## Mathematics (MATH)
*College
of Natural Sciences *
**MATH 100 Survey of Mathematics (3)** Selected topics designed
to acquaint nonspecialists with examples of mathematical reasoning. May
not be taken for credit after 215 or higher. **FS**
**MATH 111 Math for Elementary Teachers I (3)** Understanding,
communicating, and representing mathematical ideas, problem solving, and
reasoning. Operations on sets, natural numbers, integers, fractions, reals,
and functions. The properties of these operations; patterns and algebra.
Prospective elementary education majors only.
**MATH 112 Math for Elementary Teachers II (3)** Representations
of and operations on the natural numbers, integers, rationals and reals;
properties of those operations. Connections to other parts of mathematics
and applications. Pre: 111 or consent.** FS**
**MATH 140 Trigonometry and Analytic Geometry (3)** Functions,
with special attention to polynomial, rational, exponential, logarithmic,
and trigonometric functions, plane trigonometry, polar coordinates, conic
sections. Pre: two years of high school algebra, one year of plane geometry,
and precalculus assessment. **FS **
**MATH 190 Fortran Programming (1) **An introduction to
numerical algorithms and programming in Fortran (and possibly Basic).
Pre: one semester of calculus (or concurrent), or consent.
**MATH 203 Calculus for Business and Social Sciences (3)**
Basic concepts; differentiation and integration; applications to management,
finance, economics, and the social sciences. Pre: two years high school
algebra, one year plane geometry, and precalculus assessment. **FS**
**MATH 207 History of Mathematics (3) **The historical development
of mathematical thought. Pre: one year of calculus. Recommended: 311 or
321.
**MATH 215 Applied Calculus I (4)** Basic concepts; differentiation,
differential equations and integration with applications directed primarily
to the life sciences. Pre: C or better in 140 or precalculus assessment.
**FS **
**MATH 216 Applied Calculus II (3)** 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.
**MATH 241 Calculus I (4)** Basic concepts; differentiation
with applications; integration. Pre: a grade of C or better in 140 or
215 or precalculus assessment. **FS **
**MATH 242 Calculus II (4)** Integration techniques and
applications, series and approximations, differential equations. Pre:
a grade of C or better in 241 or 251 or a grade of B or better in 215;
or consent.
**MATH 243 Calculus III (3) **Vector algebra, vector-valued
functions, differentiation in several variables, and optimization. Pre:
a grade of C or better in 242 or 252; or consent.
**MATH 244 Calculus IV (3)** Multiple integrals; line integrals
and Green’s Theorem; surface integrals, Stokes’s and Gauss’s
Theorems. Pre: 243 or consent.
**MATH 251A Accelerated Calculus I (4)** Basic concepts;
differentiation with applications; integration. Compared to 241, topics
are discussed in greater depth. Pre: a grade of A in 140 or precalculus
assessment and consent.
**MATH 252A Accelerated Calculus II (4)** Integration techniques
and applications, series and approximations, differential equations, introduction
to vectors. Pre: a grade of B or better in 241 and consent.
**MATH 253A Accelerated Calculus III (4)** Vector calculus;
maxima and minima in several variables; multiple integrals; line integrals,
surface integrals and their applications. Pre: 252A.
**MATH 301 Introduction to Discrete Mathematics (3)** 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 and one semester programming;
or consent. Recommended: 190.
**MATH 302 Introduction to Differential Equations I (3)**
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.
**MATH 303 Introduction to Differential Equations II (3)**
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.
**MATH 307 Linear Algebra and Differential Equations (3)**
Introduction to linear algebra, application of eigenvalue techniques to
the solution of differential equations. Students may receive credit for
only one of 307 and 311. Pre: 243 or 253 (or concurrent) or consent.
**MATH 311 Introduction to Linear Algebra (3)** Algebra
of matrices, linear equations, real vector spaces and transformations.
Students may receive credit for only one of 307 and 311. Pre: 243 or 253A
(or concurrent) or consent.
**MATH 321 Introduction to Advanced Mathematics (3)** Formal
introduction to the concepts of logic, finite and infinite sets, functions,
methods of proof and axiomatic systems. Pre: 243 or 253A (or concurrent)
or consent.
**MATH 331 Introduction to Real Analysis (3)** A rigorous
axiomatic development of one variable calculus. Completeness, topology
of the plane, limits, continuity, differentiation, integration. Pre: 242
or 252, and 321; or consent.
**MATH 351 Foundation of Euclidean Geometry (3)** Axiomatic
Euclidean geometry and introduction to the axiomatic method. Pre: 243
or 253A, and 321 (or concurrent); or consent.
**MATH 352 Non-Euclidean Geometries (3) **Hyperbolic, other
non-Euclidean geometries. Pre: 351 or consent.
**MATH 371 Elementary Probability Theory (3) **Sets, discrete
sample spaces, problems in combinatorial probability, random variables,
mathematical expectations, classical distributions, applications. Pre:
216, 242 or 252A, or consent.
**MATH 373 Elementary Statistics (3)** Estimation, tests
of significance, the concept of power. Pre: 371 or consent.
**MATH 402 Partial Differential Equations I (3)** 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.
**MATH 403 Partial Differential Equations II (3)** Laplace’s
equation, Fourier transform methods for PDEs, higher dimensional PDEs,
spherical harmonics, Laplace series, special functions and applications.
Pre: 402 or consent.
**MATH 405 Ordinary Differential Equations (3)** 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.
**MATH 407 Numerical Analysis (3) **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: 311
and one semester programming; or consent.
**MATH 408 Numerical Analysis (3)** 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.
**MATH 411 Linear Algebra (3) **Vector spaces over arbitrary
fields, minimal polynomials, invariant subspaces, canonical forms of matrices;
unitary and Hermitian matrices, quadratic forms. Pre: a grade of B or
better in 311 or consent.
**MATH 412 Introduction to Abstract Algebra (3)** Introduction
to basic algebraic structures. Groups, finite groups, abelian groups,
rings, integral domains, fields, factorization, polynomial rings, field
extensions, quotient fields. (These topics are covered in the year sequence
412–413.) Pre: 311 or consent.
**MATH 413 Introduction to Abstract Algebra (3) **Continuation
of 412. This is the second course of a year sequence and should be taken
in the same academic year as 412. Pre: 412 or consent.
**MATH 414 Operations Research: Discrete Models (3)** Techniques
of mathematical programming. Topics may include linear programming, integer
programming, network analysis, dynamic programming, and game theory. Pre:
307 or 311, or consent.
**MATH 416 Operations Research: Probabilistic Models (3)**
Queuing theory, inventory theory, Markov chains, simulation. Pre: 307
or 311, and 371, or consent.
**MATH 420 Introduction to the Theory of Numbers (3) **Congruences,
quadratic residues, arithmetic functions, distribution of primes. Pre:
311 or consent.
**MATH 421 Topology (3)** Geometric and combinatorial topology.
Surfaces, homology, Euler characteristics, winding numbers. Jordan curve
theorem. Pre: two courses from 311, 321, 351, 411, 412, or 420; or consent.
**MATH 431 Advanced Calculus (3)** Topology of Rn, continuous
functions, Riemann integration, sequences and series, uniform convergence,
implicit function theorems, differentials and Jacobians. (These topics
are covered in the year sequence 431–432.) Pre: 311 and 331; or
consent.
**MATH 432 Advanced Calculus (3)** Continuation of 431.
This is the second course of a year sequence and should be taken in the
same academic year as 431. Pre: 431 or consent.
**MATH 442 Vector Analysis (3)** 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 253, and 307 or 311, or consent.
**MATH 443 Differential Geometry (3)** Properties and fundamental
geometric invariants of curves and surfaces in space; applications to
the physical sciences. Pre: 244 or 253A, and 311; or consent.
M**ATH 444 Complex Variable (3)** Analytic functions, complex
integration, introduction to conformal mapping. Pre: 244 or 253A; recommended
307, 311, 321 or 331; or consent.
**MATH 449 (Alpha) Topics in Undergraduate Mathematics (3)**
Advanced topics from various areas: algebra, number theory, analysis,
and geometry. Repeatable. Pre: consent.
**MATH 454 Axiomatic Set Theory (3)** 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. Not open to mathematics graduate students.
**MATH 455 Mathematical Logic (3)** 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: 454 or consent.
**MATH 471 Probability (3)** Probability spaces, random
variables, distributions, expectations, moment-generating and characteristic
functions, limit theorems. Continuous probability emphasized. Pre: 244
or 253A (or concurrent), and 371; or consent.
**MATH 472 Statistical Inference (3)** Sampling and parameter
estimation, tests of hypotheses, correlation, regression, analysis of
variance, sequential analysis, rank order statistics. Pre: 471 or consent.
M**ATH 475 Combinatorial Mathematics (3)** 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.
**MATH 480 Senior Seminar (V)** A seminar for senior mathematics
majors, including an introduction to methods of research. CR/NC only.
Pre: one 400-level mathematics course or consent.
**MATH 499 Directed Reading (V)** Limited to advanced students
who must arrange with an instructor before enrolling. Repeatable one time
for a maximum of 3 credits each.
**MATH 500 Master’s Plan B/C Studies (1)** Enrollment
for degree completion. Pre: Master’s Plan B or C candidate and consent.
**MATH 602 Ordinary and Partial Differential Equations (3)**
Classical existence and uniqueness theory for ODEs and PDEs, qualitative
properties, classification, boundary value and initial value problems,
fundamental solutions, other topics. (These topics are covered in the
year sequence 602–603.) Pre: 402, 431, 432, 442, or consent.
**MATH 603 Ordinary and Partial Differential Equations (3)**
Continuation of 602. This is the second course of a year sequence and
should be taken in the same academic year as 602. Pre: 602.
**MATH 607 Numerical Analysis (3)** Perron-Frobenius theory,
nonnegative matrices, matrix numerical analysis, iterative methods, discrete
approximation to partial differential equations. Pre: 402, 407, 408, or
consent.
**MATH 611 Modern Algebra (3)** 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.) Pre: consent.
**MATH 612 Modern Algebra (3)** Continuation of 611. This
is the second course of a year sequence and should be taken in the same
academic year as 611. Pre: 611.
**MATH 613 Group Theory (3)** Sylow theorems, solvable groups,
nilpotent groups, extension theory, representation theory, additional
topics. Pre: consent.
**MATH 615 Ring Theory (3)** Ideal theory in Noetherian
rings, localization, Dedekind domains, the Jacobson radical, the Wedderburn-Artin
theorem, additional topics. Pre: consent.
**MATH 618 Lattice Theory (3)** Introduction with applications
to general algebra. Partially ordered sets, decomposition theory, representations
of lattices, varieties and free lattices, coordinatization of modular
lattices. Pre: 612 or consent.
**MATH 619 Universal Algebra (3)** Introduction to basic
techniques, including subalgebras, congruences, automorphisms and endomorphisms,
varieties of algebras, Mal’cev conditions. Pre: 612 or consent.
**MATH 621 Topology (3)** 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.) Pre: consent.
**MATH 622 Topology (3)** Continuation of 621. This is the
second course of a year sequence and should be taken in the same academic
year as 621. Pre: 621.
**MATH 625 Differentiable Manifolds I (3)** Differentiable
structures on manifolds, tensor fields, Frobenius theorem, exterior algebra,
integration of forms, Poincare Lemma, Stoke’s theorem. Pre: 411,
432, and 622; or consent.
**MATH 631 Theory of Functions of a Real Variable (3) **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.) Pre: consent.
**MATH 632 Theory of Functions of a Real Variable (3)**
Continuation of 631. This is the second course of a year sequence and
should be taken in the same academic year as 631. Pre: 631.
**MATH 633 Functional Analysis (3)** Linear topological
spaces, normed spaces, Hilbert spaces, function spaces, function algebras,
operator theory. Pre: consent.
**MATH 637 Calculus of Variations (3)** Simple variational
problems, first and second variation formulas. Euler-Lagrange equation,
direct methods, optimal control. Pre: 432 or consent.
**MATH 644 Analytic Function Theory (3)** Conformal mapping,
residue theory, series and product developments, analytic continuation,
special functions. (These topics are covered in the year sequence 644–645.)
Pre: consent.
**MATH 645 Analytic Function Theory (3)** Continuation of
644. This is the second course of a year sequence and should be taken
in the same academic year as 644. Pre: 644.
**MATH 649 (Alpha) Topics in Mathematics (3)** Commutative
rings, function theory, geometric topology, transformation groups, etc.
Repeatable. Pre: consent.
**MATH 655 Set Theory (3)** Axiomatic development, ordinal
and cardinal numbers, recursion theorems, axiom of choice, continuum hypothesis,
consistency and independence results. Pre: consent.
**MATH 657 Recursive Functions and Complexity (3)** Recursive,
r.e., Ptime, and Logspace classes. Nondeterminism, parallelism, alternation,
and Boolean circuits. Reducibility and completeness. Pre: 455, ICS 441,
or consent.
**MATH 671 Advanced Probability (3)** Independence and conditioning,
martingales, ergodic theory, Markov chains, central limit theorem. Pre:
631 or consent.
**MATH 672 Stochastic Processes (3)** Stationary, Gaussian,
and Markov processes. Pre: 671 or consent.
**MATH 681 Graph Theory (3)** Connected graphs and digraphs.
Graph embeddings. Connectivity and networks. Factors and factorizations.
Coverings. Coloring. Applications. Pre: 311 or consent.
**MATH 699 Directed Reading and Research (V)** Maximum of
3 credit hours. Repeatable three times. Pre: graduate standing and consent.
**MATH 700 Thesis Research (V)** Research for master’s
thesis. Pre: consent.
**MATH 799 Apprenticeship in Teaching (V)** 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. CR/NC only. Pre: graduate standing in mathematics
and consent.
**MATH 800 Dissertation Research (V) **Research for doctoral
dissertation. |