Math 5090 Topics in the Foundation of Mathematics
Math 5100 Seminar in Elementary School Mathematics
Math 5110 Modeling Flow Transport in Soil and Groundwater Systems
Math 5140 Numbers, Operations, and Patterns for the Middle-Level Learner
Math 5150 Seminar in Secondary School Mathematics
Math 5160 Social and Historical Issues in Mathematics and the Middle-Level Learner
Math 5170 Connecting Geometry with Problem Solving for the Middle-Level Learner
Math 5190 Mathematics of Change and the Middle-Level Learner
Math 5200 Real Variables I
Math 5205 Real Variables II
Math 5230 Complex Variables I
Math 5235 Complex Variables II
Math 5270 Functional Analysis I
Math 5275 Functional Analysis II
Math 5290 Topics in Analysis
Math 5310 Computational Methods in Applied Sciences I
Math 5320 Mathematics Modeling of Processes
Math 5340 Computational Methods in Applied Sciences II
Math 5345 Computational Methods in Applied Sciences III
Math 5390 Topics in Numerical Analysis
Math 5400 Methods of Applied Mathematics I
Math 5405 Methods of Applied Mathematics II
Math 5430 Ordinary Differential Equations II
Math 5440 Partial Differential Equations II
Math 5460 Theory of Approximation
Math 5490 Topics in Applied Mathematics
Math 5500 Advanced Linear Algebra
Math 5510 Combinatorial Theory
Math 5530 The Theory of Groups
Math 5550 Abstract Algebra I
Math 5555 Abstract Algebra II
Math 5570 Matrix Theory and Combinatorics
Math 5590 Topics in Algebra
Math 5600 Point-Set Topology
Math 5605 Topology II
Math 5640 Differential Geometry
Math 5690 Topics in Topology
Math 5700 Topics in Combinatorics
Math 5800 Seminar in Mathematics

Prerequisite: MATH 3000

Course Description: A course to give graduate students in mathematics education, or inservice 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 MSNS, MST degrees.

Prerequisites: 6 hours of MATH  4100  or consent of instructor (1-4 hours)

Course Description: Mathematical models will be formulated and applied to simulate water flow and chemical transport in soil and underground water systems. Soil spatial variability and heterogeneity will be considered in the modeling processes. Using and comparing models, students will obtain the capability to transfer a physical problem to a mathematical model, to use numerical methods such as the finite element method, to solve the mathematical problem, and to correctly interpret the numerical outputs. Students will develop and program numerical solutions for select problems and will utilize existing codes for modeling a variety of other comprehensive problems.
Prerequisites: MATH  2310 . Familiar with programming in either FORTRAN or Pascal.

Course Description: 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.
Prerequisites: Admission to the UW Graduate School, in either degree or non-degree seeking status, and acceptance into the Middle-level Mathematics Program. Identical to NASC 5140. (3 hours)

MATH 5150: Seminar in Secondary School Mathematics

Course Description: A course to give graduate students in mathematics education, or inservice teachers, an in-depth view of new contents, materials, and strategies for teaching mathematics in secondary schools. The course is primarily designed to meet the needs of students working towards MSNS, MST degrees.

Prerequisites: 6 hours of MATH  4150  or consent of instructor (1-4 hours)

Course Description: Empowers teachers of middle-level mathematics to design more engaging experiences. It will emphasize symbols, tools, personalities, and classic problems.
Prerequisites: Admission to the UW Graduate School, in either degree or non-degree seeking status, and acceptance into the Middle-level Mathematics Program. Identical to NASC 5160. (3 hours)

Course Description: 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.
Prerequisites: Admission to the UW Graduate School, in either degree or non-degree seeking status, and acceptance into the Middle-level Mathematics Program. Identical to NASC 5170. (3 hours)

Course Description: Students gain a solid understanding of data and functions in the service of calculus. Hands-on, project driven course that focuses on the essential concepts of functions and calculus and their role in middle-level mathematics. Emphasis is on using writing and technology (calculators and probeware).
Prerequisites: Admission to the UW Graduate School, in either degree or non-degree seeking status, and acceptance into the Middle-level Mathematics Program. Identical to NASC 5190. (3 hours)

MATH 5200: Real Variables I

Course Description: 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.  Lebseque theory is an important application of this development.

Prerequisites: MATH 4205 (3 hours)

MATH 5205: Real Variables II

Course Description: A continuation of MATH 5200.

Prerequisites: MATH 5200 (3 hours)

MATH 5230: Complex Variables I

Course Description: 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, topology of spaces of holomorphic functions.

Prerequisites: MATH 4205 or consent of instructor (3 hours)

MATH 5235: Complex Variables II

Course Description: A continuation of MATH 5230.

Prerequisites: MATH 5230 (3 hours)

MATH 5270: Functional Analysis I

Course Description: Topics in this course 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.

Prerequisites: MATH 5200 or consent of instructor. (3 hours)

MATH 5275: Functional Analysis II

Course Description: A continuation of MATH 5270. Topics may include discussion of topological vector spaces, locally convex spaces, F-spaces, sepectral theory of non-compact operators on Hilbert spaces, semigroups or evolution operators, distribution theory, and applications to differential equations and Sobolev spaces.
Prerequisites: MATH 5270 (3 hours)

MATH 5290: Topics in Analysis

Prerequisites: MATH 5200 or consent of instructor. (1-6 hours)

Course Description: 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.
Prerequisites: Math 3310 and COSC 1010. Identical to COSC 5310, CHE 5140, ME 5140, and CE 5140. (3 hours)

MATH 5320: Mathematical Modeling of Processes

Course Description: 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.
Prerequisites: MATH 5310 and Graduate Standing. Identical to CHE 5879 and PETE 5870. (3 hours)

MATH 5340: Computational Methods in Applied Sciences II

Course Description: 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; and semi/fully discrete methods for time-dependent PDEs.
Prerequisites: MATH 5310. Cross listed with COSC 5340. (3 hours)

MATH 5345: Computational Methods in Applied Sciences III

Course Description: 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 equation).
Prerequisites: MATH 5340. Cross listed with COSC 5345. (3 hours)

MATH 5390: Topics in Numerical Analysis

Prerequisites: MATH 5340, MATH 5345 or consent of instructor. (3 hours)

MATH 5400: Methods of Applied Mathematics I

Course Description: First semester of a one-year survey of topics and methods of applied mathematics; 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.
Prerequisites: MATH 2250, 4200 or 4400, and 2310 or 4430. (3 hours)

MATH 5405: Methods of Applied Mathematics II

Course Description: A continuation of MATH 5400.
Prerequisites: MATH 5400.

MATH 5430: Ordinary Differential Equations II

Course Description: Differential equations constitute the mathematical language for problems of continuous change. ODEs deals 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.
Prerequisites: MATH 4200 and MATH 4430 or consent of instructor. (3 hours)

MATH 5440: Partial Differential Equations II

Course Description: 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.
Prerequisites: MATH 4200 and MATH 4400 or consent of instructor. (3 hours)

MATH 5460: Theory of Approximation

Course Description: Successive approximations and general iterative methods for systems of algebraic differential and integral equations. Approximation of functions by trigonometric polynomials, algebraic polynomials, rational functions, polynomial operators and Tchebycheff polynomials, asymptotic and orthogonal expansions, real and complex interpolation, uniform approximation, and functional equations.
Prerequisites: MATH 4205 or consent of instructor. (3 hours)

MATH 5490: Topics in Applied Mathematics

Prerequisites: Consent of instructor. (1-6 hours)

MATH 5500: Advanced Linear Algebra

Course Description: 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.
Prerequisites: MATH   3000 or  3200 and  4500 or consent of instructor. (3 hours)

MATH 5510: Combinatorial Theory

Course Description: An introduction to combinatorics covering both classical and contemporary topics. The course will include some of the following: generating functions, recursion formulas, partially ordered sets, inclusion-exclusion, partitions, graph theory, Ramsey theory, combinatorial optimization, Latin squares, finite geometries, and design theory.
Prerequisites: MATH  3500  or  3550

MATH 5530: The Theory of Groups

Course Description: 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.
Prerequisites: MATH 5550 or consent of instructor. (3 hours)

MATH 5550: Abstract Algebra I

Course Description: Studies the structure of groups, rings, and fields. For each, concepts of substructures, quotient structures, extensions, homomorphism and isomorphism are discussed.
Prerequisites:  MATH  3500 or 3550 or consent of instructor.

MATH 5555: Abstract Algebra II

Course Description: A continuation of MATH 5550, examining in depth selected topics from the theory of rings, fields and algebras, including Galois theory.
Prerequisites: MATH 5550 or consent of instructor. (3 hours)

Course Description: 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 theorems for (0,1)-matrices, network flows, spectra of graphs, and the permanent.
Prerequisites: MATH 5550 or consent of instructor. (3 hours)

MATH 5590: Topics in Algebra

Prerequisites: MATH 5550 or consent of instructor. (3 hours)

MATH 5600: Point-Set Topology

Course Description: Topics considered are metric spaces, open spheres, open sets, closed sets, continuous functions, limit points, topological spaces, homeomorphisms, compactness, connnectedness, 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 topology for a set. Certain applications to analysis and geometry are indicated.
Prerequisites: MATH  3000 and  4200 .

Course Description: Topics in algebraic topology, including simplical homology groups and their topological invariance, the Eilenberg-Steenrod axioms, singular homology theory, and cohomology.
Prerequisite: MATH  5600

Course Description: Curve theory, theory of surfaces, and geometrics on a surface.

Prerequisites: MATH 4200 or MATH 4400 or consent of instructor.

MATH 5690: Topics in Topology

Prerequisites: Consent of instructor (1-6 hours)

MATH 5700: Topics in Combinatorics

Prerequisite: Consent of instructor. (1-3 hours)

MATH 5800: Seminar in Mathematics

Prerequisite: Consent of instructor. (1-3 hours)