## Mathematics (MATH)

**1000 Level | 2000 Level | 3000 Level | 4000 Level | 5000 Level**

USP Codes are listed in brackets by the 2003 USP code followed by the 2015 USP code (i.e. [QB<>Q]).

**1000. Problem Solving. 3. [QA<>Q]** For students not planning to enroll in MATH 1400, 1450 or a calculus course. Examines modern topics chosen for their applicability and accessibility. Provides students with mathematical and logical skills needed to formulate, analyze and interpret quantitative arguments in a variety of settings. Introduces statistics and stresses the use of a calculator. Note: MATH 1000 is neither a prerequisite nor suitable preparation for MATH 1400 (College Algebra). *Prerequisite:* grade of C or better in Math 0921 or Level 2 on the Math Placement Exam or Math ACT of 21 or Math SAT of 600.

**1050. Finite Mathematics. 3. [QB<>Q]** Introduces finite mathematics for majors not requiring calculus. Includes matrix algebra, Gaussian elimination, set theory, permutations, probability and expectation. *Prerequisite:* grade of C or better in MATH 1000, 1400 or 1105 or Level 4 on the Math Placement Exam or Math ACT of 26 or Math SAT of 600.

**1080. Bridge Program Math. 3.** This class is to assist students in refreshing their skills in Mathematics, from the fundamentals of arithmetic through college level algebra and trigonometry. If time and preparation allow, basic calculus concepts will be discussed. This class is largely self-paced, but with intrusive faculty support.

**1100. Number and Operations for Elementary School Teachers. 3. [QA<>{none}]** For prospective elementary school teachers; purpose is to prepare students to be competent in teaching the major concepts and skills related to the real number system and four arithmetic operations. Includes asking and answering critical questions about subsets of the real number system, including natural, integer, and rational numbers. *Prerequisite:* grade of C or better in MATH 0921 or Level 2 on the Math Placement Exam or Math ACT of 21 or Math SAT of 600.

**1101. First-Year Seminar. 3. [{none}<>FYS]**

**1105. Data, Probability, and Algebra for Elementary School Teachers. 3. [QB<>Q]** Continuation of MATH 1100 for prospective elementary teachers; emphasis is on asking and answering critical questions about our world through algebra, probability, and data analysis to prepare students to be competent in teaching these major concepts. Explorations focus on representing, analyzing, and generalizing patterns and the chances of future events. *Prerequisite:* grade of C or better in MATH 1100.

**1123. Math, Music and Acoustics. 3.** **[{none}<>Q]** For music majors and minors not planning to enroll in MATH 1400, 1450 or a calculus course. Serves as an introduction to the mathematics and physics underlying music and develops quantitative reasoning in a musical context. Topics include the wave nature of sound, intervals, scales, temperment, acoustics and psychoacoustics. Note: MATH 1123 is neither a prerequisite nor suitable preparation for MATH 1400. *Prerequisites:* grade of C or better in MATH 0921 or Level 2 on the Math Placement Exam or Math ACT of 21 or Math SAT of 600 and grade of C or better in MUSC 1030.

**1400. College Algebra. 3. [QA<>Q]** Emphasizes aspects of algebra important in the study of calculus. Includes notation of algebra, exponents, factoring, theory of equations, inequalities, functions, graphing and logarithms. For students who plan to enroll in a calculus course (MATH 2200 or 2350). Students receiving credit for MATH 1450 may not receive credit for this course. *Prerequisite:* grade of C or better in Math 0925 or Level 3 on the Math Placement Exam or Math ACT of 23 or Math SAT of 600.

**1405. Trigonometry. 3. [QA<>Q]** Emphasizes aspects of trigonometry important in the study of calculus. Interplay between trigonometric expressions and their graphs. Students are expected to use a graphing calculator in the course and on exams. See instructor for specifications. Topics include: angle measurement, trigonometric functions, graphing, laws of sines and cosines, identities, equations, polar equations and graphs, vectors, complex numbers, DeMoivre's theorem. This course is designed for students with little or no prior knowledge of trigonometry who plan to enroll in MATH 2200. Students receiving credit for MATH 1450 may not receive credit for this course. *Prerequisite:* grade of C or better in MATH 1400 or Level 4 on the Math Placement Exam or Math ACT of 25 or Math SAT of 600.

**1450. Algebra and Trigonometry. 5. [QA<>Q]** Emphasizes aspects of algebra, trigonometry and problem solving important in the study of calculus. Functions and their applications to real world problems. Classes of functions including polynomial, exponential, logarithmic and trigonometric functions. Intuitive introduction to the idea of limit and sequence which are developed further in the calculus sequence. For the student with considerable prior exposure to trigonometry and algebra. Graphing calculators are used frequently in class and on assignments. See instructor for specifications. Students with both MATH 1400 and 1405 credit may not receive credit for this course. *Prerequisite:* grade of C or better in MATH 0925 or Level 3 on the Math Placement Exam or Math ACT of 23 or Math SAT of 600.

**2120. Geometry and Measurement for Elementary School Teachers. 3.** Continuation of MATH 1105 for prospective elementary teachers; emphasis is on asking and answering critical questions about spatial reasoning as evident in the real world. Includes investigations of two- and three-dimensional shapes and their properties, measurements, constructions, and transformations to prepare students to be competent in teaching these concepts. *Prerequisite:* grade of C or better in MATH 1105.

**2200. Calculus I. 4. [QB<>Q]** Emphasizes physical science applications. Includes plane analytic geometry, differentiation, applications of the derivative, differential equations, integration and applications. *Prerequisite:* grade of C or better in MATH 1405 or 1450 or Level 5 on the Math Placement Exam or Math ACT of 27 or Math SAT of 600.

**2205. Calculus II. 4.** **[{none}<>Q]** Continues MATH 2200. Includes elementary functions, derivatives, integrals, analytical geometry, infinite series and applications. *Prerequisite:* grade of C or better in MATH 2200 or Advanced Placement credit in MATH 2200.

**2210. Calculus III. 4.** Continues MATH 2200, 2205. Includes vectors and solid analytic geometry, partial differentiation and multiple integration. *Prerequisite:* grade of C or better in MATH 2205 or Advanced Placement credit in MATH 2205.

**2250. Elementary Linear Algebra. 3.** Studies linear equations and matrices, vector spaces, linear transformations, determinants, orthogonality, eigenvalues and eigenvectors. *Prerequisite:* grade of C or better in MATH 2200 or 2350.

**2300. Discrete Structures. 3.** Introduces the mathematical concepts that serve as foundations of computer science: logic, set theory, relations and functions, graphs (directed and undirected), inductively defined structures (lists and trees), and applications of mathematical induction. Provides an introduction to abstract and rigorous thinking in advanced mathematics and computer science. Cross listed with COSC 2300. *Prerequisite:* grade of C or better in COSC 1030, MATH 2200 or 2350.

**2310. Applied Differential Equations I. 3.** Combines with MATH 3310 for a one-year series in applied mathematics. Includes solution of ordinary differential equations, integral transforms. Emphasizes construction of mathematical models arising in physical science and other areas. *Prerequisite:* grade of C or better in MATH 2205. (Note: MATH 2210 is required for the sequel.)

**2350. Business Calculus. 4. [QB<>Q]** Combines with MATH 2355 for a one-year series in business math, primarily for students in the College of Business. Includes review of functions, their graphs and algebra; derivatives and their applications; exponential and logarithmic functions; integration and applications; and applications are generally geared to business problems. *Prerequisite:* grade of C or better in MATH 1400 or Level 4 on the Math Placement Exam or Math ACT of 26 or Math SAT of 600.

**2355. Mathematical Applications for Business. 4.** Continues business and economic applications of mathematics. Also includes linear equations and programming, finance, probability and statistics. Mandatory computer lab using spreadsheet software will meet one day per week. *Prerequisite:* grade of C or better in MATH 2200 or 2350.

**2800. Mathematics Major Seminar. 2.** Introduces mathematics majors and minors to mathematical investigation and discovery. Typically, a range of topics are covered; may include reading assignments and group or individual work on projects for presentation. Offered S/U only.

**2850 [3800]. Putnam Team Seminar. 2 (Max. 8).** Preparation for the William Lowell Putnam Mathematical Competition. Problem solving strategies and mathematical content appropriate for the Putnam Exam are emphasized with problem sets taken from previous Putnam or other international math contests. Offered S/U only. *Prerequisites:* MATH 2200, 2205. (Offered fall semester)

**3205. Analysis I: Elementary Real Analysis. 3.** An introduction to rigorous analysis in one real variable. Includes a rigorous reconsideration of the elements of calculus: the real number system, numerical sequences and series, limits, continuity, differentiability, and Riemann integrability for function of one variable. Proof and mathematical writing are emphasized. *Prerequisite:* Grade of C or better in MATH 2205 and 2800.

**3310. Applied Differential Equations II. 3.** Continues MATH 2310. Includes partial differential equations, Fourier series, boundary value problems, series solutions of ordinary differential equations, linear algebra, linear systems of equations and numerical methods. *Prerequisite:* grade of C or better in MATH 2210 and 2310.

**3340. Introduction to Scientific Computing. 3.** Introduces basic numerical methods to solve scientific and engineering problems. Topics include: code structure and algorithms, basic numerical methods for linear systems, eigenvalue problems, interpolation and data fitting, nonlinear systems, numerical differentiation and integration. Cross listed with COSC 3340. *Prerequisite:* grade of C or better in MATH 2210.

**3500. Algebra I: Introduction to Rings and Proofs. 3.** Begins with common features of integers, rational numbers, and polynomials, leading to study of rings in general. Topics include divisibility, factorization, and modular arithmetic for integers and polynomials, and homomorphisms and ideals for rings. Proof techniques include direct proof, proof by contrapostive, mathematical induction, and proof by contradiction. *Prerequisites:* MATH 2800 and grade C or better in MATH 2250 or concurrent registration in MATH 2250.

**3700. Combinatorics. 3.** Provides an introduction to combinatorics and combinatorial algorithms, with applications to areas such as computer science and probability. Topics include general counting methods, recurrence relations, generating functions, inclusion-exclusion, partial orders, and graph theory. *Prerequisite:* grade of C or better in Math 2250. (Offered fall semester)

**4000. History of Mathematics. 3.** Explores the roots of mathematics and the people who made significant contributions to it. Mathematical subjects typically include algebra, calculus and number theory; both chronological and topical approaches are employed. *Prerequisite:* grade of C or better in MATH 2205. (Offered spring semester)

**4100. Mathematics in the Elementary School. 1‑6 (Max. 6).** Acquaints prospective or experienced teachers of mathematics with newer developments in mathematics curriculum and materials. Emphasizes mathematical basis for courses in an elementary mathematics curriculum; organization and design of mathematics programs for grades K-7; and design and construction of curriculum and/or materials to meet specific needs of the teacher or school district. *Prerequisites:* grade of C or better in MATH 1105 and consent of instructor.

**4150. Secondary School on Campus. 1‑4 (Max. 8).** Provides prospective teachers opportunity to study mathematics as it relates to the secondary school. Topics may vary from semester to semester. Emphasizes current trends and concerns of secondary school mathematics education. *Prerequisites:* grade of C or better in MATH 2205 and concurrent with EDSE 4271. (Offered fall semester)

**4200. Analysis 2: Advanced Analysis. 3.** A second course in analysis. Includes metric space topology, sequences and series of functions, and analysis in R^n. *Prerequisites:* grade of C or better in MATH 2210, 2250 and 3205. (Offered fall semester)

**4205. Analysis 3: Undergraduate Topics in Analysis. 3.** Special topics in analysis. Content varies. May be repeated for credit. *Prerequisite:* grade of C or better in MATH 4200. (Offered spring semester)

**4230. Introduction to Complex Analysis. 3.** Develops the theory of functions of one complex variable. Topics include the algebra and geometry of complex numbers, functions of one complex variable, elementary functions, limits, continuity and differentiation. Differentiability leads to the Cauchy theorem, integral theorems, power series, residue theory and applications to integration theory and boundary value problems. *Prerequisite:* grade of C or better in MATH 2210. (Offered spring semester)

**4255 [4250]. Mathematical Theory of Probability. 3.** Calculus-based. Introduces mathematical properties of random variables. Includes discrete and continuous probability distributions, independence and conditional probability, mathematical expectation, multivariate distributions and properties of normal probability law. Cross listed with STAT 4255. *Prerequisite:* grade of C or better in MATH 2210.

**4265 [4260, 4010]. Introduction to the Theory of Statistics.** Presents derivations of theoretical and sampling distributions. Introduces theory of estimation and hypothesis testing. Cross listed with STAT 4265. *Prerequisite:* MATH 4255.

**4300. Introduction to Mathematical Modeling. 3.** A model of a real world problem captures the essential features of the problem, while scaling it down to a manageable size. In this course, symbolic tools and mathematical techniques are used to construct, analyze and interpret various mathematical models which arise from problems in the physical, biological and social sciences. *Prerequisite:* grade of C or better in MATH 2250 or 3310. (Offered fall semester)

**4340. Numerical Methods for Ordinary and Partial Differential Equations. 3.** Further develops the skills needed for computational problem solving and numerical analysis. Topics addressed include: one-step and linear multistep methods for solving initial value problems; truncation errors, stability analysis, and convergence of the numerical methods; finite difference approximation for elliptic equations and initial boundary value problems; iterative methods for sparse linear systems. Students typically complete a final project in this course. Cross listed with COSC 4340. *Prerequisites:* grade of C or better in MATH 2310 and MATH 3340.

**4400. Topics in Applied Math. 3.** Presents topics in applied mathematics that are of importance for a variety of disciplines in science and engineering. Content will vary and may include: mathematical biology, vector calculus, mathematics for finance, dimensional analysis and perturbation methods and the calculus of variations. *Prerequisites:* grade of C or better in MATH 2210 and 2250. (Offered fall semester)

**4420. Advanced Logic. 3.** Studies advanced topics in mathematical logic. Takes up such topics as: uninterpreted calculi and the distinctive contributions of syntax and semantics; metatheory, including completeness and consistency proofs; modal logic and semantics; logic as a philosophical tool. Dual listed with MATH 5420; cross listed with COSC/PHIL 4420. *Prerequisite:* PHIL 3420 or equivalent.

**4440. Introduction to Partial Differential Equations I. 3.** Survey of analytic methods for solving partial differential equations. Topics include: method of characteristics for solving first-order linear and quasi-linear equations; classification of second-order equations and canonical forms; background to separation of variables with applications; transform methods and Green functions; elliptic equations; heat and wave equations in one dimension. *Prerequisites:* grade of C or better in MATH 2210 and MATH 2310.

**4500. Matrix Theory. 3.** Continuation from MATH 2250 of the study of matrices, an important tool in statistics, physics, engineering and applied mathematics in general. Concentrates on the structure of matrices, including diagonalizability; symmetric, hermitian and unitary matrices; and canonical forms such as Jordan form. *Prerequisite:* grade of C or better in MATH 2250. (Offered fall semester)

**4510. Algebra II: Introduction to Group Theory. 3.** An introduction to the fundamental properties of groups including: binary operations, groups, permutation groups, subgroups, homomorphisms, and quotient groups. *Prerequisite:* grade of C or better in MATH 3500. (Offered spring semester)

**4520. Algebra III: Topics in Abstract Algebra. 3.** Further examples and structure of rings and fields. Finite fields and number fields. Special topics. *Prerequisite:* grade of C or better in MATH 4510. (Offered fall semester)

**4550. Theory of Numbers. 3.** Studies topics in mathematics which are motivated by questions about integers. Topics include divisibility, congruences, diophantine equations, quadratic residues, primitive roots, primes, and representations of positive integers. *Prerequisite:* grade of C or better in MATH 3500. (Offered spring semester)

**4600. Foundations of Geometry. 3.** Broadens the student's understanding of the many faces of geometry and provides a context for the specific case of Euclidean geometry. Various approaches will be presented, including axiomatic, synthetic, coordinate, and transformational methods. *Prerequisite:* grade of C or better in MATH 3205 or 3500. (Offered fall semester)

**4800. Seminar in Mathematics. 1‑3 (Max. 6).** Exposes students to problems and thinking in mathematics which would otherwise be unavailable. *Prerequisite:* consent of instructor.

**4970. Professional Development in Teaching. 1.** Undergraduate student will assist in classroom and discussion section teaching under the guidance of an instructor in Mathematics. Does not count towards Mathematics degree requirements. Offered Satisfactory/Unsatisfactory only. *Prerequisite:* Consent of instructor.

**5090. Topics in the Foundations of Mathematics. 1-6 (Max. 9).** *Prerequisites:* MATH 3000 and consent of instructor.

**5100. Seminar in Elementary School Mathematics. 1-4 (Max. 8).** 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.

**5110. Modeling Flow Transport in Soil and Groundwater Systems. 4.** 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.

**5140. Numbers, Operations, and Patterns for the Middle-level Learner. 3.** 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 the Middle-level Mathematics Program.

**5150. Seminar in Secondary School Mathematics. 1-4 (Max. 18).** Seminar in Secondary School Mathematics. *Prerequisite:* 6 hours of MATH 4150.

**5160. Social and Historical Issues in Mathematics and the Middle-Level Learner. 3.** Empowers teachers of middle-level mathematics 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.

**5170. Connecting Geometry with Problem-Solving for the Middle-Level Learner. 3.** 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.

**5190. Mathematics of Change and the Middle-Level Learner. 3.** 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.

**5200. Real Variables I. 3.** 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 4200.

**5205. Real Variables II. 3.** A continuation of MATH 5200. *Prerequisite:* MATH 5200.

**5230. Complex Variables I. 3.** 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 4200.

**5235. Complex Variables II. 3.** A continuation of MATH 5230. *Prerequisite:* MATH 5230.

**5255. Mathematical Theory of Probability. 3.** 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.

**5265. Introduction to the Theory of Statistics. 3.** 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.

**5270. Functional Analysis I. 3.** 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.

**5275. Functional Analysis II. 3.** 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.

**5290. Topics in Analysis. 1-6 (Max. 18). Topics** in analysis. *Prerequisite:* consent of instructor.

**5310. Computational Methods in Applied Sciences I. 3.** 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. *Prerequisites:* MATH 3310, COSC 1010.

**5320. Mathematical Modeling Processes. 3.** 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. *Prerequisites:* MATH 5310 and graduate standing.

**5340. Computational Methods II. 3.** 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.

**5345. Computational Methods III. 3.** 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.

**5390. Topics in Numerical Analysis. 1-6. (Max 18).** Topics in numerical analysis. *Prerequisite:* consent of instructor.

**5400. Methods of Applied Mathematics I. 3.** 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.

**5405. Methods of Applied Mathematics II. 3.** A continuation of MATH 5400. *Prerequisite:* MATH 5400.

**5420. Advanced Logic. 3.** Studies advanced topics in mathematical logic. Takes up such topics as: uninterpreted calculi and the distinctive contributions of syntax and semantics; metatheory, 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.

**5430. Ordinary Differential Equations II. 3.** 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.

**5440. Partial Differential Equations II. 3.** 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.

**5490. Topics in Applied Mathematics. 1-6 (Max. 18).** *Prerequisite:* consent of instructor.

**5500. Advanced Linear Algebra. 3.** 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 3500 and MATH 4500.

**5510. Combinatorial Theory. 3.** 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:* consent of instructor.

**5530. The Theory of Groups. 3.** 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.

**5550. Abstract Algebra I.** **3.** 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 5500.

**5555. Abstract Algebra II. 3.** A continuation of MATH 5550, examining in depth selected topics from the theory of rings, fields, and algebras, including Galois theory. *Prerequisite:* MATH 5550.

**5570. Matrix Theory and Combinatorics. 3.** 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.

**5590. Topics in Algebra. 1-6 (Max. 18).** Topics in algebra. *Prerequisite:* Consent of instructor.

**5600. Point-Set Topology. 3.** 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 3205.

**5605. Algebraic Topology. 3.** Topics in algebraic topology, including simplicial homology groups and their topological invariance, the Eilenberg-Steenrod axioms, singular homology theory, and cohomology. *Prerequisite:* MATH 3500.

**5640. Differential Geometry. 3.** Curve theory, theory of surfaces, and geometrics on a surface. *Prerequisite:* MATH 4200.

**5690. Topics in Topology. 1-6 (Max. 9).** *Prerequisite:* consent of instructor.

**5700. Topics in Combinatorics. 1-6 (Max. 18).** Selected topics in combinatorial analysis.

**5800. Seminar in Mathematics. 1-3 (Max. 8).** *Prerequisite:* consent of instructor.

**5900. Practicum in College Teaching. 1-3 (Max. 3).** Work in classroom with a major professor. Expected to give some lectures and gain classroom experience. *Prerequisite:* graduate status.

**5920. Continuing Registration: On Campus. 1-2 (Max. 16).** *Prerequisite:* advanced degree candidacy.

**5940. Continuing Registration: Off Campus. 1-2 (Max. 16).** *Prerequisite:* advanced degree candidacy.

**5959. Enrichment Studies. 1-3 (Max. 99).** 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.

**5960. Thesis Research. 1-12 (Max. 24).** 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.

**5980. Dissertation Research. 1-12 (Max. 48).** 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.

**5990. Internship. 1-12 (Max. 24).** *Prerequisite:* graduate standing.