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MATHEMATICS

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**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 100 Algebra (5)**

Similar to the first three terms of high school algebra. Assumes no previous experience in algebra. Open only to students [1] in the Educational Opportunity Program or [2] admitted with an entrance deficiency in mathematics. Offered: A.

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**MATH 102 Algebra (5)**

Similar to the first three terms of high school algebra. Assumes no previous experience in algebra. Open only to students [1] in the Educational Opportunity Program or [2] admitted with an entrance deficiency in mathematics. Prerequisite: minimum grade of 2.0 in MATH 100. Offered: AW.

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**MATH 103 Introduction to Elementary Functions (5)**

Continues the study of algebra begun in MATH 100 and MATH 102 with emphasis on functions (polynomial, rational, logarithmic, exponential, and trigonometric). Open only to students who have completed MATH 102. Prerequisite: minimum grade of 2.0 in MATH 102. Offered: WSp.

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**MATH 108 International Baccalaureate (IB) Mathematical Studies (5) NW**

Course awarded based on International Baccalaureate (IB) score. Consult the Admissions Exams for Credit website for more information.

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**MATH 109 International Baccalaureate (IB) Standard Level Mathematics (5) NW**

Course awarded based on International Baccalaureate (IB) score. Consult the Admissions Exams for Credit website for more information.

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**MATH 111 Algebra with Applications (5) NW, QSR**

Use of graphs and algebraic functions as found in business and economics. Algebraic and graphical manipulations to solve problems. Exponential and logarithm functions; various applications to growth of money. Recommended: completion of Department of Mathematics' Guided Self-Placement. Offered: AWS.

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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 115 Study Abroad Mathematics 1 (1-10, max. 15)**

Mathematics courses taken through a UW approved study abroad program. Content varies and must be individually evaluated.

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**MATH 120 Precalculus (5) NW**

Basic properties of functions, graphs; with emphasis on linear, quadratic, trigonometric, exponential functions and their inverses. Emphasis on multi-step problem solving. Recommended: completion of Department of Mathematics' Guided Self-Placement. Offered: AWSpS.

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**MATH 124 Calculus with Analytic Geometry I (5) NW, QSR**

First quarter in calculus of functions of a single variable. Emphasizes differential calculus. Emphasizes applications and problem solving using the tools of calculus. Recommended: completion of Department of Mathematics' Guided Self-Placement.
Offered: AWSpS.

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**MATH 125 Calculus with Analytic Geometry II (5) NW**

Second quarter in the calculus of functions of a single variable. Emphasizes integral calculus. Emphasizes applications and problem solving using the tools of calculus. Prerequisite: either minimum grade of 2.0 in MATH 124, score of 3 on AB advanced placement test, or score of 3 on BC advanced placement test. Offered: AWSpS.

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**MATH 126 Calculus with Analytic Geometry III (5) NW**

Third quarter in calculus sequence. Introduction to Taylor polynomials and Taylor series, vector geometry in three dimensions, introduction to multivariable differential calculus, double integrals in Cartesian and polar coordinates. Prerequisite: either a minimum grade of 2.0 in MATH 125, or a score of 4 on BC advanced placement test. Offered: AWSpS.

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**MATH 134 Accelerated [Honors] Calculus (5) NW, QSR**

Covers the material of MATH 124, MATH 125, MATH 126; MATH 307, MATH 308. First year of a two-year accelerated sequence. May receive advanced placement (AP) credit for MATH 124 after taking MATH 134. For students with above average preparation, interest, and ability in mathematics. Offered: A.

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**MATH 135 Accelerated [Honors] Calculus (5) NW**

Covers the material of MATH 124, MATH125, MATH 126; MATH 307, MATH 308. First year of a two-year accelerated sequence. May receive advanced placement (AP) credit for MATH 125 after taking MATH 135. For students with above average preparation, interest, and ability in mathematics. Prerequisite: Minimum grade of 2.0 in MATH 134. Offered: W.

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**MATH 136 Accelerated [Honors] Calculus (5) NW**

Covers the material of MATH 124, MATH 125, MATH 126; MATH 307, MATH 308. First year of a two-year accelerated sequence. May not receive credit for both MATH 126 and MATH 136. For students with above average preparation, interest, and ability in mathematics. Prerequisite: minimum grade of 2.0 in MATH 135. Offered: Sp.

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**MATH 197 Problem Solving in Mathematics (2, max. 4) NW**

Lectures and problem sessions in mathematics with applications. Enrollment restricted to EOP students only. Credit/no-credit only. Offered: AWSp.

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**MATH 198 Special Topics in Mathematics (1-5, max. 15)**

Independent reading in math. Does not count as credit toward a math major. Credit/no-credit only. Offered: AWSpS.

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**MATH 215 Study Abroad Mathematics 2 (1-10, max. 15)**

Mathematics courses taken through a UW approved study abroad program. Content varies and must be individually evaluated.

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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 301 Elementary Number Theory (3) NW**

Brief introduction to some of the fundamental ideas of elementary number theory. Prerequisite: minimum grade of 2.0 in MATH 126 and MATH 300, or minimum grade of 2.0 in MATH 136, or minimum grade of 2.0 in MATH 334.

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**MATH 307 Introduction to Differential Equations (3) NW**

Introductory course in ordinary differential equations. Includes first- and second-order equations and Laplace transform. Prerequisite: minimum grade of 2.0 in MATH 125. Offered: AWSpS.

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**MATH 308 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 309 Linear Analysis (3) NW**

First order systems of linear differential equations, Fourier series and partial differential equations, and the phase plane. Prerequisite: either a minimum grade of 2.0 in both MATH 307 and MATH 308 or minimum grade of 2.0 in MATH 136. Offered: AWSpS.

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**MATH 315 Study Abroad Mathematics 3 (1-10, max. 15)**

Mathematics courses taken through a UW approved study abroad program. Content varies and must be individually evaluated.

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**MATH 324 Advanced Multivariable Calculus I (3) NW**

Topics include double and triple integrals, the chain rule, vector fields, line and surface integrals. Culminates in the theorems of Green and Stokes, along with the Divergence Theorem. Prerequisite: minimum grade of 2.0 in either MATH 126 or MATH 136. Offered: AWSpS.

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**MATH 327 Introductory Real Analysis I (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 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 300, MATH 309, MATH 324, MATH 327, MATH 328, and MATH 427. Second year of an accelerated two-year sequence; prepares students for senior-level mathematics courses. Prerequisite: either minimum grade of 2.0 in MATH 136, or minimum grade of 3.0 in all MATH 126 and MATH 307 and 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 300, MATH 309, MATH 324, MATH 327, MATH 328, and MATH 427. Second year of an accelerated two-year sequence; prepares students for senior-level mathematics courses. Prerequisite: 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 300, MATH 309, MATH 324, MATH 327, MATH 328, and MATH 427. Second year of an accelerated two-year sequence; prepares students for senior-level mathematics courses. Prerequisite: 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: minimum grade of 2.0 in MATH 334 or 2.0 in each of MATH 300 and MATH 308.

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**MATH 342 Art of Problem Solving (3) NW**

Explores the artful side of problem-solving, with examples from various fields across mathematics, including combinatorics, number theory, algebra, geometry, probability, and analysis. Offered: A.

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**MATH 380 Intermediate Topics in Undergraduate Mathematics (3, max. 12) NW**

Covers intermediate topics in undergraduate mathematics.

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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: minimum grade of 2.0 in either CSE 142, CSE 143, or AMATH 301; minimum grade of 2.0 in either MATH 136 or MATH 308. Offered: AW.

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**MATH 394 Probability I (3) NW**

Axiomatic definitions of probability; random variables; conditional probability and Bayes' theorem; expectations and variance; named distributions: binomial, geometric, Poisson, uniform (discrete and continuous), normal and exponential; normal and Poisson approximations to binomial. Transformations of a single random variable. Markov and Chebyshev's inequality. Weak law of large numbers for finite variance. Prerequisite: either a minimum grade of 2.0 in MATH 126, or a minimum grade of 2.0 in MATH 136. Offered: jointly with STAT 394; AWS.

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**MATH 395 Probability II (3) NW**

Jointly distributed random variables; conditional distributions and densities; conditional expectations and variance; covariance, correlation, and Cauchy-Schwarz inequality; bivariate normal distribution; multivariate transformations; moment generating functions; sums of independent random variables; Central Limit Theorem; Chernoff's inequality; Jensen's inequality. Prerequisite: either a minimum grade of 2.0 in MATH 394/STAT 394, or a minimum grade of 2.0 in STAT 340. Offered: jointly with STAT 395; WSpS.

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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 308; 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 398 Special Topics in Mathematics (1-5, max. 15)**

Independent reading in math. Does not count as credit toward a math major. Credit/no-credit only. Offered: AWSpS.

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**MATH 399 Undergraduate Research (1-5, max. 15)**

Research opportunity for undergraduates. Offered: AWSp.

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**MATH 402 Introduction to Modern Algebra (3) NW**

Elementary theory of groups: Cosets and Lagrange's theorem. Homomorphisms, normal subgroups, quotient groups, and the fundamental isomorphism theorems. Cyclic and symmetric groups. Orders and Cauchy's theorem. Direct products. Automorphisms. Prerequisite: either minimum grade of 2.0 in MATH 300 and MATH 308, or MATH 334.
Offered: AS.

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**MATH 403 Introduction to Modern Algebra (3) NW**

Elementary theory of rings and fields: polynomial rings. Ideals, homomorphisms, quotients, and fundamental isomorophism theorems. Fields and maximal ideals. Euclidean rings. Field extensions. Algebraic extensions. Vector spaces and degrees of extensions. Adjoining roots of polynomials. Finite fields. Straight edge and compass constructions. Prerequisite: minimum grade of 2.0 in MATH 402. Offered: W.

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**MATH 404 Introduction to Modern Algebra (3) NW**

Topics in algebra chosen from Galois theory, theory of modules, geometric group actions, and the theory of rings and fields. Specific content determined by instructor. Prerequisite: minimum grade of 2.0 in MATH 403. Offered: Sp.

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**MATH 407 Linear Optimization (3) NW**

Maximization and minimization of linear functions subject to constraints consisting of linear equations and inequalities; linear programming and mathematical modeling. Simplex method, elementary games and duality. Prerequisite: minimum grade of 2.0 in either MATH 136, MATH 308, or AMATH 352. Offered: AW.

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**MATH 408 Nonlinear Optimization (3) NW**

Maximization and minimization of nonlinear functions, constrained and unconstrained; nonlinear programming problems and methods. Lagrange multipliers; Kuhn-Tucker conditions, convexity. Quadratic programming. Prerequisite: minimum grade of 2.0 in MATH 407 or MATH 464; minimum grade of 2.0 in either MATH 327 or MATH 334. Offered: W.

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**MATH 409 Discrete Optimization (3) NW**

Maximization and minimization problems in graphs and networks (shortest paths, minimum spanning trees, maximum flows, minimum cost flows); transportation and trans-shipment problems, NP-completeness. Prerequisite: minimum grade of 2.0 in MATH 407; and either a minimum grade of 2.0 in MATH 300, or a minimum grade of 2.0 in MATH 334. Offered: Sp.

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**MATH 411 Introduction to Modern Algebra with Applications I (3) NW**

Basic concepts of abstract algebra with an emphasis on problem solving, constructing proofs, and communication of mathematical ideas. Cannot be taken for credit if credit received for MATH 402 or MATH 403. Prerequisite: a minimum grade of 2.0 in either MATH 136 or MATH 308. Offered: AS.

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**MATH 412 Introduction to Modern Algebra with Applications II (3) NW**

Basic concepts of abstract algebra with an emphasis on problem solving, constructing proofs, and communication of mathematical ideas. Cannot be taken for credit if credit received for MATH 402 or MATH 403. Prerequisite: a minimum grade of 2.0 in MATH 411. Offered: WS.

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**MATH 415 Study Abroad Mathematics 4 (1-10, max. 15)**

Mathematics courses taken through a UW approved study abroad program. Content varies and must be individually evaluated.

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**MATH 420 History of Mathematics (3) NW**

Survey of the development of mathematics from its earliest beginnings through the first half of the twentieth century. Prerequisite: minimum grade of 2.0 in either MATH 126 or MATH 136. Offered: S.

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**MATH 424 Fundamental Concepts of Analysis (3) NW**

The real number system; field, order, and LUB axioms. Metric spaces: Euclidean space. Bolzano-Weierstrass property. Sequences and limits of sequences. Cauchy sequences and completeness. The Heine-Borel Theorem. Uniform continuity. Connected sets and the intermediate value theorem. Prerequisite: either minimum grade of 2.0 in MATH 327, or minimum grade of 2.0 in MATH 335. Offered: AWSpS.

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**MATH 425 Fundamental Concepts of Analysis (3) NW**

One-variable differential calculus: chain rule, inverse function theorem, Rolle's theorem, intermediate value theorem, Taylor's theorem, and intermediate value theorem for derivatives. Multivariable differential calculus: mean value theorem, inverse and implicit function theorems, and Lagrange multipliers. Prerequisite: minimum grade of 2.0 in either MATH 136 or MATH 308; and minimum grade of 2.0 in either MATH 335 or MATH 424 Offered: W.

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**MATH 426 Fundamental Concepts of Analysis (3) NW**

Lebesgue measure on the reals. Construction of the Lebesgue integral and its basic properties. Monotone Convergence Theorem, Fatou's Lemma, and Dominated Convergence Theorem. Integration of series. Continuity and differentiability theorems for functions defined by integrals. Introduction to general measures and integration. Prerequisite: minimum grade of 2.0 in MATH 425. Offered: Sp.

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**MATH 427 Complex Analysis (3) NW**

Complex numbers; analytic functions; sequences and series; complex integration; Cauchy integral formula; Taylor and Laurent series; uniform convergence; residue theory; conformal mapping. Topics chosen from: Fourier series and integrals, Laplace transforms, infinite products, complex dynamics; additional topics chose by instructor. Prerequisite: minimum grade of 2.0 in either MATH 327 or MATH 335. Offered: AS.

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**MATH 428 Complex Analysis (3) NW**

Continuation of MATH 427. Prerequisite: either minimum grade of 2.0 in MATH 427 or MATH 336 Offered: W.

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**MATH 441 Topology (3) NW**

Metric and topological spaces, convergence, continuity, finite products, connectedness, and compactness. Prerequisite: minimum grade of 2.0 in either MATH 327 or MATH 335. Offered: AS.

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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 335, or a minimum grade of 2.0 in MATH 308 and a minimum grade of 2.0 in MATH 324; and minimum grade of 2.0 in MATH 441. Offered: W.

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**MATH 443 Differential Geometry (3) NW**

Further 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: minimum grade of 2.0 in MATH 442. Offered: Sp.

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**MATH 444 Geometry for Teachers (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. Designed for teaching majors. Cannot be used as elective credit for either BS program in mathematics. Prerequisite: either minimum grade of 2.0 in MATH 334, or minimum grade of 2.0 in MATH 300 and MATH 308. Offered: WS.

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**MATH 445 Geometry for Teachers (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. Designed for teaching majors. Cannot be used as elective credit for either BS program in mathematics. Prerequisite: 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: minimum grade of 2.0 in MATH 334, or minimum grade of 2.0 in MATH 300 and minimum grade of 2.0 in either MATH 136 or MATH 308.

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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: minimum grade of 2.0 in either MATH 136, MATH 308, 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: minimum grade of 2.0 in either MATH 136, both MATH 307 and MATH 308, or MATH 335.

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**MATH 480 Advanced Topics in Undergraduate Mathematics (3, max. 12)**

Covers advanced topics in undergraduate mathematics.

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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 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 492.

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**MATH 496 Honors Senior Thesis (1-5) NW**

Problem seminar for Honors students. Cannot be repeated for credit. Offered: AWSp.

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**MATH 497 Special Topics in Mathematics for Teachers (2-9, max. 9) NW**

Study of selected areas of mathematics. Designed for the improvement of teachers of mathematics. Offered: jointly with EDC&I 478.

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**MATH 498 Special Topics in Mathematics (1-5, max. 15)**

Reading and lecture course intended for special needs of advanced students. Offered: AWSpS.

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**MATH 499 Undergraduate Research (8) NW**

Summer research opportunity for undergraduates. Credit/no-credit only. Offered: S.

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**MATH 504 Modern Algebra (5)**

First quarter of a three-quarter sequence covering group theory; field theory and Galois theory; commutative rings and modules, linear algebra, theory of forms; representation theory, associative rings and modules; commutative algebra and elementary algebraic geometry. Prerequisite: MATH 404 or equivalent.

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**MATH 505 Modern Algebra (5)**

Continuation of MATH 504. Prerequisite: MATH 504.

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**MATH 506 Modern Algebra (5)**

Continuation of MATH 505. Prerequisite: MATH 505.

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**MATH 507 Algebraic Structures (3)**

First quarter of a three-quarter sequence covering homological 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 cohomology, local duality, triangulated and derived categories, group cohomology, 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 homological 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 cohomology, local duality, triangulated and derived categories, group cohomology, and structure and representation. Prerequisite: MATH 506.

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**MATH 509 Algebraic Structures (3)**

Third quarter of a three-quarter sequence covering homological 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 cohomology, local duality, triangulated and derived categories, group cohomology, and structure and representation. Prerequisite: MATH 506.

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**MATH 510 Seminar in Algebra (2-5, max. 12)**

Prerequisite: permission of Graduate Program Coordinator. Credit/no-credit only.

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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 308 or AMATH 352 any additional 400-level mathematics 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 516 Numerical Optimization (3)**

Methods of solving optimization problems in finitely many variables, with or without constraints. Steepest descent, quasi-Newton methods. Quadratic programming and complementarity. Exact penalty methods, multiplier methods. Sequential quadratic programming. Cutting planes and nonsmooth optimization. Offered: jointly with AMATH 516.

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**MATH 518 Theory of Optimal Control (3)**

Trajectories from ordinary differential equations with control variables. Controllability, optimality, maximum principle. Relaxation and existence of solutions. Techniques of nonsmooth analysis. Prerequisite: real analysis on the level of MATH 426; background in optimization corresponding to MATH 515. Offered: jointly with AMATH 518.

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**MATH 521 Advanced Probability (3)**

Measure theory and integration, independence, laws of large numbers. Fourier analysis of distributions, central limit problem and infinitely divisible laws, conditional expectations, martingales. Prerequisite: either MATH 426 or MATH 576. Offered: jointly with STAT 521; A.

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**MATH 522 Advanced Probability (3)**

Measure theory and integration, independence, laws of large numbers. Fourier analysis of distributions, central limit problem and infinitely divisible laws, conditional expectations, martingales. Prerequisite: either MATH 426 or MATH 576. Offered: jointly with STAT 522; W.

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**MATH 523 Advanced Probability (3)**

Measure theory and integration, independence, laws of large numbers. Fourier analysis of distributions, central limit problem and infinitely divisible laws, conditional expectations, martingales. Prerequisite: either MATH 426 or MATH 576. Offered: jointly with STAT 523; Sp.

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**MATH 524 Real Analysis (5)**

First quarter of a three-quarter sequence covering the theory of measure and integration, point set topology, Banach spaces, Lp spaces, applications to the theory of functions of one and several real variables. Additional topics to be chosen by instructor. Prerequisite: MATH 426 or equivalent.

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**MATH 525 Real Analysis (5)**

Continuation of MATH 524. Prerequisite: MATH 524.

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**MATH 526 Real Analysis (5)**

Continuation of MATH 525. Prerequisite: MATH 525.

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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 528 Functional Analysis (3)**

Continuation of MATH 527. Prerequisite: MATH 527.

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**MATH 529 Functional Analysis (3)**

Continuation of MATH 528. Prerequisite: MATH 528.

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**MATH 530 Seminar in Analysis (2-5, max. 12)**

Prerequisite: permission of graduate program coordinator. Credit/no-credit only.

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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 535 Complex Analysis (5)**

Continuation of MATH 534. Prerequisite: MATH 534.

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**MATH 536 Complex Analysis (5)**

Continuation of MATH 535. Prerequisite: MATH 535.

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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, homogeneous spaces, tensor fields, differential forms, Stokes's theorem, deRham cohomology. Prerequisite: MATH 404 and MATH 426 or equivalent.

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**MATH 545 Topology and Geometry of Manifolds (5)**

Continuation of MATH 544. Prerequisite: MATH 544.

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**MATH 546 Topology and Geometry of Manifolds (5)**

Continuation of MATH 545. Prerequisite: MATH 545.

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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 548 Geometric Structures (3, max. 9)**

Continuation of MATH 547. Prerequisite: MATH 547.

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**MATH 549 Geometric Structures (3, max. 9)**

Continuation of MATH 548. Prerequisite: MATH 548.

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**MATH 550 Seminar in Geometry (2-5, max. 12)**

Prerequisite: permission of Graduate Program Coordinator. Credit/no-credit only.

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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 555 Linear Analysis (5)**

Continuation of MATH 554. Prerequisite: MATH 554.

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**MATH 556 Linear Analysis (5)**

Continuation of MATH 555. Prerequisite: MATH 555.

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**MATH 557 Introduction to Partial Differential Equations (3)**

First quarter of a three-quarter sequence. Reviews the theory of distribution theory, weak derivatives, and Fourier transform; Laplace, heat, wave, Schrodinger equations; and notion of Euler-Lagrange equation and variational derivative. Prerequisite: either MATH 526 or MATH 556.

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**MATH 558 Introduction to Partial Differential Equations (3)**

Continuation of MATH 557. Covers Sobolev spaces; boundary value problems; additional topics may include: Cauchy-Kowalevski theorem, first order equations, initial value problems, and variational methods. Prerequisite: MATH 557.

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**MATH 559 Introduction to Partial Differential Equations (3)**

Continuation of MATH 558. Covers selected topics such as: introduction to microlocal analysis, Lax parametrix construction, Schauder estimates, Calderon-Zygmund theory, energy methods, and boundary regularity on rough domains. Prerequisite: MATH 558.

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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 564 Algebraic Topology (3)**

First quarter of a three-quarter sequence covering classical and modern approaches; complexes and their homology theory; applications; fixed points, products and Poincare duality; axiomatic approach. Prerequisite: MATH 506 and MATH 544, or equivalent.

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**MATH 565 Algebraic Topology (3)**

Continuation of MATH 564. Prerequisite: MATH 564.

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**MATH 566 Algebraic Topology (3)**

Continuation of MATH 565. Prerequisite: MATH 565.

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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, cohomology, genus, and the Riemann-Roch theorem; and related topics. Prerequisite: MATH 506.

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**MATH 568 Algebraic Geometry (3)**

Continuation of MATH 567. Prerequisite: MATH 567.

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**MATH 569 Algebraic Geometry (3)**

Continuation of MATH 568. Prerequisite: MATH 568.

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**MATH 570 Seminar in Topology (2-5, max. 12)**

Prerequisite: permission of graduate program coordinator. Credit/no-credit only.

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**MATH 574 Fundamental Concepts of Analysis (3)**

Sets, real numbers, topology of metric spaces, normed linear spaces, multivariable calculus from an advanced viewpoint. Introduction to Lebesque measure and integration. Intended for students in biostatistics and related fields; does not fulfill requirements for degrees in mathematics.

View course details in MyPlan: MATH 574

**MATH 575 Fundamental Concepts of Analysis (3)**

Sets, real numbers, topology of metric spaces, normed linear spaces, multivariable calculus from an advanced viewpoint. Introduction to Lebesque measure and integration. Intended for students in biostatistics and related fields; does not fulfill requirements for degrees in mathematics.

View course details in MyPlan: MATH 575

**MATH 576 Fundamental Concepts of Analysis (3)**

Sets, real numbers, topology of metric spaces, normed linear spaces, multivariable calculus from an advanced viewpoint. Introduction to Lebesque measure and integration. Intended for students in biostatistics and related fields; does not fulfill requirements for degrees in mathematics.

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**MATH 577 Lie Groups and Lie Algebras (3, max. 9)**

Topics chosen from: root systems and reflection groups; the structure, classification, and representation theory of complex semisimple Lie algebras, compact Lie groups, or semisimple Lie groups; algebraic groups; enveloping algebras; infinite-dimensional representation theory of Lie groups and Lie algebras; harmonic analysis on Lie groups. Prerequisite: MATH 506; MATH 526 or MATH 546.

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**MATH 578 Lie Groups and Lie Algebras (3, max. 9)**

Topics chosen from: root systems and reflection groups; the structure, classification, and representation theory of complex semisimple Lie algebras, compact Lie groups, or semisimple Lie groups; algebraic groups; enveloping algebras; infinite-dimensional representation theory of Lie groups and Lie algebras; harmonic analysis on Lie groups. Prerequisite: MATH 506; MATH 526 or MATH 546.

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**MATH 579 Lie Groups and Lie Algebras (3, max. 9)**

Topics chosen from: root systems and reflection groups; the structure, classification, and representation theory of complex semisimple Lie algebras, compact Lie groups, or semisimple Lie groups; algebraic groups; enveloping algebras; infinite-dimensional representation theory of Lie groups and Lie algebras; harmonic analysis on Lie groups. Prerequisite: MATH 506; MATH 526 or MATH 546.

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**MATH 580 Current Topics in Mathematics (2, max. 12)**

Discussion of current research topics in mathematics, with emphasis on current departmental research projects and interests. Offered: AWSp.

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**MATH 581 Special Topics in Mathematics (1-5, max. 36)**

Advanced topics in various areas of mathematics. Offered: A.

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**MATH 582 Special Topics in Mathematics (1-5, max. 36)**

Advanced topics in various areas of mathematics. Offered: W.

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**MATH 583 Special Topics in Mathematics (1-5, max. 36)**

Advanced topics in various areas of mathematics. Offered: Sp.

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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 585 Numerical Analysis of Boundary Value Problems (5)**

Numerical methods for steady-state differential equations. Two-point boundary value problems and elliptic equations. Iterative methods for sparse symmetric and non-symmetric linear systems: conjugate-gradients, preconditioners. Prerequisite: either AMATH 581, AMATH 584/MATH 584, or permission of instructor. Offered: jointly with AMATH 585; W.

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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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**MATH 590 Seminar in Probability (2-5, max. 12)**

Prerequisite: permission of instructor. Credit/no-credit only.

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**MATH 597 Seminar on Teaching Math (1, max. 3)**

Issues in the teaching and learning of college mathematics, such as discovering and working with student background and expectations, increasing student engagement with course material, and evaluating student achievement. For graduate students who are, or soon will be, teaching mathematics courses on their own. Credit/no-credit only.

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**MATH 600 Independent Study or Research (*-)**

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**MATH 700 Master's Thesis (*-)**

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**MATH 800 Doctoral Dissertation (*-)**

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