# Graduate Courses in Mathematics

*Notes:*

- Courses numbered 400 to 499 are open to undergraduate and graduate students.
- Courses numbered 500 to 599 are open only to graduate students.
- MT 501 to MT 530 are not applicable to the Master of Science program.

420. PROBABILITY AND STATISTICS I 4 cr. Prerequisite: MT 233. Combinatorial probability, discrete and continuous distributions, simulation of sampling distributions and the central limit theorem, introduction to data analysis, estimation and hypothesis testing; with use of CAS and statistical software.

421. PROBABILITY AND STATISTICS II 3 cr. Prerequisite: MT 420. Mathematical treatment of estimation and hypothesis testing; including one and two-factor analysis of variance, simple regression and correlation, and nonparametric analyses.

422. APPLIED STATISTICS 3 cr. Prerequisites: MT 342, MT 420. Categorical data analysis, multiple regression, analysis of variance of various designs, introduction to design of experiments. Use of statistical software.

425. OPERATIONS RESEARCH 3 cr. Prerequisite: MT 342. Linear programming, sensitivity analysis and duality, queuing theory, and topics from networks, decision making, game theory, Markov chains, dynamic programming, and simulation.

431. ADVANCED CALCULUS OF ONE VARIABLE 3 cr. Prerequisites: MT 233, MT 341. Real number system, limits, continuity, differentiability, Riemann integral, properties of continuous and differentiable functions, sequences and series of functions.

432. ADVANCED CALCULUS OF SEVERAL VARIABLES 3 cr. Prerequisites: MT 233, MT 342. Development of and motivation for vector-valued functions, calculus of functions of several variables, implicit functions and Jacobians, multiple integrals, line integrals.

436. INTRODUCTION TO COMPLEX ANALYSIS 3 cr. Prerequisite: MT 341 or MT 342 or permission of department chair. Complex number plane, analytic functions, integration of complex functions, sequences and series. Residue theorem, evaluation of real integrals.

438. ORDINARY LINEAR DIFFERENTIAL EQUATIONS 3 cr. Prerequisites: MT 233, MT 342. Linear equations and systems, existence and uniqueness theorems, oscillation theory. Autonomous equations and systems, their solutions and qualitative properties.

441. ABSTRACT ALGEBRA 3 cr. Prerequisite: MT 341. Groups, rings, domains, fields, extension fields, introduction to Galois Theory.

442. LINEAR ALGEBRA 3 cr. Prerequisite: MT 342. Vector spaces, linear transformations, characteristic values and applications.

450. EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY 3 cr. Prerequisite: MT 341 or MT 342 or permission of department chair. Alternative ways of investigating the Euclidean plane including transformational geometry; examination of the parallel postulate and how it can be changed to create new geometries; hyperbolic geometry.

452. ELEMENTARY TOPOLOGY 3 cr. Prerequisite: MT 341. Topological spaces, homeomorphisms, connected spaces, compact spaces, regular and normal spaces, metric spaces.

456. FRACTAL GEOMETRY 3 cr. Prerequisites: MT 233, MT 341. Topics from metric spaces, transformations, iterated function systems, dynamical systems, fractal dimension, Julia sets, and Mandelbrot sets.

468. THEORY OF NUMBERS 3 cr. Prerequisite: MT 341. Divisibility theorems, number-theoretic functions, primitive roots, quadratic congruences and reciprocity, partitions.

469. HISTORY OF MATHEMATICS 3 cr. Prerequisite: MT 341. Study of mathematics from its origins to its present state. Topics include the development and impact of geometry, algebra, number theory, irrational numbers, analytic geometry, calculus, non-Euclidean geometry, and infinite sets.

478. FORMAL LANGUAGES (CS 478) 3 cr. Prerequisites: MT 341 or MT 379. Finite and push-down automata and Turing machines. Regular languages, context-free grammars, recursive and recursively enumerable languages. Other topics chosen from Church’s thesis, Gödel numbering, decidability, and recursive functions.

479. COMBINATORICS AND GRAPH THEORY 3 cr. Prerequisite: MT 341 or MT 342 or equivalent. Pigeonhole principle, inclusion and exclusion, recurrence relations and generating functions, combinatorial designs, the theory of graphs, graphical optimization problems.

480. SPECIAL TOPICS cr. TBA. Reading, reports on, and investigation of selected material and topics.

501. MATHEMATICAL STRUCTURES 3 cr. Topics selected from set theory, cardinality, axiomatic and constructive approaches to the number systems, algebraic structures. M.A. program only.

502. DISCRETE MATHEMATICS 3 cr. Matrices, graph theory, iterative processes, game theory, and applications. M.A. program only.

503. MODERN GEOMETRY 3 cr. Euclidean and non-Euclidean geometric systems, the synthetic and metric approaches to geometry, geometric constructability. M.A. program only.

504. CURVES, SURFACES AND SPACE 3 cr. Examination of the shape and size of mathematical spaces via topology and geometry; soap bubble problems, knots, nonorientable surfaces, curvature, elementary differential geometry, shape and measurement in the universe. M.A. program only.

505. TOPICS IN CALCULUS 3 cr. Alternative approaches to selected topics in the traditional calculus course. Designed for the teacher of calculus who wishes to deepen and broaden his/her understanding of this area. M.A. program only.

507. STATISTICAL LITERACY 3 cr. Graphical approach to data analysis, probability, art and techniques of simulation, surveys and information from samples, confidence intervals and tests of hypotheses; emphasis is on material applicable to the high school curriculum. M.A. program only.

509. GREAT MOMENTS IN MATHEMATICS 3 cr. Survey of some of the more important historical developments in the history of mathematics, with emphasis on those with connections to the secondary curriculum. M.A. program only.

510. MATHEMATICAL POTPOURRI 3 cr. Topics in and about mathematics, including famous problems, enrichment and appreciation material, and the use of these topics in the high school curriculum. M.A. program only.

512. TECHNOLOGY IN THE TEACHING OF MATHEMATICS 3 cr. Seminar/lab course in the use of graphing calculators and computer software in teaching high school mathematics. Students will collaborate in developing classroom and laboratory activities for use in the secondary curriculum. M.A. program only.

513. COMPUTER SCIENCE FOR HIGH SCHOOL TEACHERS 3 cr. An introduction to programming, algorithms and data structures using C++. Covers material included in the high school Advanced Placement Computer Science course (AB level) and other topics as time permits. M.A. program only.

514. PROBLEMS IN MATHEMATICS 3 cr. Old and new problems from various areas of mathematics, chosen to be applicable to co-curricular high school activities such as mathematics clubs and contests. M.A. program only.

515. CHAOS AND FRACTALS IN THE CLASSROOM 3 cr. Overview of chaotic dynamical systems and associated

fractals; computerized explorations of chaos and fractals, and their use in the high school curriculum. M.A. program only.

517. MATHEMATICAL MODELING IN THE HIGH SCHOOL CLASSROOM 3 cr. Exploration of mathematical modeling for use within high school classroom contexts. Topics include theory of measurement, dynamical systems, probability, network analysis. Applications include population growth, biomechanics, financial models, social networks and ecology. Emphasis on the use of modeling as a necessary and sufficient requirement for excellent mathematical pedagogy.

519. SPECIAL TOPICS IN MATHEMATICS cr. TBA Supervised study of special topics. M.A. program only.

531. REAL ANALYSIS I 3 cr. Topics to be chosen from: real number system, completion of metric space. Stone Weierstrass and Ascoli theorems, equicontinuity, functions of bounded variation, nowhere differentiable functions.

532. REAL ANALYSIS II 3 cr. Prerequisite: MT 531. Topics to be selected from: Borel sets, Baire functions, ordinal numbers, Lebesgue measure, absolute continuity, Lebesgue-Stieljes integral, signed measures, Radon-Nikodym theorem, product measures and Fubini’s theorem.

536. COMPLEX ANALYSIS 3 cr. Prerequisite: MT 431. The topology of the complex plane, analytic functions, integration theory, Riemann Mapping Theorem, analytic continuation, Riemann surfaces, harmonic functions.

538. FUNCTIONAL ANALYSIS 3 cr. Prerequisite: MT 452. Topics to be selected from: normed spaces, linear functionals, Hahn-Banach theorem, dual space, inner product space, Riesz-Fischer theorem, linear operators.

541. ALGEBRA I 3 cr. Groups, homomorphism, group actions, Sylow theorems, rings and ideals, polynomials, and p.i.d.’s.

542. ALGEBRA II 3 cr. Prerequisite: MT 541. Topics to be selected from: projective and injective modules, structure of semigroups, rings, radicals and Galois Theory.

552. GENERAL TOPOLOGY 3 cr. Prerequisite: MT 452. Topics to be selected from: topological spaces and mappings, topological and homotopic invariants, product and quotient spaces, topological constructions, separation axioms, metrization, generalized convergence, fundamental group.

557. DIFFERENTIAL GEOMETRY 3 cr. Prerequisite: MT 431. Local and global properties of curves and surfaces; Gauss map, curvature, Theorema Egregium, covariant derivative, geodesics, Gauss-Bonnet Theorem, generalizations to manifolds.

580. SPECIAL TOPICS cr. TBA. Reading, reports on, and investigation of selected material and topics.