# Mathematics Courses

## Mathematics courses

## Undergraduate courses

**MATH 005. WR College Algebra I. 4 crs.** An intensive college algebra course that emphasizes manipulative algebra, solutions of equations and inequalities, and graphs and analysis of linear, quadratic, exponential and logarithmic functions. Three lecture hours and two recitation hours per week. Prerequisite: none.

**MATH 006. College Algebra I. 3 crs.** An intensive college algebra course that emphasizes manipulative algebra, solutions of equations and inequalities, and graphs and analysis of linear, quadratic, exponential and logarithmic functions. Prerequisite: Satisfactory score on mathematics placement examination.

**Caution:** MATH 005 and MATH 006 contain the same material. A student cannot get credit for both MATH 005 and MATH 006. A student who wishes to take a succeeding course in math after MATH 005 or MATH 006 should take one of the following: MATH 007, 009, 010, or 012. Science students (and those who would like to take Calculus I) should take MATH 007. Business students often take MATH 010 and Applied Calculus (MATH 026). Social science students may want to take Introduction to Statistics (MATH 009). Liberal Arts majors often take Patterns in Math (MATH 012).

**MATH 007. Precalculus. 4 crs.** Graphing and analysis of higher-order polynomial and rational functions; trigonometry (including unit circle, trigonometric identities, and inverse trig functions), and systems of equations. Students planning to take 156 should take this course. It is not intended for those students planning to take 026; they should take 010 instead. Prerequisite: 005 or 006, or satisfactory score on the Mathematics Placement Examination.

**MATH 009. Introduction to Statistics. 4 crs.** A first course in statistics, which includes descriptive and inferential statistics, data collection and organization, measures of central tendency, variation, and position, probability, normal distributions, and confidence intervals. This is an introductory course that may be followed by more specialized statistics courses offered by other departments of the University. Not intended for students who have taken calculus; students with a calculus background should take 189. Prerequisite: 005 or 006, or satisfactory score on the Mathematics Placement Examination.

**MATH 010. College Algebra II. 4 crs.** Exponential and logarithmic functions; matrix theory, combinatorics, and probability. Students planning to take 026 should take this course. It is not intended for students planning to take 156, who should, instead, follow 006 with 007. Prerequisite: 005 or 006, or satisfactory score on the Mathematics Placement Examination.

**MATH 012. Patterns in Mathematics. 3 crs.** Introduction to the art, nature and applications of mathematics. Emphasis is placed on mathematical patterns occurring in real life situations. The course is not intended for students planning to take any Calculus course Prerequisite: 005 or 006, or satisfactory score on the Mathematics Placement Examination.

**MATH 020. Fundamental Concepts of Mathematics for Education I. 3 crs.** Fundamental concepts of mathematics needed by elementary school teachers. Prerequisite: 005 or 006, or satisfactory score on the Mathematics Placement Examination.

**MATH 021. Fundamental Concepts of Mathematics for Education II. 3 crs.** Algebra, expressions and solving equations. Visualization. Properties of angles, circles, spheres, triangles, and quadrilaterals. Measurement, length, area, and volume. Transformations, congruence, and similarity. Basic descriptive statistics and probability. Required of all students in an elementary school certification program. Offered every spring semester. Prerequisite: 020.

**MATH 026. Applied Calculus. 4 crs. **Limits; differentiation; integration; introduction to differential equations; and functions of several variables. Prerequisite: 007 or 010, or outstanding score on Mathematics Placement Examination.

**MATH 084, 085. Directed Readings in Honors for Sophomores. 1 cr. ea.** This set of courses (084, 085, 088, 089, 092, 093) is designed for students in the honor's program, and is designed to help students writing an honor's thesis. Others may take the courses with consent of the instructor.

**MATH 088, 089. Directed Readings in Honors for Juniors. 1 cr. ea.**

**MATH 092, 093. Senior Departmental Honors. 3 crs. ea. **

**MATH 101. Proof and Problem Seminar I. 1 cr. **This course and 102 are designed to help mathematics majors make the transition from the Calculus sequence to more advanced and abstract courses, and is to be taken early when a student declares a major. The topics are sets, relations, functions, proofs by induction and contradiction, complex numbers, and binomial coefficients. Prerequisite: 156.

**MATH 102. Proof and Problem Seminar II. 1 cr.** A continuation of 101. The topics of 101 are reinforced by going more deeply into one of number theory, dynamics, probability, graph theory, or modeling. Prerequisite: 101.

**MATH 150. Modern Geometry. 3 crs.** Deductive reasoning through the study of selected topics from Euclidean and non-Euclidean geometrics. Prerequisite: 157.

**MATH 156. Calculus I. 4 crs. **Limits, continuity, and the derivative and integral of functions of one variable, with applications, and the fundamental theorem of calculus. Prerequisite: 007 or outstanding score on Mathematics Placement Examination.

**MATH 157. Calculus II. 4 crs.** Continuation of 156, including more integration, sequences, series, Taylor's theorem, improper integrals, and L'Hospital's rule. Prerequisite: 156.

**MATH 158. Calculus III. 4 crs. **Continuation of 157, including calculus of functions of several variables, through Green’s Theorem and Stokes’s Theorem, with applications. Prerequisite: 157.

**MATH 159. Differential Equations. 4 crs. **Elementary techniques of ordinary differential equations, including slope fields, equilibria, separation of variables, linear differential equations, homogeneous differential equations, undetermined coefficients, bifurcations, power series, Laplace transforms, systems, and numerical methods. Prerequisite: 157.

**MATH 160. Advanced Calculus for Science and Engineering. 3 crs.** Vector calculus in *n *dimensions. Generalizations of the fundamental theorem of calculus. Stokes theorem divergence theorem. Inverse and implicit functions theorems. Use of Jacobians. Prerequisite: 158.

**MATH 161, 162. Seminar 1-3 crs. each.** Offered on demand; seminars in various topics in mathematics.

**MATH 164. Introduction to Numerical Analysis. 3 crs.** Treats numerical integration and numerical solution of differential equations; numerical linear algebra, matrix inversion, characteristic values; error propagation; and stability. Prerequisite: 159 and SYCS 135.

**MATH 165, 166. Directed Readings. 1-3 crs. each.** Readings under a faculty member whose approval is required for admission to course.

**MATH 168. Actuarial Science Laboratory I. 1 cr.** Systematic methods and approaches for rapid and accurate solutions of problems arising in elementary algebra, calculus, and analysis. Prerequisite: Consent of instructor or 158 and 189.

**MATH 169. Actuarial Science Laboratory II. 1 cr.** Continuation of 168 with the problems to be solved coming from mathematical statistics. Prerequisite: Consent of instructor or 190.

**MATH 180. Introduction to Linear Algebra. 3 crs.** Vector Spaces, linear transformations, the Gram-Schmidt process, determinants, eigenvectors and eigenvalues, diagonalization and applications. Prerequisite: 157.

**MATH 181. Discrete Structures. 3 crs. **Algebraic structures applicable to computer science; semigroups, graphs, lattices, Boolean algebras, and combinatorics. Prerequisite: 157. [NOTE: There is no computer science co-requisite.]

**MATH 183. Intermediate Differential Equations. 3 crs.** Initial value problems, existence and uniqueness of solutions, properties of solutions boundary value problems, Sturm-Liouville systems, and orthogonal expansions. Prerequisites: 159 and 180.

**MATH 184. Introduction to Number Theory. 3 crs.** Elements of algebraic number theory. Prerequisite: 197.

**MATH 185. Introduction to Complex Variables. 3 crs.** Complex numbers and their geometry, plane topology, limits, continuity, differentiation, Cauchy-Riemann equations, analytic functions, series, Cauchy theorems, contour integration, and residue theory. Prerequisite: 195.

**MATH 186. Introduction to Differential Geometry. 3 crs.** Calculus in Euclidean space, vector fields, geometry of surfaces, and curves. Prerequisites: 158 and 180.

**MATH 187. Introduction to Algebraic Topology. 3 crs.** Complexes, homology, surface topology, and the classical groups. Prerequisite: 197 and 199.

**MATH 189. Probability and Statistics I. 3 crs.** Samples spaces, random variables, distributions, expectation, independence, law of large numbers.

Prerequisite: 157.

**MATH 190. Probability and Statistics II. 3 crs. ** Continuation of 189. Includes estimation, order statistics, sufficient statistics, test of hypotheses, and analysis of variance. Prerequisite: 189.

**MATH 191. Foundations of Applied Mathematics. 3 crs.** Introduction to the concepts and methods of applied mathematics, including gravitational motion, calculus of variations, Lagrange's and Hamilton's equations; approximation techniques, partial differential equations, Fourier series, and Fourier integrals. Prerequisites: 159.

**MATH 192. Topics in Applied Mathematics. 3 crs.** Topics are selected from the following areas: combinatorics, computer science, control theory, fluid dynamics, game theory, information theory, mathematical biology, and statistical mechanics. Prerequisite: 191. Prerequisite: permission of instructor.

**MATH 193. Actuarial Science Seminar. 3 crs.** Treats life contingency, or the theory of interest, or other applications of mathematics to actuarial science as required. Prerequisite: 190.

**MATH 194. Introduction to Set Theory. 3 crs. ** Axiomatic foundations; relations and functions; ordered and well-ordered sets; ordinals and cardinals and axiom of choice with its equivalents. Prerequisite: 195.

**MATH 195 or 795. Introduction to Analysis I. 3 crs.** Set theory, logic, real and complex numbers, introductory topology, and continuous functions. Required for mathematics majors. 795 is the version of the course that fulfills the writing requirement. Prerequisite: 157.

**MATH 196. Introduction to Analysis II. 3 crs.** Sequences; series; limits; continuity; uniform continuity and convergence; differentiation and integration of functions of one variable. Prerequisite: 195.

**MATH 197. Introduction to Modern Algebra I. 3 crs.** Groups, rings, fields and homomorphisms. Prerequisite: 180.

**MATH 198. Introduction to Modern Algebra II. 3 crs.** Continuation of 197, including isomorphism theorems, Cayley's theorem, the Sylow theorems, p-groups, abelian groups, unique factorization domains, and Galois theory. Prerequisite: 197.

**MATH 199. Introduction to General Topology. 3 crs. **Topological spaces; relative topology; and subspaces, finite product spaces; quotient spaces; continuous and topological maps; compactness; connectedness; and separation axioms. Prerequisite: 157 and 195.

**MATH 795. Introduction to Analysis. Writing across the curriculum. **See 195. This version of the course fulfills an undergraduate writing requirement.

## Graduate courses

**MATH-204. Graduate Tutorial**. 3 crs.

**MATH-205.** **Graduate Tutorial.** 3 crs.

**MATH-208. Introduction to Modern Algebra I**. 3 crs. Groups, subgroups, cyclic groups, quotient groups, Lagranges Theorem, permutation groups, homomorphism and isomorphism theorems, Cayley's theorem, rings, subrings, ideals, fields, homomorphism and isomorphism theorems.

**MATH-209. Introduction to Modern Algebra II**. 3 crs. Sylow's theorems for finite groups, p-groups, abelian groups, group action on sets, domains, prime and maximal ideals, unique factorization domain. Prereq.: MATH-208

**MATH-210. Modern Algebra I.** 3 crs. Groups, group actions on sets, structure of finitely generated abelian groups, category theory, exact sequences, rings, principal ideal domains, modules, projective, injective and free modules.

**MATH-211. Modern Algebra ll.** 3 crs. Structure of finitely generated modules over principal ideal domains, fields, Galois theory, vector spaces and classical groups *G*(*n*, *R*), algebras over a field.

**MATH-214. Number Theory I.** 3 crs. Congruences; primitive roots and indices; quadratic residues; number-theoretic functions; primes; sums of squares; Pell's theorem; and rational approximations.

**MATH-215. Number Theory II. **3 crs. Continuation of MATH-214, including binary quadratic forms; algebraic numbers; rational number theory, irrationality and transcendence; Dirichlet's theorem; and the prime number theorem. Prereq.: MATH-214.

**MATH-218. Mathematical Logic I.** 3 crs. Axiomatic and formal mathematics; consistency and completeness; recursive functions; undecidability and intuitionism. Prereq: Graduate status.

**MATH-219.** **Mathematical Logic ll.** 3 crs. Continuation of MATH-218, including model theory and first-order set theory. Prereq.: MATH-218.

**MATH-220. Introduction to Analysis I.** 3 crs. Logical connectives, qualifiers, mathematical proof, basic set operations, relations, functions, cardinality, axioms of set theory, natural number and induction, ordered fields. The completeness axiom, topology of the reals, Heine-Borel theorem, convergence Bolzano-Weierstrass theorem, limit theorems, monotone sequence and Cauchy sequence, subsequences, infinite series and convergence criterion, convergence tests, power series.

**MATH-221.** **Introduction to Analysis II.** 3 crs. Limits of functions, continuity, uniform continuity, differentiation, the mean value theorem, Rolle's theorem, L'Hospital's rule, Taylor's theorem, Riemann Integral, properties of the Riemann Integral, the fundamental theorem of calculus, pointwise and uniform convergence, applications of uniform convergence. Prereq.: MATH-220.

**MATH-222. Real Analysis I**. 3 crs. Topology of n-dimensional Euclidean space, functions of bounded variation, absolute continuity, differentiation, Riemann-Stieltjes integration. Lebesgue measure and integration theory; Lp spaces, separability, completeness, duality, L-spaces and the Riesz-Fischer theorem.

**MATH-223.** **Real Analysis II.** 3 crs. Continuation of MATH-222. Abstract measures, mappings of measure spaces, integration sets and product spaces, the Fubini, Tonelli and Radon-Nikodym theorems, the Riesz representation theorem, Haar measures on locally compact groups.

**MATH-224. Applications of Analysis.** 3 crs. Operators defined by convolution, maximal functions, Fourier transform in classical spaces of functions, distributions; harmonic and subharmonic functions; applications to P.D.E and probability theory, Bochner theorem and central limit theorem. Prereq.: MATH-223.

**MATH-229. Complex Analysis I.** 3 crs. Linear fractional transformations, conformal mapping, holomorphic functions, Cauchy's theorem (including the homotopic version), properties of holomorphic functions, the argument principle, residues, power series, Laurent series, meromorphic functions.

**MATH-230. Complex Analysis II.** 3 crs. Continuation of MATH-229. Montel's theorem, normal families, Riemann Mapping Theorem, Picard's theorem, Mittag-Leffler's theorem, Weierstrass' theorem, simply connected domains, Riemann surfaces, meromorphic functions on compact Riemann surfaces.

**MATH-231. Functional Analysis I.** 3 crs. Banach spaces; the dual topology and weak topology; the Hahn-Banach, Krein-Milman and Alaoglu theorems; the Baire category theorem; the closed graph theorem; the open mapping theorem; the Banach-Steinhaus theorem; elementary spectral theory; and differential equations. Prereq.: Graduate status.

**MATH-232. Functional Analysis II.** 3 crs. Continuation of MATH- 231, including topological vector spaces; bounded operators; Banach algebras; spectra and symbolic calculus; Gelfand and Fourier transforms; and distributions. Prereq.: MATH-231.

**MATH-234. Advanced Ordinary Differential Equatlons I.** 3 crs. Existence, uniqueness, and representation of solutions of ordinary differential equations; periodic solutions, singular points, oscillation theorems, and boundary value problems. Prereq.: Graduate status.

**MATH-235. Advanced Ordinary Differential Equations II.** 3 crs. Continuation of MATH-234. including qualitative theory stability and Liapunov functions; focal, nodal, and saddle points; limit sets: and the Poincare-Bendixson theorem. Prereq.: MATH-234.

**MATH-236. Partial Differential Equations I.** 3 crs. First-order partial differential equations, method of characteristics; Cauchy-Kovalevskaya theorem; second-order equations, classification existence, and uniqueness results; formulation of some of the classical problems of mathematical physics. Prereq.: Graduate status.

**MATH-237. Partial Differential Equations II.** 3 crs. Continuation of MATH-236, showing applications of functional analysis to differential equations including distributions, generalized functions, semigroups of operators, the variational method, and the Riesz-Schauder theorem. Prereq.: MATH-236.

**MATH-239. Fourier Series and Boundary Value Problems**. 3 crs. Fourier analysis, Bessel's inequality, Parseval's relation, Hilbert spaces, compact operators, eigenfunction expansions, and Sturm-Liouville problems. Prereq.: Graduate status.

**MATH-240. Mathematics Statistlcs I.** 3 crs. Probability; random variables; distributions; moment generating functions; limit theorems; parametric families of distributions; sampling distributions; sufficiency; and likelihood functions. Prereq.: Graduate status.

**MATH-241.** **Mathematical Statistics II.** 3 crs. Continuation of MATH-240 including point and interval estimations; hypothesis testing; decision functions; regression; non-parametric inferences; and analysis of categorical data.

**MATH-242. Stochastic Processes.** 3 crs. Continuation of MATH-241 including conditional probability, conditional expectation, normal processes, covariance, stationary processes, renewal equations, and Markov chains. Prereq.: MATH-241.

**MATH-243. Dynamical System I.** 3 crs. Systems of differential equations existence, uniqueness and continuity of solutions, linear systems, including constant coefficients, asymptotic behavior, periodic coefficients; stability of linear and almost linear systems, the Poincare-Bendix theorem; global stability (Lyapunov method); differential equations and dynamical systems, including closed orbits, structural stability, and 2-dimensional flow. Prereq.: Graduate status.

**MATH-244. Dynamical Systems II.** 3 crs. Introduction to Chaos; local bifurcations, center manifolds, normal forms, equilibria, and periodic orbits; averaging and perturbation, Poincare maps, Hamiltonian In systems and Melnikov's methods; hyperbolic sets, symbolic dynamics and strange attractors; Smale Horseshoe, invariant sets, Markov partitions and statistical properties; global bifurcations; Lorentz and Hopf bifurcations; Chaos in discrete dynamical system. Prereq.: MATH-243.

**MATH-245. Methods of Applied Mathematics I.** Principles and techniques of modern applied mathematics with case studies involving deterministic problems, random problems, and Fourier analysis. Prereq.: Graduate status.

**MATH-246.** **Methods of Applied Mathematics II.**. 3 crs. Asymptotic sequences and series, special functions, asymptotic expansions of integrals and solutions of ordinary differential equations, and singular perturbations. Prereq.: MATH-245.

**MATH-247. Numerical Analysis I.** 3 crs. Numerical solutions of ordinary and partial differential equations including convergence stability, and consistence of schemes. Prereq.: Graduate status.

**MATH-248. Numerical Analysis II.** 3 crs. Continuation of MATH-247 including numerical methods for partial differential equations using functional analysis techniques; the Lax equivalence theorem; Courant-Friedrich Levy condition; Kreiss matrix theorem; and finite element methods. Prereq.: MATH-247.

**MATH-250. Topology I.** 3 crs. Topological basis, continuous, open closed topological maps, product spaces, connectedness, compactness, local connectedness, local compactness; identification and weak topologies, separation axioms, metrizable spaces, covering spaces, homotopy, fundamental groups.

**MATH-251. Topology II.** 3 crs. Compactifications, Baire spaces, function spaces, topological vector spaces.

**MATH-252. Algebraic Topology I.** 3 crs. Homotopy, covering spaces, fibrations, polyhedra, simplicial complexes, simplicial and singular homology, and Eilenberg-Steenrod axioms. Prereq.: MATH-251.

**MATH-253. Algebraic Topology II.** 3 crs. Continuation of MATH-252 including products; cohomology; homotopy, CW spaces, obstructions; sheaf theory; and spectral sequences. Prereq.: MATH-252.

**MATH-259.** **Differential Geometry I.** 3 crs. Differential manifolds, tensors, affine connections, and Riemannian manifolds. Prereq.: Graduate status.

**MATH-260.** **Differential Geometry II.** 3 crs. Continuation of MATH-259 including Riemannian geometry; submanifolds; variations of the length integral; the Morse index theorem; complex manifolds; Hermitian vector bundles; and characteristic classes. Prereq.: MATH-259.

**MATH-270.** **Several Complex Variables I.** 3 crs. Basic facts about holomorphic functions; zero sets of holomorphic functions, analytic sets and Weierstrass' Preparation theorem; domains of holomorphy, convexity with respect to holomorphic curves, plurisubharmonic functions, pseudoconvexity, Levi problem; holomorphic convexity, Stein domains and complete Reinhardt domains; differential forms; complex manifolds, complex structure on TpM, almost complex structures, exterior derivatives forms of the (p,q)-type, cohomology. Prereq.: MATH-229, MATH-230.

**MATH-271.** **Several Complex Variables II.** 3 crs. Holomorphic convexity, Stein domains and complete Reinhardt domains; differential forms; complex manifolds, complex manifolds, complex structure on TpM, almost complex structures, exterior derivative forms of the (p,q)-type, cohomology.

**MATH-273. Combinatorics I.** 3 crs. Topics include: basic counting, generating functions, sets and multisets, Stirling numbers, q-enumeration, the twelvefold way, permutation statistics, integer partitions, labeled trees, ordered trees, special counting sequences, a brief survey of graph theory, sieve methods, partially ordered sets, and Möbius inversion. Prereq.: Calculus II and familiarity with linear and abstract algebra, or permission of instructor.**MATH-274. Combinatorics II.** 3 crs. We look in greater depth at some of the topics from Combinatorics I and add material including Lagrange inversion, Polya-Redfield theory, symmetric functions, Young tableaux, Tutte polynomial, the Riordan group, permutation patterns, and asymptotic methods. Beyond this, the material may vary to reflect research interests in the department related to combinatorics.* *Prereq.: Combinatorics I or permission of instructor.

**MATH-280. Topics in History of Mathematics**. 3 crs. Topic to be selected by the instructor. Prereq.: Graduate status.

**MATH-290. Reading in Mathematics**. 3 crs. Topic to be selected by the instructor. Prereq.: Graduate status.

**MATH-300.** **Graduate Seminar.** 3 crs. Topic to be selected by the instructor. Prereq.: Graduate status.

**MATH-350. M.S. Thesis.** 6 crs. Topic to be selected by mutual consent of the student and the instructor. Prereq.: Consent of graduate chairperson.

**MATH-410, 419. Topics in Algebra. **3 crs. ea. Further topics in algebra to be selected by the instructor. Prereq.: Consent of instructor.

**MATH-430, 439. Topics in Analysis.** 3 crs. ea. Further topics in real and complex analysis to be selected by the instructor. Prereq.: Consent of instructor.

**MATH-450, 459. Topics in Applied Mathematics**. 3 crs. ea. Further topics in applied mathematics to be selected by the instructor. Prereq.: Consent of instructor.

**MATH-470, 479.** **Topics in Topology and Geometry**. 3 crs. ea. Further topics in geometry and topology to be selected by the instructor. Prereq.: Consent of instructor.

**MATH-500, 501. Graduate Seminar**. 3 crs. ea. Topics to be selected by the instructor. Prereq.: Consent of instructor.

**MATH-550. Ph.D. Dissertation. **12 crs. Prereq.: Consent of Ph.D. adviser.