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MATH 5090. Topics in the Foundations of Mathematics. Various topics. Prerequisite: MATH 3000 and consent of instructor.
MATH 5100. Seminar in Elementary School Mathematics. A course to give graduate students in mathematics education, or in-service teachers, an in-depth view of new contents, materials, and strategies for teaching mathematics in elementary schools. The course is primarily designed to meet the needs of students working towards M.S.N.S., M.S.T., M.A.T. degrees. Prerequisite: 6 hours of MATH 4100.
MATH 5110. Modeling Flow Transport in Soil and Groundwater Systems. Mathematical models are formulated and applied to simulate water flow and chemical transcript in soil and groundwater systems. Soil spatial variability and heterogeneity are considered in the modeling processes. Using and comparing models, students obtain the capability to transfer a physical problem to a mathematical model, to use numerical methods, such as the finite element methods, to solve the mathematical problem, and to correctly interpret the numerical outputs. Students develop and program numerical solutions for select problems and utilize existing codes for modeling a variety of comprehensive problems. Cross listed with SOIL 5110.
MATH 5140. Numbers, Operations, and Patterns for the Middle-level Learner. Provides working middle-level mathematics teachers opportunities to understand and discuss numbers, their representations, and operations on them from an abstract perspective that includes elegant proof. Also emphasized is the role of language and purpose in composing definitions. Cross listed with NASC 5140. Prerequisites: admission to a university graduate program, in either degree or non-degree seeking status, and acceptance into Middle-level Mathematics Program.
MATH 5150. Seminar in Secondary School Mathematics. Seminar in Secondary School Mathematics. Prerequisite: 6 hours of MATH 4150.
MATH 5160. Social and Historical Issues in Mathematics and the Middle-Level Learner. Empowers teachers of middle-level mathematicics to design more engaging experiences. Emphasizes the historical context for the development of mathematics, especially its symbols, tools, personalities, and classic problems. Cross listed with NASC 5160. Prerequisites: admission to a UW graduate program, in either degree or non-degree seeking status, and acceptance into the Middle-level Mathematics Program.
MATH 5170. Connecting Geometry with Problem- Solving for the Middle-Level Learner. Showcases two aspects of 2D and 3D geometry: measurement and transformation. Emphasis reflects current state and national standards for middle-level mathematics classroom and teacher preparation, especially appropriate uses of technology, geometric tools, mathematical language, and problem-solving strategies. Cross listed with NASC 5170. Prerequisites: admission to a university graduate program, in either degree or non-degree seeking status, and acceptance into the Middle-level Mathematics Program.
MATH 5190. Mathematics of Change and the Middle-Level Learner. Students gain a solid understanding of data and functions in the service of calculus. Course is hands-on, project-driven and focuses on the essential concepts of functions and calculus and their role in middle-level mathematics. Emphasis is on writing and technology (calculators and probeware). Cross listed with NASC 5190. Prerequisites: admission to a UW graduate program, in either degree or non-degree seeking status, and acceptance into the Middle-level Mathematics Program.
MATH 5200. Real Variables I. Develops the theory of measures, measurable functions, integration theory, density and convergence theorems, product measures, decomposition and differentiation of measures, and elements of function analysis on Lp spaces. Lebesgue theory is an important application of this development. Prerequisite: MATH 4205.
MATH 5205. Real Variables II. A continuation of MATH 5200. Prerequisite: MATH 5200.
MATH 5230. Complex Variables I. Develops the function theory of holomorphic (analytic) and harmonic functions. Topics covered include the Cauchy-Riemann equations, Cauchy-Goursat theorem, Cauchy integral theorem, Morera’s theorem, maximum modulus theorem, Liouville’s theorem, power series representation, harmonic functions, theory of singularities of functions of one complex variable, contour integration, analytic continuation, Riemann mapping theorem and topology of spaces of holomorphic functions. Prerequisite: MATH 4205.
MATH 5235. Complex Variables II. A continuation of MATH 5230. Prerequisites: MATH 5230.
MATH 5255. Mathematical Theory of Probability. Calculus-based. Introduces mathematical properties of random variables. Includes discrete and continuous probability distributions, independence, and conditional probability distributions, independence and conditional probability, mathematical expectation, multivariate distributions and properties of normal probability law. Dual listed with MATH 4255, cross listed with STAT 5255. Prerequisites: grade of C or better in MATH 2210 or 2355.
MATH 5265. Introduction to the Theory of Statistics. Presents derivations of theoretical and sampling distributions. Introduces theory of estimation and hypothesis testing. Dual listed with MATH 4265, cross listed with STAT 5265. Prerequisites: STAT 4250/5250, MATH 4250.
MATH 5270. Functional Analysis I. Topics include the geometry of Hilbert spaces, linear functions and operators on Hilbert spaces, spectral theory of compact normal operators, Banach space theory, the open mapping theorem, Hahn-Banach theorem, Banach- Steinhaus theorem, duality and linear operators on Banach spaces, and different topologies on Banach spaces and their duals. Prerequisite: MATH 5200.
MATH 5275. Functional Analysis II. Topics may include discussion of topological vector spaces, locally convex spaces, F-spaces, spectral theory of non-compact operators on Hilbert spaces, semigroups or evolution operators, distribution theory, and applications to differential equations and Sobolev spaces. Prerequisite: MATH 5270.
MATH 5290. Topics in Analysis. Topics in analysis. Prerequisite: MATH 5200.
MATH 5310. Computational Methods in Applied Sciences I. First semester of a three-semester computational methods series. Review of iterative solutions of linear and nonlinear systems of equations, polynomial interpolation/approximation, numerical integration and differentiation, and basic ideas of Monte Carlo methods. Comparison of numerical techniques for programming time and space requirements, as well as convergence and stability. Identical to COSC 5310. Prerequisite: MATH 3310, COSC 1010.
MATH 5320. Mathematical Modeling Processes. Introduction to techniques in the process of constructing mathematical models. Application of the techniques to areas such as petroleum reservoir simulation, chemical process industry operations, and plant start-up. Identical to CHE 5870. Prerequisite: MATH 5310 and graduate standing.
MATH 5340: Computational Methods II. Second semester of a three-semester computational methods series with emphasis on numerical solution of differential equations. Topics include explicit and implicit methods, methods for stiff ODE problems, finite difference, finite volume, and finite element methods for time-independence PDEs semi/fully discrete methods for time-dependent PDEs. Cross listed with COSC 5340. Prerequisite: MATH 5310.
MATH 5345: Computational Methods III. Third semester of a three-semester computational methods series with emphasis on numerical solution of problems displaying sharp fronts and interfaces (nonlinear conservation laws, Hamilton-Jacobi equations). Cross listed with COSC 5345. Prerequisite: MATH 5340.
MATH 5390: Topics in Numerical Analysis. Topics in numerical analysis. Prerequisite: MATH 5340 or 5345.
MATH 5400: Methods of Applied Mathematics I. First semester of a one-year survey of topics and methods of applied mathematics, with emphasis on applications from physics and engineering. The full sequence includes introductions to mathematical aspects of mechanics (e.g., conservation laws), asymptotic expansions, systems of ODE and stability, integral equations and calculus of variations, PDE with boundary value problems and generalized solutions (including wave, heat, and potential equations), numerical methods and stability. Prerequisite: MATH 2250, 4200 or 4400, and 2310 or 4430.
MATH 5405: Methods of Applied Mathematics II. A continuation of MATH 5400. Prerequisite: MATH 5400.
MATH 5420: Advanced Logic. Studies advanced topics in mathematical logic. Takes up such topics as: uninterpreted calculi and the distinctive contributions of syntax and semantics; methatheory, including completeness and consistency proofs; modal logic and semantics; logic as a philosophical tool. Dual listed with MATH 4420; cross listed with COSC/PHIL 5420. Prerequisite: PHIL 3420 or equivalent; graduate standing.
MATH 5430: Ordinary Differential Equations II. Differential equations constitute the mathematical language for problems of continuous change. ODEs deal with evolutionary processes involving one independent variable. This course revisits solution techniques but emphasizes the theoretical framework. Topics include: existence and uniqueness, linear and nonlinear differential systems, asymptotics and perturbations, and stability. Prerequisite: MATH 4200, 4430.
MATH 5440: Partial Differential Equations II. The theory of PDEs is important for abstract mathematics, applied science, and mathematical modeling. This course covers solution techniques but emphasizes the theoretical framework. Topics include: first order systems; characteristics; hyperbolic, elliptic and parabolic equations; separations of variables; series and transforms; integral relations; Green’s functions, maximum principles; variational methods. Prerequisite: MATH 4200 and 4440.
MATH 5490: Topics in Applied Mathematics. Prerequisite: consent of instructor.
MATH 5500: Advanced Linear Algebra. An introduction to the theory of abstract vector spaces and linear transformations from an axiomatic point of view, with applications to matrix theory. Topics include vector spaces, dimension, linear transformations, dual spaces and functionals, inner product spaces, and structure theorems. Prerequisite: MATH 3000 or 3200, and 4500.
MATH 5510: Combinatorial Theory. An introduction to combinatorics covering both classical and contemporary topics. Includes some of the following: generating functions, recursion formulas, partially ordered sets, inclusion-exclusion, partitions, graph theory, Ramsey theory, combinational optimization, Latin squares, finite geometries, and design theory. Prerequisite: MATH 3500 or 3550.
MATH 5530: The Theory of Groups. An in-depth study of various aspects of group theory, building on MATH 5550. Topics include some of the following: classical theory of finite groups (both Abelian and non-Abelian), infinite Abelian groups, free groups, permutation groups, group representations, endomorphism, extensions, and cohomology. Prerequisite: MATH 5550.
MATH 5550: Abstract Algebra I. Studies the structure of groups, rings, and fields. For each, concepts of substructures, quotient structures, extensions, homomorphism, and isomorphism are discussed. Prerequisite: MATH 3500 or 3550.
MATH 5555: Abstract Algebra II. A continuation of MATH 5550, examining in depth selected topics from the theory of rings, fields, and algebras, including Galois theory. Prerequisite: MATH 5550.
MATH 5570: Matrix Theory and Combinatorics. An overview of matrix theory and its applications to combinatorics. Topics include Smith normal form, the Perron-Frobenius theory of non-negative matrices, location and perturbation of eigenvalues, and interlacing of eigenvalues. Applications include structure theorums for (0,1)-matrices, network flows, spectra of graphs, and the permanent. Prerequisite: MATH 5500.
MATH 5590: Topics in Algebra. Topics in algebra. Prerequisites: MATH 5555.
MATH 5600: Point-Set Topology. Topics considered are metric spaces, open spheres, open sets, closed sets, continuous functions, limit points, topological spaces, homeomorphisms, compactness, connectedness, and separability. The familiar notion of distance on the real number line is generalized to the notion of a metric for an arbitrary set, which is in turn generalized to the concept of a set topology for a set. Certain applications to analysis and geometry are indicated. Prerequisite: MATH 3000 and 4200.
MATH 5605: Topology II. Topics in algebraic topology, including simplicial homology groups and their topological invariance, the Eilenberg-Steenrod axioms, singular homology theory, and cohomology. Prerequisite: MATH 5600.
MATH 5640: Differential Geometry. Curve theory, theory of surfaces, and geometrics on a surface. Prerequisite: MATH 4200 or 4400.
MATH 5690: Topics in Topology. Prerequisite: consent of instructor.
MATH 5700: Topics in Combinatorics. Selected topics in combinatorial analysis.
MATH 5800: Seminar in Mathematics. Prerequisite: consent of Instructor.
MATH 5900: Practicum in College Teaching. Work in classroom with a major professor. Expected to give some lectures and gain classroom experience. Prerequisite: graduate status.
MATH 5920: Continuing Registration: On Campus. Prerequisite: advanced degree candidacy.
MATH 5940: Continuing Registration: Off Campus. Prerequisite: advanced degree candidacy.
MATH 5959: Enrichment Studies. Designed to provide an enrichment experience in a variety of topics. Note: credit in this course may not be included in a graduate Program of Study for degree purposes.
MATH 5960: Thesis Research. Graduate level course designed for students who are involved in research for their thesis project. Also used for students whose coursework is complete and are writing their thesis. Prerequisite: enrollment in a graduate degree program.
MATH 5980: Dissertation Research. Graduate level course designed for students who are involved in research for their dissertation project. Also used for students whose coursework is complete and are writing their dissertation. Prerequisite: enrollment in a graduate level degree program.
MATH 5990: Internship. Prerequisite: graduate standing.