MATH 098 Intermediate Algebra (0)
Intermediate algebra equivalent to third semester of high school algebra. Includes linear equations and models, linear systems in two variables, quadratic equations, completing the square, graphing parabolas, inequalities, working with roots and radicals, distance formula, functions and graphs, exponential and logarithmic functions. Course awarded as transfer equivalency only. Consult the Admissions Equivalency Guide website for more information.
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MATH 112 Application of Calculus to Business and Economics (5) NW, QSR
Rates of change, tangent, derivative, accumulation, area, integrals in specific contexts, particularly economics. Techniques of differentiation and integration. Application to problem solving. Optimization. Credit does not apply toward a mathematics major. Prerequisite: minimum grade of 2.0 in MATH 111. Offered: WSp.
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MATH 208 Matrix Algebra with Applications (3) NW
Systems of linear equations, vector spaces, matrices, subspaces, orthogonality, least squares, eigenvalues, eigenvectors, applications. For students in engineering, mathematics, and the sciences. Prerequisite: minimum grade of 2.0 in MATH 126. Offered: AWSpS.
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MATH 300 Introduction to Mathematical Reasoning (3) NW
Mathematical arguments and the writing of proofs in an elementary setting. Elementary set theory, elementary examples of functions and operations on functions, the principle of induction, counting, elementary number theory, elementary combinatorics, recurrence relations. Prerequisite: minimum grade of 2.0 in either MATH 126 or MATH 136. Offered: AWSpS.
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MATH 318 Advanced Linear Algebra Tools and Applications (3)
Eigenvalues, eigenvectors, and diagonalization of matrices: nonnegative, symmetric, and positive semidefinite matrices. Orthogonality, singular value decomposition, complex matrices, infinite dimensional vector spaces, and vector spaces over finite fields. Applications to spectral graph theory, rankings, error correcting codes, linear regression, Fourier transforms, principal component analysis, and solving univariate polynomial equations. Prerequisite: a minimum grade of 2.7 in either MATH 208 or MATH 308, or a minimum grade of 2.0 in MATH 136.
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MATH 327 Introductory Real Analysis I (3) NW
Covers number systems, fields, order, the least upper bound property, sequences, limits, liminf and limsup, series, convergence tests, alternating series, absolute convergence, re-arrangements of series, continuous functions of a real variable, and uniform continuity. Prerequisite: a minimum grade of 2.0 in either MATH 300 or MATH 334. Offered: AWSpS.
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MATH 328 Introductory Real Analysis II (3) NW
Limits and continuity of functions, sequences, series tests, absolute convergence, uniform convergence. Power series, improper integrals, uniform continuity, fundamental theorems on continuous functions, theory of the Riemann integral. Prerequisite: minimum grade of 2.0 in MATH 327.
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MATH 334 Accelerated [Honors] Advanced Calculus (5) NW
Introduction to proofs and rigor; uniform convergence, Fourier series and partial differential equations, vector calculus, complex variables. Students who complete this sequence are not required to take MATH 209, MATH 224, MATH 300, MATH 327, MATH 328, and MATH 427. Second year of an accelerated two-year sequence; prepares students for senior-level mathematics courses. Prerequisite: either a minimum grade of 2.0 in MATH 136, or a minimum grade of 3.0 in MATH 126 and a minimum grade of 3.0 in either MATH 207 or MATH 307 and a minimum grade of 3.0 in either MATH 208 or MATH 308. Offered: A.
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MATH 335 Accelerated [Honors] Advanced Calculus (5) NW
Introduction to proofs and rigor; uniform convergence, Fourier series and partial differential equations, vector calculus, complex variables. Students who complete this sequence are not required to take MATH 209, MATH 224, MATH 300, MATH 327, MATH 328, and MATH 427. Second year of an accelerated two-year sequence; prepares students for senior-level mathematics courses. Prerequisite: a minimum grade of 2.0 in MATH 334. Offered: W.
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MATH 336 Accelerated [Honors] Advanced Calculus (5) NW
Introduction to proofs and rigor; uniform convergence, Fourier series and partial differential equations, vector calculus, complex variables. Students who complete this sequence are not required to take MATH 209, MATH 224, MATH 300, MATH 327, MATH 328, and MATH 427. Second year of an accelerated two-year sequence; prepares students for senior-level mathematics courses. Prerequisite: a minimum grade of 2.0 in MATH 335. Offered: Sp.
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MATH 340 Abstract Linear Algebra (3) NW
Linear algebra from a theoretical point of view. Abstract vector spaces and linear transformations, bases and linear independence, matrix representations, Jordan canonical form, linear functionals, dual space, bilinear forms and inner product spaces. Prerequisite: a minimum grade of 2.0 in either MATH 334, or both MATH 208 and MATH 300.
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MATH 381 Discrete Mathematical Modeling (3) NW
Introduction to methods of discrete mathematics, including topics from graph theory, network flows, and combinatorics. Emphasis on these tools to formulate models and solve problems arising in variety of applications, such as computer science, biology, and management science. Prerequisite: a minimum grade of 2.0 in either CSE 142, CSE 143, or AMATH 301; and a minimum grade of 2.0 in either MATH 136 or MATH 208. Offered: AW.
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MATH 396 Finite Markov Chains and Monte-Carlo Methods (3) NW
Finite Markov chains; stationary distributions; time reversals; classification of states; classical Markov chains; convergence in total variation distance and L2; spectral analysis; relaxation time; Monte Carlo techniques: rejection sampling, Metropolis-Hastings, Gibbs sampler, Glauber dynamics, hill climb and simulated annealing; harmonic functions and martingales for Markov chains. Prerequisite: a minimum grade of 2.0 in MATH 208; and either a minimum grade of 2.0 in MATH 394/STAT 394 and STAT 395/MATH 395, or a minimum grade of 2.0 in STAT 340 and STAT 341, or a minimum grade of 2.0 in STAT 340 and STAT 395/MATH 395. Offered: jointly with STAT 396; Sp.
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MATH 402 Introduction to Modern Algebra (3) NW
Elementary theory of rings and fields: basic number theory of the integers, congruence of integers and modular arithmetic, basic examples of commutative and non-commutative rings, an in depth discussion of polynomial rings, irreducibility of polynomials, polynomial congruence rings, ideals, quotient rings, isomorphism theorems. Additional topics including Euclidean rings, principal ideal domains and unique factorization domains may be covered. Prerequisite: either a minimum grade of 2.0 in MATH 300 and a minimum grade of 2.0 in either MATH 208 or MATH 308, or a minimum grade of 2.0 in MATH 334. Offered: AWS.
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MATH 403 Introduction to Modern Algebra (3) NW
Elementary theory of groups: basic examples of finite and infinite groups, symmetric and alternating groups, dihedral groups, subgroups, normal subgroups, quotient groups, isomorphism theorems, finite abelian groups. Additional topics including Sylow theorems, group actions, congugacy classes and counting techniques may be covered. Prerequisite: a minimum grade of 2.0 in MATH 402. Offered: WSp.
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MATH 424 Fundamental Concepts of Analysis (3) NW
Focuses on functions of a real variable, including limits of functions, differentiation, Rolle's theorem, mean value theorems, Taylor's theorem, and the intermediate value theorem for derivatives. Riemann-Stieltjes integrals, change of variable, Fundamental Theorem of Calculus, and integration by parts. Sequences and series of functions, uniform convergence, and power series. Prerequisite: either a minimum grade of 2.0 in MATH 327, or a minimum grade of 2.0 in MATH 335. Offered: AWSpS.
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MATH 425 Fundamental Concepts of Analysis (3) NW
Introduction to metric spaces and multivariable differential calculus: Euclidean spaces, abstract metric spaces, compactness, Bolzano-Weierstrass property, sequences and their limits, Cauchy sequences and completeness, Heine-Borel Theorem, continuity, uniform continuity, connected sets and the intermediate value theorem. Derivatives of functions of several variables, chain rule, mean value theorem, inverse and implicit function theorems. Prerequisite: a minimum grade of 2.0 in either MATH 136 or MATH 208; and a minimum grade of 2.0 in either MATH 335 or MATH 424. Offered: WSp.
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MATH 442 Differential Geometry (3) NW
Examines curves in the plane and 3-spaces, surfaces in 3-space, tangent planes, first and second fundamental forms, curvature, the Gauss-Bonnet Theorem, and possible other selected topics. Prerequisite: either minimum grade of 2.0 in MATH 334, or a minimum grade of 2.0 in MATH 208 and a minimum grade of 2.0 in MATH 224; and minimum grade of 2.0 in MATH 441. Offered: W.
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MATH 444 Introduction to Geometries I (3) NW
Concepts of geometry from multiple approaches; discovery, formal and informal reasoning, transformations, coordinates, exploration using computers and models. Topics selected from Euclidean plane and space geometry, spherical geometry, non-Euclidean geometries, fractal geometry. Prerequisite: either a minimum grade of 2.0 in MATH 334, or a minimum grade of 2.0 in MATH 208 and MATH 300. Offered: WS.
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MATH 445 Introduction to Geometries II (3) NW
Concepts of geometry from multiple approaches; discovery, formal and informal reasoning, transformations, coordinates, exploration using computers and models. Topics selected from Euclidean plane and space geometry, spherical geometry, non-Euclidean geometries, fractal geometry. Prerequisite: a minimum grade of 2.0 in MATH 444. Offered: SpS.
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MATH 461 Combinatorial Theory I (3) NW
Basic counting techniques and combinatorial objects. Topics may include permutations, sets, multisets, compositions, partitions, graphs, generating functions, the inclusion-exclusion principle, bijective proofs, and recursions. Prerequisite: a minimum grade of 2.0 in MATH 334, or a minimum grade of 2.0 in MATH 300 and a minimum grade of 2.0 in either MATH 136 or MATH 208.
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MATH 462 Combinatorial Theory II (3) NW
Structural theorems and methods in combinatorics, including those from extremal combinatorics and probabilistic combinatorics. Topics may include graphs, trees, posets, strategic games, polytopes, Ramsey theory, and matroids. Prerequisite: minimum grade of 2.0 in MATH 461 or CSE 421.
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MATH 464 Numerical Analysis I (3) NW
Basic principles of numerical analysis, classical interpolation and approximation formulas, finite differences and difference equations. Numerical methods in algebra, systems of linear equations, matrix inversion, successive approximations, iterative and relaxation methods. Numerical differentiation and integration. Solution of differential equations and systems of such equations. Prerequisite: a minimum grade of 2.0 in either MATH 136, MATH 208, or MATH 335. Offered: A.
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MATH 465 Numerical Analysis II (3) NW
Basic principles of numerical analysis, classical interpolation and approximation formulas, finite differences and difference equations. Numerical methods in algebra, systems of linear equations, matrix inversion, successive approximations, iterative and relaxation methods. Numerical differentiation and integration. Solution of differential equations and systems of such equations. Prerequisite: minimum grade of 2.0 in MATH 464. Offered: W.
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MATH 466 Numerical Analysis III (3) NW
Basic principles of numerical analysis, classical interpolation and approximation formulas, finite differences and difference equations. Numerical methods in algebra, systems of linear equations, matrix inversion, successive approximations, iterative and relaxation methods. Numerical differentiation and integration. Solution of differential equations and systems of such equations. Prerequisite: a minimum grade of 2.0 in either MATH 136, both MATH 207 and MATH 208, or MATH 335.
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MATH 491 Introduction to Stochastic Processes (3) NW
Random walks, Markov chains, branching processes, Poisson process, point processes, birth and death processes, queuing theory, stationary processes. Prerequisite: minimum grade of 2.0 in MATH 394/STAT 394 and MATH 395/STAT 395, or minimum grade of 2.0 in STAT 340 and STAT 341 and MATH 396/STAT 396. Offered: jointly with STAT 491; A.
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MATH 492 Introduction to Stochastic Processes II (3)
Introduces elementary continuous-time discrete/continuous-state stochastic processes and their applications. Covers useful classes of continuous-time stochastic processes (e.g., Poisson process, renewal processes, birth and birth-and-death processes, Brownian motion, diffusion processes, and geometric Brownian motion) and shows how useful they are for solving problems of practical interest. Prerequisite: a minimum grade of 2.0 in MATH 491/STAT 491. Offered: jointly with STAT 492.
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MATH 493 Stochastic Calculus for Option Pricing (3) NW
Introductory stochastic calculus mathematical foundation for pricing options and derivatives. Basic stochastic analysis tools, including stochastic integrals, stochastic differential equations, Ito's formula, theorems of Girsanov and Feynman-Kac, Black-Scholes option pricing, American and exotic options, bond options. Prerequisite: minimum grade of 2.0 in either STAT 395/MATH 395, or a minimum grade of 2.0 in STAT 340 and STAT 341. Offered: jointly with STAT 493.
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MATH 507 Algebraic Structures (3)
First quarter of a three-quarter sequence covering hom*ological algebra, advanced commutative algebra, and Lie algebras and representation theory. Specific topics include chain complexes, resolutions and derived functors, dimension theory, Cohen-Macaulay modules, Gorenstein rings, local cohom*ology, local duality, triangulated and derived categories, group cohom*ology, and structure and representation. Prerequisite: MATH 506 or equivalent.
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MATH 508 Algebraic Structures (3)
Second quarter of a three-quarter sequence covering hom*ological algebra, advanced commutative algebra, and Lie algebras and representation theory. Specific topics include chain complexes, resolutions and derived functors, dimension theory, Cohen-Macaulay modules, Gorenstein rings, local cohom*ology, local duality, triangulated and derived categories, group cohom*ology, and structure and representation. Prerequisite: MATH 506.
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MATH 509 Algebraic Structures (3)
Third quarter of a three-quarter sequence covering hom*ological algebra, advanced commutative algebra, and Lie algebras and representation theory. Specific topics include chain complexes, resolutions and derived functors, dimension theory, Cohen-Macaulay modules, Gorenstein rings, local cohom*ology, local duality, triangulated and derived categories, group cohom*ology, and structure and representation. Prerequisite: MATH 506.
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MATH 514 Networks and Combinatorial Optimization (3)
Mathematical foundations of combinatorial and network optimization with an emphasis on structure and algorithms with proofs. Topics include combinatorial and geometric methods for optimization of network flows, matching, traveling salesmen problem, cuts, and stable sets on graphs. Special emphasis on connections to linear and integer programming, duality theory, total unimodularity, and matroids. Prerequisite: either MATH 208 or AMATH 352; and any additional 400-level MATH course. Offered: jointly with AMATH 514.
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MATH 515 Optimization: Fundamentals and Applications (5)
Maximization and minimization of functions of finitely many variables subject to constraints. Basic problem types and examples of applications; linear, convex, smooth, and nonsmooth programming. Optimality conditions. Saddlepoints and dual problems. Penalties, decomposition. Overview of computational approaches. Prerequisite: Proficiency in linear algebra and advanced calculus/analysis; recommended: Strongly recommended: probability and statistics. Desirable: optimization, e.g. Math 408, and scientific programming experience in Matlab, Julia or Python. Offered: jointly with AMATH 515/IND E 515.
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MATH 527 Functional Analysis (3)
First of three-quarter sequence. Review of Banach, Hilbert, and Lp spaces; locally convex spaces (duality and separation theory, distributions, and function spaces); operators on locally convex spaces (adjoints, closed graph/open mapping and Banach-Steinhaus theorems); Banach algebras (spectral theory, elementary applications); spectral theorem for Hilbert space operators. Working knowledge of real variables, general topology, complex variables.
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MATH 534 Complex Analysis (5)
First quarter of a three-quarter sequence covering complex numbers, analytic functions, contour integration, power series, analytic continuation, sequences of analytic functions, conformal mapping of simply connected regions, and related topics. Prerequisite: MATH 426.
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MATH 544 Topology and Geometry of Manifolds (5)
First quarter of a three-quarter sequence covering general topology, the fundamental group, covering spaces, topological and differentiable manifolds, vector fields, flows, the Frobenius theorem, Lie groups, hom*ogeneous spaces, tensor fields, differential forms, Stokes's theorem, deRham cohom*ology. Prerequisite: MATH 404 and MATH 426 or equivalent.
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MATH 547 Geometric Structures (3, max. 9)
First quarter of a three-quarter sequence covering differential-geometric structures on manifolds, Riemannian metrics, geodesics, covariant differentiation, curvature, Jacobi fields, Gauss-Bonnet theorem. Additional topics to be chosen by the instructor, such as connections in vector bundles and principal bundles, symplectic geometry, Riemannian comparison theorems, symmetric spaces, complex manifolds, Hodge theory. Prerequisite: MATH 546
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MATH 554 Linear Analysis (5)
First quarter of a three-quarter sequence covering advanced linear algebra and matrix analysis, ordinary differential equations (existence and uniqueness theory, linear systems, numerical approximations), Fourier analysis, introductions to functional analysis and partial differential equations, distribution theory. Prerequisite: MATH 426 and familiarity with complex analysis at the level of MATH 427 (the latter may be obtained concurrently).
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MATH 561 Foundations of Combinatorics (3)
First quarter of a three-quarter sequence on combinatorics, covering topics selected from among enumeration, generating functions, ordered structures, graph theory, algebraic combinatorics, geometric combinatorics, and extremal and probabilistic combinatorics. Prerequisite: familiarity with linear algebra, discrete probability, and MATH 504, 505, 506, which may be taken concurrently.
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MATH 562 Foundations of Combinatorics (3)
Second quarter of a three-quarter sequence on combinatorics, covering topics selected from among enumeration, generating functions, ordered structures, graph theory, algebraic combinatorics, geometric combinatorics, and extremal and probabilistic combinatorics. Prerequisite: MATH 561.
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MATH 563 Foundations of Combinatorics (3)
Third quarter of a three-quarter sequence on combinatorics, covering topics selected from among enumeration, generating functions, ordered structures, graph theory, algebraic combinatorics, geometric combinatorics, and extremal and probabilistic combinatorics. Prerequisite: MATH 562.
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MATH 567 Algebraic Geometry (3)
First quarter of a three-quarter sequence covering the basic theory of affine and projective varieties, rings of functions, the Hilbert Nullstellensatz, localization, and dimension; the theory of algebraic curves, divisors, cohom*ology, genus, and the Riemann-Roch theorem; and related topics. Prerequisite: MATH 506.
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MATH 584 Applied Linear Algebra and Introductory Numerical Analysis (5)
Numerical methods for solving linear systems of equations, linear least squares problems, matrix eigen value problems, nonlinear systems of equations, interpolation, quadrature, and initial value ordinary differential equations. Prerequisite: either a course in linear algebra or permission of instructor. Offered: jointly with AMATH 584; A.
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MATH 586 Numerical Analysis of Time Dependent Problems (5)
Numerical methods for time-dependent differential equations, including explicit and implicit methods for hyperbolic and parabolic equations. Stability, accuracy, and convergence theory. Spectral and pseudospectral methods. Prerequisite: either AMATH 581, AMATH 584/MATH 584, AMATH 585/MATH 585, or permission of instructor. Offered: jointly with AMATH 586/ATM S 581; Sp.
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