MAT 100

Pre-Calculus with Linear Algebra and Geometry (4 Cr.)

Prerequisites: None

Description: This course is intended for the student with a weak high school math background. It presents a review of college algebra and trigonometry. The most basic part covers a review of functions and their graphs. This course emphasizes polynomial, rational, trigonometric, exponential and logarithmic functions as well as their inverses. Topics in trigonometry include analytic trigonometry and identities, the unit circle, and trigonometric functions of a real variable. Other topics include systems of equations and conic sections. MAT 140 instead of MAT 100 is recommended for RTIS freshmen. Students may only earn credit for one of MAT 100 or MAT 140.

MAT 140

Linear Algebra and Geometry (4 Cr.)

Prerequisites: None

Description: The two main themes throughout the course are vector geometry and linear transformations. Topics from vector geometry include vector arithmetic, dot product, cross product, and representations of lines and planes in three-space. Linear transformations covered include rotations, reflections, shears and projections. Students will study the matrix representations of linear transformations along with their derivations. The curriculum also presents Affine geometry and affine transformations along with connections to computer graphics. This course also includes a review of relevant algebra and trigonometry concepts. MAT 140 is recommended for RTIS freshmen. Students may only earn credit for one of MAT 100 or MAT 140.

MAT 150

Calculus and Analytic Geometry I (4 Cr.)

Prerequisite: MAT 100 or MAT 140

Description: This course introduces the calculus of functions of a single real variable. The main topics include limits, differentiation, and integration. Limits include the graphical and intuitive computation of limits, algebraic properties of limits, and continuity of functions. Differentiation topics include techniques of differentiation, optimization, and applications to graphing. Integration includes Riemann sums, the definite integral, antiderivatives, and the Fundamental Theorem of Calculus.

MAT 200

Calculus and Analytic Geometry II (4 Cr.)

Prerequisite: MAT 150

Description: This course builds on the introduction to calculus in MAT 150. Topics in integration include applications of the integral in physics and geometry and techniques of integration. The course also covers sequences and series of real numbers, power series and Taylor series, and calculus of transcendental functions. Further topics may include a basic introduction to concepts in multivariable and vector calculus.

MAT 225

Calculus and Analytic Geometry III (3 Cr.)

Prerequisite: MAT 200

Description: 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 will focus on curvature, torsion, and the TNB-frame. Topics in vector analysis include multiple integrals, vector fields, Green’s Theorem, the Divergence Theorem and Stokes’ Theorem. Additionally, the course may cover the basics of differential equations.

MAT 250

Linear Algebra (3 Cr.)

Prerequisites: MAT 140 and MAT 150, or MAT 200

Description: 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.

MAT 256

Introduction to Differential Equations (3 Cr.)

Prerequisite: MAT 200

Description: This course introduces the basic theory and applications of first and second-order linear differential equations. The class will emphasize 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.

MAT 258

Discrete Mathematics (3 Cr.)

Prerequisites: MAT 140 and MAT 150, or MAT 200

Description: 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.

MAT 290

Linear Algebra and Geometry of Curves (3 Cr.)

Prerequisite: MAT 200

Description: This course combines material from MAT 250 and MAT 300 into a single course. Topics from linear algebra include vector spaces, linear transformations, change of basis, function spaces, and piecewise polynomials. Topics from geometry include Bezier curves, splines, interpolation, and constructive curves and surfaces. Students may not earn credit for MAT 290 if they also earned credit for either MAT 250 or MAT 300.

MAT 300

Curves and Surfaces (3 Cr.)

Prerequisites: MAT 250, MAT 258

Description: This course is an introduction to parametrized polynomial curves and surfaces with a view toward applications in computer graphics. It will discuss 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.

MAT 340

Probability and Statistics (3 Cr.)

Prerequisites: MAT 200, MAT 258

Description: 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.

MAT 350

MAT 550

Advanced Curves and Surfaces (3 Cr.)

Prerequisite: MAT 300

Description: This course is a continuation of MAT 300 with topics taken from the theory and applications of curves and surfaces. The class will treat some of the material from MAT 300 in more detail, like the mathematical foundations for nonuniform 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.

MAT 351

MAT 551

Quaternions, Interpolation and Animation (3 Cr.)

Prerequisite: MAT 300

Description: This course covers topics in abstract algebra and geometry woven together by the thread of quaternions. This particular thread is chosen with an eye toward applications in computer graphics, specifically to the interpolation of rotation operators and their relations to animation. The course will also present a self-contained summary of abstract algebra, including elements of finite groups, rings, fields, and real algebras. Students will pay attention to certain examples of these structures, including the complex numbers as an algebra, and the unit complex numbers as the rotation group in the plane. These examples in two dimensions are then extended to three and four dimensions with the study of division algebras and Hamilton’s quaternion algebra. In particular, the unit quaternion sphere is exhibited as a two-fold cover of the rotation group of threespace. The second part of the course will review basic differential geometry, continuity, and differentiability of curves in four-space. Here the students will have the opportunity to combine their knowledge of abstract algebra with their knowledge of parametrized curves and produce some interesting visual applications to three-dimensional animation.

MAT 352

MAT 552

Wavelets (3 Cr.)

Prerequisites: MAT 250, MAT 258

Description: This course presents the foundations of wavelets as a method of representing and approximating functions. It will discuss 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 multiresolution 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.

MAT 353

MAT 553

Differential Geometry (3 Cr.)

Prerequisite: MAT 300

Description: This course presents an introduction to differential geometry, with emphasis on curves and surfaces in three-space. It will include background material on the differentiability of multivariable functions. Topics covered include parametrized 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.

MAT 354

MAT 554

Discrete and Computational Geometry (3 Cr.)

Prerequisites: MAT 250, MAT 258

Description: This course gives an introduction to the basic theorems and algorithms of computational geometry, with particular attention paid to mathematical foundations. Topics include convex hulls, the theory of triangulation, Art Gallery Theorems, Voronoi diagrams, and the Delaunay graph. Further topics may include Minkowski sums, path finding, and randomized algorithms. CS 330 (Analysis of Algorithms) is recommended background for this course.

MAT 355

MAT 555

Graph Theory (3 Cr.)

Prerequisites: MAT 250, MAT 258

Description: 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.

MAT 356

MAT 556

Advanced Differential Equations (3 Cr.)

Prerequisites: MAT 250, MAT 256

Description: 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.

MAT 357

MAT 557

Numerical Analysis (3 Cr.)

Prerequisites: MAT 250, MAT 258

Description: This course covers the numerical techniques arising in many areas of computer science and applied mathematics. Such techniques provide essential tools for obtaining approximate solutions to nonlinear equations arising from the construction of mathematical models of real-world phenomena. Topics of study include root finding, interpolation, approximation of functions, cubic splines, integration, and differential equations. Further topics may include stability, iterative methods for solving systems of equations, eigenvalue approximation, and the Fast Fourier Transform.

MAT 359

MAT 559

Computational Algebraic Geometry (3 Cr.)

Prerequisite: MAT 300

Description: 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 will 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.

MAT 361

MAT 561

An Introduction to Number Theory and Cryptography (3 Cr.)

Prerequisites: MAT 250, MAT 258

Description: This course introduces topics from classical number theory and discusses the applications of some of these topics to the subject of cryptography. Topics from classical number theory include divisibility, the Euclidean Algorithm, congruences, and quadratic reciprocity. Further topics in number theory may include finite fields, number fields, and arithmetic of elliptic curves. Topics from cryptography include factoring algorithms, public key cryptosystems, the discrete log problem, zero knowledge protocols, the RSA algorithm, and primality testing.

MAT 362

Fuzzy Sets and Logic (3 Cr.)

Prerequisites: MAT 250 & MAT258

Description: This course introduces the basic theory of Fuzzy Sets and Fuzzy Logic and explores some of their applications. Topics include: Classical Sets and their Operations, Fuzzy Sets and their Operations, Membership Functions, Fuzzy Relations, Fuzzification and Defuzzification, Classical Logic, Fuzzy Logic, Fuzzy Reasoning, Fuzzy Systems, Decision Making with Fuzzy Information, Classical Groups, Fuzzy Groups. Applications include: Approximate Reasoning, Fuzzy Control Systems, Fuzzy Behavior and Interaction in Computer Games.

MAT 390

MAT 590

Special Topics (3 Cr.)

Prerequisite: Permission of instructor

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

MAT 399

MAT 599

Independent Study (3 Cr.)

Prerequisite: Permission of instructor

Description: An independent study allows a student to take a course by meeting regularly one-on-one with an instructor instead of attending scheduled lectures. Through reading and self-study in lieu of lectures, this type of study places the onus of covering new material on the student. Because of this, an independent study course is usually more difficult than a regular class, and only strong math students should expect to get faculty permission for an independent study.

MAT 400

Introductory Analysis I (3 Cr.)

Prerequisite: MAT 250

Description: This course will introduce 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 will also cover the convergence of sequences and series of real numbers and the uniform convergence of sequences of real valued functions.

MAT 410

Introductory Analysis II (3 Cr.)

Prerequisite: MAT 400

Description: 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.

MAT 450

Abstract Algebra I (3 Cr.)

Prerequisites: MAT 250, MAT 258

Description: This course provides an introduction to the foundations of abstract algebra. The fundamental objects of study are groups, rings, and fields. The student will build 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 will extend these ideas to the study of rings and fields. Topics in ring theory include polynomial rings and ideals in rings. The course will also cover fields, their construction from rings, finite fields, basic theory of equations, and Galois theory.

MAT 460

Abstract Algebra II (3 Cr.)

Prerequisite: MAT 400

Description: This course builds on the foundations established in MAT 450. It will extend the fundamental objects of groups, rings, and fields to include modules over rings and algebras. The course will give the basic ideas of linear algebra a more rigorous treatment and extend scalars to elements in a commutative ring. In this context, students will study the general theory of vector spaces and similarity of transformations. The curriculum will also discuss 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.