical methods used in many areas of computer science, applied mathematics, science and engineering. Topics include: solutions of non-linear equations, interpolation, approximation of functions, quadrature rules, numerical solutions of ordinary differential equations, and numerical methods in linear algebra. Further topics may include Fourier series, wavelets, and stability theory.

This course introduces computational algebra as a tool to study the geometry of curves and surfaces in affine and projective space. The central objects of study are affine varieties and polynomial ideals, and the algebra-geometry dictionary captures relations between these two objects. The precise methods of studying polynomial ideals make use of monomial orderings, Grobner bases, and the Buchberger algorithm. Students have opportunities to program parts of these algorithms and to use software packages to illustrate key concepts. Further topics may include resultants, Zariski closure of algebraic sets, intersections of curves and surfaces, and multivariate polynomial splines.

This course explores topics in linear algebra and abstract algebra. Topics in linear algebra include: vector spaces, transformations, canonical forms, and complex inner product spaces. Topics in abstract algebra include: introduction to abstract groups, rings, fields, and algebras. Further topics may include: modules, multivariate polynomials, algebraic varieties, tensor products, and duality.

This course is an introduction to elementary number theory and cryptography. Among the essential tools of number theory that are covered are divisibility and congruence, Euler'92s function, Fermat'92s little theorem, Euler'92s formula, the Chinese remainder theorem, powers modulo m, kth roots modulo m, primitive roots and indices, and quadratic reciprocity. These tools are then used in cryptography, where the course discusses encryption schemes, the role of prime numbers, security and factorization, the DES algorithm, public key encryption, and various other topics, as time allows.

This course introduces the basic theory of fuzzy sets and fuzzy logic and explores some of their applications. Topics covered include classical sets and their operations, fuzzy sets and their operations, membership functions, fuzzy relations, fuzzification/ defuzzification, classical logic, multi-valued logic, fuzzy logic, fuzzy reasoning, fuzzy arithmetic, classical groups, and fuzz groups. Students will also explore a number of applications, including approximate reasoning, fuzzy control, fuzzy behavior, and interaction in computer games.

This course explores partial differential equations (PDEs) and fluid dynamics. Topics covered in this class include Fourier series, Fourier transforms, classification of PDEs, Poisson'92s equation, heat equation, wave equation, and introductory topics of fluid dynamics. Solution methods of initial and boundary value problems of various types will be investigated. Numerical methods, such as finite difference, finite volume, and finite element will be studied.

Combinatorial Game Theory studies finite two-player games in which there are no ties. Techniques from logic, combinatorics, and set theory are used to prove various properties of such games. Typical games include Domineering , Hackenbush, and Nim. The analysis of such games can also be used to study other more complex games like Dots and Boxes, impartial and partisan games, winning strategies outcome classes, algebra of games.

This course is an introduction to topology and its applications. Topics include: topological spaces, quotient and product spaces, metric and normed spaces, connectedness, compactness, and separation axioms. Further topics may include: basic algebraic topology, fixed point theorems, theory of knots, and applications to kinematics, game theory, and computer graphics.

This course introduces the basic theory of fuzzy sets and fuzzy logic, fuzzy systems, neural networks and neuro-fuzzy systems. Topics in Fuzzy Systems include: fuzzy sets and their operations, membership functions, fuzzy systems of various types, fuzzy control, and fuzzy clustering. Topics in Artificial Neural Networks include: artificial neural networks, the backpropagation algorithm, deep learning, adaptive neuro-fuzzy inference systems. Additional topics may include parameter selection and regularization for neural networks, and convolutional neural networks.

This course explores topics in mathematical analysis of real numbers and functions of real variables. Topics covered in this course include: real numbers, metric spaces, topology of metric spaces, the contraction principle, continuity of functions on metric spaces, differentiability of real-valued functions, sequences and series of functions, continuity and differentiability of functions of several variables, and Riemann integration. Additional topics may include Euclidean spaces, normed spaces, functions of bounded variation, and Riemann-Stieltjes integrals.

This course explores topics in measure theory and functional analysis. The topics covered in this course include: Lebesgue measure, Lebesgue integration, normed spaces, Banach spaces, Fourier series and wavelets, and Hilbert spaces, together with their applications. Additional topics may include Hahn-Banach theorem, bounded linear operators on Hilbert spaces, Riesz representation theorem, Sobolev spaces, and self-adjoint operators.

This course explores topics in complex analysis. Topics include: the complex number field and its geometry, complex functions, limits, complex differentiation, analytic functions, conformal mappings, contour integration, and Laurent series. Additional topics may include: Rouche'92s theorem, the maximum modulus theorem, Liouville'92s theorem, and applications.

This course is a formal introduction to stochastic processes with applications. The main topics are discrete and continuous time Markov chains, Poisson processes, random walks, branching processes, first passage times, recurrence and transience, and stationary distributions. The course also covers Brownian motion and martingales. Other topics may include renewal processes, queues, optimal stopping theory, Monte Carlo methods, and stochastic integration.

This course presents modern statistical concepts and methods developed in a mathematical framework. Topics include statistical inference, point and interval estimation, confidence intervals and hypothesis testing, sufficiency, Neyman-Pearson theory, maximum likelihood, Bayesian analysis, and large sample theory. Additional topics may include decision theory, linear models, and nonparametric statistics.

The content of this course may change each time it is offered. n It is for the purpose of offering a new or specialized course of interest to the faculty and students that is not covered by the courses in the current catalog.