Selected topics designed to acquaint nonspecialists with examples of mathematical reasoning. May not be taken for credit after 215 or higher.

Understanding, communicating, and representing mathematical ideas, problem solving, and reasoning. Number systems, place value, fractions, and properties of operations. Prospective elementary education majors only.

Understanding, communicating, and representing mathematical ideas; problem solving; reasoning and proof; and using symbolism. Patterns and algebraic thinking, place value and decimals, geometry, and mathematical modeling. Pre: 111.

Algebra review, functions with special attention to polynomial, rational exponential and logarithmic functions, composed and inverse functions, techniques of graphing. Credit not allowed for 134 and 140, or 134 and 161. Pre: two years of high school algebra, one year of plane geometry.

Studies trigonometric functions, analytic geometry, polar coordinates, vectors, and related topics. This course is the second part of the precalculus sequence. Credit allowed for one of 134, 135, or 140. Pre: 134, 135, or 161 or assessment exam.

Algebra review, functions with special attention to polynomial, rational, exponential, and logarithmic functions, algebra of functions, techniques of graphing, differentiation and integration of algebraic functions, applications in economics and social sciences. Credit allowed for only one of 134, 135, or 161. A-F only.

(3 hr) Introduction to numerical algorithms and structured programming using Fortran, MATLAB, or other appropriate language. Pre: one semester of calculus (203, 215, 241, 242, 243, 244, 251A, 252A, or 253A) (or concurrent), or consent.

Basic concepts; differentiation and integration applications to management, finance, economics, and the social sciences. Credit allowed for at most one of 203, 215, 241, 251A. Pre: 134, 135, or 161, or assessment exam.

Basic concepts; differentiation, differential equations and integration with applications directed primarily to the life sciences. Credit allowed for at most one of 203, 215, 241, 251A. Pre: 140 or assessment exam.

Differential calculus for functions in several variables and curves, systems of ordinary differential equations, series approximation of functions, continuous probability, exposure to use of calculus in the literature. Pre: 215 or consent.

Basic concepts; differentiation with applications; integration. Credit allowed for at most one of 203, 215, 241, 251A. Pre: 140 or 215 or assessment exam.

Integration techniques and applications, series and approximations, differential equations. Pre: 241 or 251A or a grade of B or better in 215; or consent.

Vector algebra, vector-valued functions, differentiation in several variables, and optimization. Pre: 242 or 252A, or consent.

Multiple integrals; line integrals and Green’s Theorem; surface integrals, Stokes’s and Gauss’s Theorems. Pre: 243 or consent.

Basic concepts; differentiation with applications; integration. Compared to 241, topics are discussed in greater depth. Credit allowed for at most one of 203, 215, 241, 251A. Pre: assessment and consent, or a grade of A in 140 and consent.

Integration techniques and applications, series and approximations, differential equations, introduction to vectors. Pre: 251A, or a grade of B or better in 241 and consent.

Vector calculus; maxima and minima in several variables; multiple integrals; line integrals, surface integrals and their applications. Pre: 252A.

The historical development of mathematical thought. Pre: 216 or 242 or 252A.

Symbolic logic, sets and relations, algorithms, trees and other graphs. Additional topics chosen from algebraic systems, networks, automata. Pre: one semester of calculus from mathematics department; or consent. Recommended: one semester programming.

First order ordinary differential equations, constant coefficient linear equations, oscillations, Laplace transform, convolution, Green’s function. Pre: 216 or 243 (or concurrent) or 253A (or concurrent), or consent.

Constant coefficient linear systems, variable coefficient ordinary differential equations, series solutions and special functions, Fourier series, partial differential equations. Pre: 302, 311 (or concurrent); or consent.

Deterministic mathematical modeling emphasizing models and tools used in the biological sciences. Topics include difference equations, qualitative behavior solutions of ODEs and reaction-diffusion equations. A computer lab may be taken concurrently. Pre: 216 or 242 or 252A, or consent.

Optional laboratory for 304. Pre: 304 (or concurrent).

Probabilistic mathematical modeling emphasizing models and tools used in the biological sciences. Topics include stochastic and Poisson processes, Markov models, estimation, and Monte Carlo simulation. A computer lab may be taken concurrently. Pre: 216 or 242 or 252A, or consent.

Optional laboratory for 305. Pre: 305 (or concurrent).

Introduction to linear algebra, application of eigenvalue techniques to the solution of differential equations. Students may receive credit for only one of 307 or 311. Pre: 242 or 252A, or consent.

Algebra of matrices, linear equations, real vector spaces and transformations. Emphasis on concepts and abstraction and instruction of careful writing. Students may receive credit for only one of 307 or 311. Pre: 242 or 252A, or consent.

Formal introduction to the concepts of logic, finite and infinite sets, functions, methods of proof and axiomatic systems. Learning mathematical expressions in writing is an integral part of the course. Pre: 243 (or concurrent) or 253A (or concurrent), or consent.

A rigorous axiomatic development of one variable calculus. Completeness, topology of the line, limits, continuity, differentiation, integration. Emphasis on teaching mathematical writing. Pre: 242 or 252A, and 321; or consent.

Axiomatic Euclidean geometry and introduction to the axiomatic method. Pre: 243 or 253A, and 321 (or concurrent); or consent.

Hyperbolic, other non-Euclidean geometries. Pre: 351 or consent.

Sets, discrete sample spaces, problems in combinatorial probability, random variables, mathematical expectations, classical distributions, applications. Pre: 216, 242, or 252A; or consent.

Problem-oriented introduction to the basic concepts of probability and statistics, providing a foundation for applications and further study. Pre: 216 or 242 or 252A or consent.

Estimation, tests of significance, the concept of power. Pre: 371 or consent.

Integral surfaces and characteristics of first and second order partial differential equations. Applications to the equations of mathematical physics. Pre: 243 or 253A, or consent. Recommended: 244 and 302.

Laplace’s equation, Fourier transform methods for PDEs, higher dimensional PDEs, spherical harmonics, Laplace series, special functions and applications. Pre: 402 or consent.

Systems of linear ordinary differential equations, autonomous systems, and stability theory applications. Optional topics include series solutions, Sturm theory, numerical methods. Pre: 302 and 311, or consent.

Numerical solution of equations, interpolation, least-squares approximation, quadrature, eigenvalue problems, numerical solution of ordinary and partial differential equations. (These topics are covered in the year sequence 407–408.) Pre: 243 or 253A, and 307 or 311, and one semester programming; or consent.

Continuation of 407. This is the second course of a year sequence and should be taken in the same academic year as 407. Pre: 407 or consent.

Vector spaces over arbitrary fields, minimal polynomials, invariant subspaces, canonical forms of matrices; unitary and Hermitian matrices, quadratic forms. Pre: 307 or 311, and 321; or consent.

Introduction to basic algebraic structures. Groups, finite groups, abelian groups, rings, integral domains, fields, factorization, polynomial rings, field extensions, quotient fields. Emphasis on writing instruction. (These topics are covered in the year sequence 412–413.) Pre: 311 and 321; or consent.

Continuation of 412. This is the second course of a year sequence and should be taken in the same academic year as 412. Emphasis on writing instruction. Pre: 412 or consent.

Techniques of mathematical programming. Topics may include linear programming, integer programming, network analysis, dynamic programming, and game theory. Pre: 307 or 311, or consent.

Congruences, quadratic residues, arithmetic functions, distribution of primes. Emphasis is on teaching theory and writing, not on computation. Pre: 321 or consent.

General topology, including compactness and connectedness; the Jordan Curve Theorem and the classification of surfaces; first homotopy or homology groups. Pre: 321 or consent.

Topology of Rn , metric spaces, continuous functions, Riemann integration, sequences and series, uniform convergence, implicit function theorems, differentials and Jacobians. Emphasis on teaching mathematical writing. (These topics are covered in the year sequence 431–432.) Pre: 311, 321, and 331; or consent.

Continuation of 431. This is the second course of a year sequence and should be taken in the same academic year as 431. Emphasis on writing instruction continues. Pre: 431 or consent.

Vector operations, wedge product, differential forms, and smooth mappings. Theorems of Green, Stokes, and Gauss, both classically and in terms of forms. Applications to electromagnetism and mechanics. Pre: 244 or 253A, and 307 or 311, or consent.

Properties and fundamental geometric invariants of curves and surfaces in space; applications to the physical sciences. Pre: 244 or 253A, and 311; or consent.

Analytic functions, complex integration, introduction to conformal mapping. Pre: 244 or 253A; recommended 331; or consent.

Advanced topics from various areas: algebra, number theory, analysis, and geometry. Repeatable unlimited times. Pre: consent.

Sets, relations, ordinal arithmetic, cardinal arithmetic, axiomatic set theory, axiom of choice and the continuum hypothesis. Pre: 321 or graduate standing in a related field or consent.

A system of first order logic. Formal notions of well-formed formula, proof, and derivability. Semantic notions of model, truth, and validity. Completeness theorem. Pre: 321 or graduate standing in a related field or consent. Recommended: 454.

Probability spaces, random variables, distributions, expectations, moment-generating and characteristic functions, limit theorems. Continuous probability emphasized. Pre: 244 (or concurrent) or 253A (or concurrent), or consent. Recommended: 305 or 371 or 372; or consent.

Sampling and parameter estimation, tests of hypotheses, correlation, regression, analysis of variance, sequential analysis, rank order statistics. Pre: 471 or consent.

Finite configurations. Topics may include counting methods, generating functions, graph theory, map coloring, block design, network flows, analysis of discrete algorithms. Pre: 311 or consent.

Seminar for senior mathematics majors, including an introduction to methods of research. Significant portion of class time is dedicated to the instruction and critique of oral presentations. All students must give the equivalent of three presentations. CR/NC only. Pre: one 400-level mathematics course or consent.

Reports on research in mathematical biology, reviews of literature, and research presentation. Required for Certificate in Mathematical Biology. Repeatable one time. Pre: junior standing or higher and consent. (Cross-listed as BIOL 490)

Limited to advanced students who must arrange with an instructor before enrolling. Repeatable one time, up to six credits.

Enrollment for degree completion. Pre: master’s Plan B or C candidate and consent.

Practicing teachers develop and improve their problem-solving skills by working on challenging mathematical tasks. Students improve their mathematics content knowledge by working on problems and learning to design challenge problems for their own classes. Practicing teachers in grades K-12 only. Repeatable unlimited times. CR/NC only. All 600-courses prerequisites graduate standing or consent.

Continuous and discrete dynamical systems; bifurcation theory; chaotic maps. Additional topics from PDEs and linear algebra. Graduate students only.

Classical existence and uniqueness theory for ODEs and PDEs, qualitative properties, classification, boundary value and initial value problems, fundamental solutions, other topics.

Continuation of 602. This is the second course of a year sequence and should be taken in the same academic year as 602.

Numerical linear algebra including iterative methods, SVD, and other matrix factorizations, locating eigenvalues, discrete approximation to partial differential equations. Recommended: 407, 411, or consent.

Modules, Sylow theorems, Jordan-Holder theorem, unique factorization domains, Galois theory, algebraic closures, transcendence bases. (These topics are covered in the year sequence 611–612.)

Continuation of 611. This is the second course of a year sequence and should be taken in the same academic year as 611.

Sylow theorems, solvable groups, nilpotent groups, extension theory, representation theory, additional topics.

Ideal theory in Noetherian rings, localization, Dedekind domains, the Jacobson radical, the Wedderburn-Artin theorem, additional topics.

Introduction with applications to general algebra. Partially ordered sets, decomposition theory, representations of lattices, varieties and free lattices, coordinatization of modular lattices.

Introduction to basic techniques, including subalgebras, congruences, automorphisms and endomorphisms, varieties of algebras, Mal’cev conditions.

Properties of topological spaces; separation axioms, compactness, connectedness; metrizability; convergence and continuity. Additional topics from general and algebraic topology. (These topics are covered in the year sequence 621–622.)

Continuation of 621. This is the second course of a year sequence and should be taken in the same academic year as 621.

Geometric, topological, and dynamical methods in the study of finitely generated infinite groups. Graduate students only. Pre: 621 (with a minimum grade of B-).

Differentiable structures on manifolds, tensor fields, Frobenius theorem, exterior algebra, integration of forms, Poincare Lemma, Stoke’s theorem.

Lebesgue measure and integral, convergence of integrals, functions of bounded variation, Lebesgue-Stieltjes integral and more general theory of measure and integration. (These topics are covered in the year sequence 631–632.)

Continuation of 631. This is the second course of a year sequence and should be taken in the same academic year as 631.

Linear topological spaces, normed spaces, Hilbert spaces, function algebras, operator theory. Pre: consent.

Simple variational problems, first and second variation formulas. Euler-Lagrange equation, direct methods, optimal control.

Conformal mapping, residue theory, series and product developments, analytic continuation, special functions. (These topics are covered in the year sequence 644–645.)

Continuation of 644. This is the second course of a year sequence and should be taken in the same academic year as 644.

(B) logic; (D) analysis; (E) commutative rings; (F) function theory; (G) geometric topology; (H) operator theory; ((I) probability; (J) algebra; (K) special; (M) lattice theory and universal algebra; (N) noncommutative rings; (O) transformation groups; (P) partial differential equations; (Q) potential theory; (R) algebraic topology; (S) functional analysis; (T) number theory and combinatorics; (U) differentiable manifolds II. Repeatable up to nine credits for (U); unlimited times for the other alphas.

Model theory, computability theory, set theory. In particular syntax and semantics of first order logic; incompleteness, completeness, and compactness theorems; Loewenheim-Skolem theorems; computable and computably enumerable sets; axioms of set theory; ordinals and cardinals. Graduate students only.

Axiomatic development, ordinal and cardinal numbers, recursion theorems, axiom of choice, continuum hypothesis, consistency and independence results.

Recursive, r.e., Ptime, and Logspace classes. Nondeterminism, parallelism, alternation, and Boolean circuits. Reducibility and completeness.

Number fields and rings of integers; primes, factorization, and ramification theory; finiteness of the class group; Dirichlet’s Unit Theorem; valuations, completions, and local fields. Further topics. Graduate students only. Pre: 611 (with a minimum grade of B-).

Independence and conditioning, martingales, ergodic theory, Markov chains, central limit theorem. A-F only. Pre: 631 (with a minimum grade of B) or consent. (Alt. years)

Stationary, Gaussian, and Markov processes. A-F only. Pre: 671 (with a minimum grade of B) or consent. (Alt. years)

Connected graphs and digraphs. Graph embeddings. Connectivity and networks. Factors and factorizations. Coverings. Coloring. Applications.

Maximum of 3 credit hours. Repeatable two times. Graduate standing in MATH. A-F only.

Maximum of 3 credit hours. Repeatable unlimited times.

Research for master’s thesis. Repeatable unlimited times. Pre: consent.

An experience-based introduction to college-level teaching; students serve as student teachers to professors; responsibilities include supervised teaching and participation in planning and evaluation. Open to graduate students in mathematics only. Repeatable one time, up to six credits. CR/NC only. Pre: graduate standing in mathematics and consent.

Research for doctoral dissertation. Repeatable unlimited times.