MATHEMATICS (MATH) College of Science and Mathematics MATH 10 (3) Pre-Algebra

Review of the properties of natural numbers and integers, including prime factorization. Introduction to the language of Algebra: translating between word phrases and algebraic expressions, evaluating algebraic expressions, and estimating the value of an algebraic expression. Includes working with fractions, including fractions with variable expressions in the numerator and denominator, ratios, proportions, and percent as well as their use in solving common applied problems. Concludes with the interpretation of graphs and calculation of elementary statistical measures. Note: MATH 10 does not count toward any graduation requirement to be completed at CSUSM, but it is counted towards financial aid and VA benefits. Enrollment restricted to students who have not satisfied the Entry Level Mathematics (ELM) requirement and whose highest ELM examination score is below 30. Students who have satisfied the ELM requirement may not enroll.

MATH 20 (3) Beginning Algebra

Review of the use of algebraic expressions for the solution of linear equations and geometric problems. Introduction to the rectangular coordinate system and its use for graphing functions and representing information. Solution of linear systems of equations, linear systems of inequalities, equations, and inequalities involving the absolute value. Introduction to polynomial expressions, their properties, and the solution of polynomial equations. Note: MATH 20 does not count toward any graduation requirement to be completed at CSUSM, but it is counted towards financial aid and VA benefits. Enrollment Requirement: Highest Entry-Level Mathematics (ELM) exam score from 30 to 38, or completion of MATH 10 with a grade of C (2.0) or better. Students who have satisfied the ELM requirement may not enroll.

MATH 22 (1) Supplemental Instruction in MATH 30

Peer-facilitated learning session focused on development of skills needed to succeed in MATH 30 and subsequent math courses. Intended to provide a collaborative learning environment where students can work on problem solving, general study skills, and MATH 30 course content. May be repeated for a total of two (2) units. Graded Credit/No Credit. Enrollment restricted to students who have obtained consent of instructor.

MATH 30 (3) Entry Level Mathematics

Review of the rectangular coordinate system, lines, linear equation systems, and polynomial expressions and arithmetic. Algebraic subjects include: simplification of algebraic expressions, solution of quadratic and rational equations, properties of exponents, and arithmetic operations involving rational exponents. Geometric subjects include: determination of angles, basic geometric figures and their uses, properties of triangles, circles, polygons, and applications of the Pythagorean Theorem. Note: MATH 30 does not count toward any graduation requirement to be completed at CSUSM, but it is counted towards financial aid and VA benefits. Enrollment Requirement: Highest Entry-Level Mathematics (ELM) exam score from 40 to 48, or completion of MATH 20 with a minimum grade of C (2.0) or better. Students who have satisfied the ELM requirement may not enroll. Students who complete MATH 30 with a grade of C (2.0) or better will satisfy the ELM requirement.

MATH 30C (3) Computer Aided Entry Level Math

Review of the rectangular coordinate system, lines, linear equation systems, and polynomial expressions and arithmetic. Algebraic subjects include: simplification of algebraic expressions, solution of quadratic and rational equations, properties of exponents, and arithmetic operations involving rational exponents. Geometric subjects include: determination of angles, basic geometric figures

and their uses, properties of triangles, circles, polygons, and applications of the Pythagorean Theorem. Content is identical to MATH 30, and part of the content is taught with the help of computer software. Note: MATH 30C does not count toward any graduation requirement to be completed at CSUSM, but it is counted towards financial aid and VA benefits. Enrollment Requirement: Highest Entry-Level Mathematics (ELM) exam score from 40 to 48, or completion of MATH 20 with a minimum grade of C (2.0) or better. Students who have satisfied the ELM requirement may not enroll. Students who complete MATH 30C with a grade of C (2.0) or better will satisfy the ELM requirement.

MATH 100 (3) Mathematical Ideas Basic mathematical concepts such as logic, number theory, number systems, algebra, geometry, functions, graphs, counting methods, probability, and statistics together with related cultural and historical perspectives. Applications of mathematics will be emphasized. May not be taken for credit by students who have received credit for GEM 100. Enrollment restricted to students who have completed the Entry-Level Mathematics (ELM) requirement.

MATH 110 (3) Critical Thinking

Critical thinking in decision-making. Formal and informal fallacies of language and thought; the often unreliable guide of common-sense reasoning; analysis and criticism of ideas; distinction between fact and judgment, belief and knowledge; inductive and deductive arguments; and effective techniques of decision-making. Students will learn critical thinking skills to apply to common issues of everyday life.

MATH 115 (3) College Algebra

Equations and inequalities, functions, graphs, polynomials, exponential and logarithmic functions, conics, sequences and series, counting principles, binomial theorem, and systems of linear equations. Students preparing to take MATH 160 should take MATH 125 instead of this course. May not be taken for credit by students who have received credit for MATH 120 or MATH 125. Enrollment restricted to students who have completed the Entry-Level Mathematics (ELM) requirement.

MATH 125 (4) Pre-Calculus

Designed for students preparing to take MATH 160. Equations and inequalities, functions, graphs, polynomial and rational functions, trigonometric functions, exponential and logarithmic functions, systems of linear equations, conics, sequences and series, and the binomial theorem. May not be taken for credit by students who have received credit for MATH 115. Enrollment restricted to students who have completed the Entry-Level Mathematics (ELM) requirement.

MATH 132 (3) Survey of Calculus

Basic calculus concepts with applications to business, economics, and the social sciences. Differential calculus for algebraic, exponential, and logarithmic functions; optimization, linearization, and other applications of derivatives; introduction to integral calculus. Includes use of graphing calculators. Enrollment Requirement: MATH 115 with a grade of C (2.0) or better.

MATH 160 (5) Calculus with Applications, I

Differential and integral calculus of functions of one variable: analytic geometry, limits, continuity, derivatives, analysis of curves, integrals, and applications; algebraic, trigonometric, logarithmic, and exponential functions; and historical perspectives. Includes a laboratory experience using either computers or graphing calculators. Enrollment Requirement: A strong background in high school mathematics (Algebra I and II, Geometry, and Trigonometry) or MATH 125 with a grade of C (2.0) or better.

MATH 162 (4)

MATH 264 (3)

Calculus with Applications, II

Introduction to Linear Algebra

A continuation of differential and integral calculus: inverse trigonometric and hyperbolic functions, integration methods, indeterminate forms, coordinate systems, planes and lines in space, sequences and series, applications, and historical perspectives. Includes a laboratory experience using either computers or graphing calculators. Prerequisite: MATH 160 with a grade of C (2.0) or better.

Matrix algebra, systems of linear equations, vector spaces, independence, linear transformations, eigenvalues and eigenvectors, and applications. This course is not currently offered at Cal State

MATH 210 (3)

MATH 270 (3)

Math for K-8 Teachers I: Number Sense

Basic Discrete Mathematics

Designed to reinforce mathematical concepts for those teaching in grades K-8. Emphasis on numeric concepts: sets, logic, counting numbers, integers, rational numbers, real numbers, some number theory, and measurement and estimation, appropriate use of technology, and historical/cultural perspectives. Credit may not be

Exposure to fundamental discrete mathematical skills and knowledge: basic logic and applications in computer science, methods of proof, functions, relations, set, basic counting techniques, graphs, trees, and applications in computer science.

counted toward the mathematics major. Enrollment restricted to students who have completed the Entry-Level Mathematics (ELM) requirement.

MATH 212 (3) Mathematics for K-8 Teachers II: Geometry, Measurement, and Reasoning

Designed to reinforce mathematical concepts for those teaching in grades K-8. Emphasis on patterns and functional relationships; geometric concepts in two- and three-dimensional space: points, lines, planes, curves, triangles, convex figures, parallelism, congruence, similarity, symmetry, perimeter, area, and volume; problem-solving strategies; appropriate use of technology; and historical/cultural perspectives. Credit may not be counted toward the mathematics major. Enrollment restricted to students who have completed the Entry-Level Mathematics (ELM) requirement. Prerequisite: MATH 210 with a grade of C (2.0) or better.

MATH 242 (3) Introduction to Statistics

Types of data, measures of central tendency and variation, visualizing data, counting principles, standard random variables, probability, conditional probability, standard discrete probability distributions, normal probability distribution, tests for normality, sampling distribution, central limit theorem, hypothesis tests for means and proportions, correlation, and regression. May include computer software such as Excel, Minitab, or courseware. Credit may not be counted toward the mathematics major. Enrollment restricted to students who have completed the Entry-Level Mathematics (ELM) requirement. Enrollment Requirement: MATH 115.

MATH 260 (4) Calculus with Applications, III

Differential and integral calculus of functions of several variables: three dimensional analytic geometry, vector calculus, partial derivatives, multiple integrals, line integrals, applications, and historical perspectives. Includes a computer laboratory experience. Prerequisite: MATH 162 with a grade of C (2.0) or better.

MATH 262 (3) Introduction to Differential Equations

Models involving first-order equations, higher-order linear equations, systems of equations, numerical methods, and applications. Combines theoretical ideas with hands-on experience using appropriate computer software packages. This course is not currently offered at Cal State San Marcos. It is listed only for transfer credit and course equivalency purposes. Enrollment Requirement: MATH 162 with a grade of C (2.0) or better.

San Marcos. It is listed only for transfer credit and course equivalency purposes. Enrollment Requirement: MATH 115 with a grade of C (2.0) or better.

Prerequisite: MATH 160 with a grade of C (2.0) or better.

MATH 303 (3) Themes for Society

Descriptive overviews of selected areas of mathematics which play a visible role in the modern world. Topics include management science and operations research, political science, statistics, computer science, biology, and some late 20th Century advancements in pure mathematics. Credit may not be counted toward the mathematics major. Enrollment restricted to students who have completed the Lower-Division General Education requirement in Mathematics/ Quantitative Reasoning (B4).

MATH 304 (3) Women and Mathematics

Examination of the social phenomena that have led to the small number of women in the mathematical profession. Exploration of the controversy concerning research on the comparative mathematical ability of boys and girls. Study of the lives, times, and works of women mathematicians. Enrollment restricted to students who have completed the Lower-Division General Education requirement in Mathematics/Quantitative Reasoning (B4).

MATH 308 (3) Non-Statistical Mathematics in the Social Sciences

Themes involving applications of mathematics in the social sciences such as: proportional representation, voting rules and aggregation of individual preferences, spatial models of election competition, power in weighted voting systems, power indices in politics, balance theory and social inequalities, measurement theory, game theory, static models of animal dominance, rumor and information networks. May not be taken for credit by students who have received credit for MATH 404. Enrollment restricted to students who have completed the Lower-Division General Education requirement in Mathematics/Quantitative Reasoning (B4).

MATH 311 (3) Mathematics for K-8 Teachers III: Algebra, Probability, Statistics, and Data Analysis

Designed to reinforce mathematical concepts for those teaching grades K-8. Emphasis on linear and quadratic equations and inequalities; collection, organization, and representation of data; inferences, predictions, and arguments based on data; basic notions of chance and probability; appropriate use of technology; and historical/cultural perspectives. May not be taken for credit by students who have received credit for MATH 311B. Credit may not be counted toward the mathematics major. Prerequisite: MATH 212 with a grade of C (2.0) or better.

MATH 311B (3)

MATH 260. Prerequisite: MATH 162 with a grade of C (2.0) or better.

Mathematics for K-8 Teachers III: Algebra, Probability, Statistics, and Data Analysis

MATH 350 (3)

Designed to reinforce mathematical concepts for those teaching grades K-8. Emphasis on linear and quadratic equations and inequalities; collection, organization, and representation of data; inferences, predictions, and arguments based on data; basic notions of chance and probability; appropriate use of technology; and historical perspectives. Credit may not be counted toward the mathematics major. May not be taken for credit by students who have received credit for MATH 311. Prerequisite: MATH 212 with a grade of C (2.0) or better. Corequisite: EDMS 512B, EDMS 522B, EDMS 543B. Enrollment restricted to students participating in the Integrated Credential Program.

MATH 314 (1) Workshop for Future Mathematics Educators

Foundations for Theoretical Mathematics

Bridge course between computation-driven mathematics and theoretical mathematics. Designed to familiarize the student with the language and process of rigorous mathematical thought, speech, and writings through the introduction of typical and important examples from algebra, analysis, combinatorics, and geometry. Covers elementary logic, methods of proof, mathematical induction, sets, relations, including order relations and equivalent relations, functions and inverse functions, and binary operations. Mathematics majors are encouraged to take this course as early as possible. Prerequisite: MATH 160 with a grade of C (2.0) or better.

MATH 362 (3) Differential Equations

Provides a discussion forum for students pursuing the Mathematics Single-Subject CSET Waiver Program, while co-enrolled in EDUC 350, Foundations of Teaching as a Profession. Discussions focus on various mathematical subject matter typically presented in secondary-level classrooms, which students visit and observe in EDUC 350. Students link their observations from the field experience to their own study of relevant mathematical skills and knowledge. Credit may not be

Analysis and application of ordinary differential equations: linear and nonlinear equations, existence and uniqueness theorems, analytic methods, qualitative analysis of solutions, and numerical methods. Combines theoretical ideas along with hands-on experience using appropriate computer software. Prerequisite: MATH 162 with a grade of

counted towards the mathematics major. Prerequisite: MATH 162 with a grade of C (2.0) or better. Corequisite: EDUC 350.

Systems of linear equations, vector spaces, independence, bases, dimension, orthogonality, least squares, determinants, eigenvalues and eigenvectors, positive definiteness, computation, and linear programming. Combines theoretical ideas with hands-on experience using appropriate computer software packages. Prerequisite: MATH

MATH 315 (3) Finite Mathematics

Sets, permutations, combinations, probability, linear equations and inequalities, matrices, linear programming, and finance. Credit may not be counted toward the mathematics major. May not be taken for credit by students who have received credit for MATH 130. Enrollment restricted to students who have completed the lower-division General Education requirement in Mathematics/Quantitative Reasoning (B4).

C (2.0) or better.

MATH 374 (3) Linear Algebra

160 with a grade of C (2.0) or better.

MATH 378 (3) Number Systems

MATH 340 (3)

Numbers: natural, rational, real, and complex. Algebraic laws: commutative, associative, and distributive. Brief introduction to groups, rings, and fields. Divisibility and unique factorization for integers and polynomials. Integers modulo n as finite rings and fields. The rational numbers as a non-complete countable ordered field. The real numbers as a complete uncountable ordered field. Sequences and limits including Cauchy sequences, lim inf and lim sup. Complex numbers including De Moivre’s theorem and related trigonometric identities. Factoring polynomials over the various number systems. The Fundamental Theorem of Algebra.

Stochastic Modeling in Business and Economics

Prerequisite: MATH 350 or MATH 370 with a grade of C (2.0) or better.

MATH 330 (3) Introduction to the History of Mathematics

Major currents in the evolution of mathematical thought from early civilization to modern times. Prerequisite: MATH 160 with a grade of C (2.0) or better.

Introduction to stochastic modeling with emphasis on application in business and economics. Discrete probability distributions including uniform, Bernoulli, binomial, hypergeometric, multinomial, and geometric. Random variables, expected value, and standard deviation. Joint distributions, conditional distributions, independence, and conditional expected value. Laws of large numbers. Discrete time Markov chains and martingales. Applications to include queuing models, cash and inventory management models, and stock option pricing. May not be taken for credit by students who have received credit for MATH 440, 441, or 571. Prerequisites: MATH 132 or 160 or 264 and MATH 315 or 374 with a grade of C (2.0) or better.

MATH 390 (1) Mathematics Colloquium

Guest lecturers present seminars on mathematical topics, e.g., recent advances in mathematics research, interesting applications of mathematics, or fun and challenging math problems. Students must attend each seminar, prepare a journal summarizing the content of each presentation, and write a follow up paper on one of the topics that they found particularly interesting. May be repeated for credit for a total of three (3) units. Graded Credit/No Credit. Enrollment Requirement: MATH 162 with a grade of C (2.0) or better.

MATH 410 (3) Modern Geometry

MATH 346 (3) Mathematical Methods for Physics

Survey of mathematical methods applicable to physics. Includes series, complex analysis, ordinary and partial differential equations, and special functions and transforms. Recommended Preparation:

Critical review of the foundations and basic structure of plane and solid Euclidean geometry, non-Euclidean geometries, incidence and affine geometries; convexity and applications. Prerequisite: MATH 350 or 370 with a grade of C (2.0) or better.

MATH 422 (3)

numerical results. Enrollment Requirement: MATH 160 with a grade of C

Introduction to Number Theory

(2.0) or better

Divisibility, Euclidean algorithm, unique factorization, congruences, and quadratic reciprocity. May also cover some of the following: included primitive roots and indices, continued fractions, sum of squares, introduction to Diophantine equations, prime numbers, pseudo-primes, the prime number theorem, and factorization and primality-testing algorithms. May not be taken for credit by students who have received credit for MATH 372. Prerequisite: MATH 378 with a grade of C (2.0) or better.

MATH 430 (3) Foundations of Analysis

A classical treatment of the basic concepts of calculus of one variable: the real number system, limits, continuity, differentiability, the Riemann integral, and sequences and series of numbers and functions. May not be taken for credit by students who have received credit for MATH 360. Prerequisite: MATH 378 with a grade of C (2.0) or better.

MATH 440 (4) Introduction to Mathematical Probability and Statistics

Basic concepts of probability: axiomatic formulation, combinatorics, conditional probability, independence, standard discrete and continuous random variables, expectation, variance, joint distributions, limit theorems. Statistical inference: tests of significance, point estimation methods, confidences intervals, simple linear regression. Combines theoretical ideas with hands-on experience using appropriate computer software packages. Enrollment Requirement: MATH 260 with a grade of C (2.0) or better.

MATH 441 (3) Introduction to Probability

Discrete and continuous probability spaces, axiomatic formulation, combinatorics, conditional probability and independence, standard discrete and continuous probability distributions (including uniform, Bernoulli, binomial, Poisson, geometric, normal, and exponential), random variables, expectation and variance, joint distributions, and limit theorems. Emphasis on modeling. Simple proofs required. Additional topics may include random walks, branching processes, and generating functions. Prerequisite: MATH 162 with a grade of C (2.0) or better.

MATH 442 (3) Introduction to Mathematical Statistics

Data analysis and inferential statistics: random samples, estimation, sufficient statistics, confidence intervals, hypothesis tests, curve fitting, linear regression, least squares, and goodness of fit. Covers both theory and applications, with emphasis on applications. Simple proofs required. Prerequisite: MATH 441 with a

MATH 464 (3) Numerical Analysis and Computing

Computer arithmetic, solution of a single algebraic equation, solution of systems of equations interpolating polynomials, numerical integration, numerical solution of ordinary differential equations, error analysis, and computational effort of numerical algorithms. Combines theoretical ideas with hands-on laboratory experience. Also offered as CS 464. Students may not receive credit for both. Prerequisite: CS 111 and MATH 162.

MATH 470 (3) Introduction to Abstract Algebra

An introduction to the theory of groups, rings, and fields, with abstract ideas reinforced by concrete and important examples, such as permutation groups, polynomial rings, and finite fields. The power of the axiomatic systems introduced will be illustrated via several applications to concrete and classical problems. Prerequisite: MATH 378 with a grade of C (2.0) or better.

MATH 472 (3) Introduction to Graph Theory

Fundamental concepts of undirected and directed graphs, trees, connectivity and traversability, planarity, colorability, networks, and matchings; emphasis on modern applications. Prerequisite: MATH 350 or 370 with a grade of C (2.0) or better.

MATH 474 (3) Introduction to Combinatorics

Introduction of the basic tools of combinatorics and their applications. Permutations, combinations, occupancy problems, generating functions, recurrences, inclusion/ exclusion, graph theory, pigeonhole principle, experimental design, and coding theory. Prerequisite: MATH 350 or 370 with a grade of C (2.0) or better.

MATH 480 (3) Introduction to Optimization

Modern study of linear programming with an emphasis on model formulation, solution, and interpretation of software output. Applications in work-scheduling, diet, capital budgeting, blending, production process, transportation, assignment, transshipment, and flow problems. Programming methods include the simplex method and its specialized variations, Big M Method, goal programming, and integer programming. Theoretical aspects include optimality conditions, sensitivity analysis, and duality. Requires using industry-standard software to strengthen the ideas and concepts. Also offered as CS 480. Students may not receive credit for both. Prerequisite: MATH 374.

grade of C (2.0) or better.

MATH 490 (3) MATH 448 (3)

Senior Seminar

Mathematical Models and Methods in Biology

Presentation and discussion of selected areas of mathematics in order to supplement available offerings. Sample areas include differential forms, complex variables, partial differential equations, and a second course in analysis, abstract algebra, or discrete math.

Introduces mathematical models in Biology and their analysis. Both one dimensional models, including the Malthusian Model and the logistic model, and multi-dimensional models, including structured population models and predator-prey models, are studied, as are matrix models for base substitution in DNA, phylogenetic trees, and sequence alignment. Mathematical concepts and methods to formulate and analyze these models include limits, derivatives, matrix algebra, eigenvectors, probability theory, and dynamic programming. Software is used to simulate models and visualize the

May be repeated twice as course content changes, with consent of the program, for a maximum of nine (9) units of credit from MATH 490 and 491. Enrollment Requirement: Twelve (12) units of upper-division mathematics. Other requirements to be determined by instructor.

MATH 491 (3)

MATH 521 (3)

Senior Seminar with Lab

Computational and Applied Algebra

Presentation and discussion of selected areas of mathematics in order to supplement available offerings. Sample areas include mathematical modeling and a second course in numerical analysis, optimization, or statistics. This course meets for four hours per week. May

Introduction to algebraic tools and ideas that have applications in such fields as cryptography, coding theory, number theory, algebraic geometry, integer programming, computing modeling, and robotics. Includes some of the following: finite fields, Gröbner bases, resultants, algebraic curves, and their codes.

be repeated for a maximum of nine (9) units of credit for MATH 490 and 491. Enrollment Requirement: Twelve (12) units of upper-division mathematics. Other requirements to be determined by instructor.

Prerequisite for undergraduate students and enrollment requirement for graduate students: MATH 470.

MATH 495 (1-3)

MATH 522 (3)

Internship in Mathematics

Number Theory

Faculty-sponsored academic internship in business, industrial, government, research firm, or university labs and centers.

Introduction to number theory from the algebraic and/or analytic point of view. Includes some of the following: congruences, finite fields and rings, and quadratic reciprocity; quadratic forms and Diophantine equations; elliptic curves; the Gaussian integers, the Eisenstein integers, and unique factorization in these rings; other quadratic and cyclotomic fields and ideal factorization; and introduction to analytic number theory, primes in arithmetic progressions, and the prime number theorem.

Enrollment restricted to students who have obtained consent of instructor.

MATH 498A (1) 498B (2) 498C (3) Individual Study in Mathematics

Individually directed reading and study in mathematical sciences literature. May be repeated for a maximum of three (3) units. Enrollment Requirement: Twelve (12) units of upper-division Mathematics. Enrollment restricted to students who have obtained consent of instructor.

MATH 499A (1) 499B (2) 499C (3) Independent Research in Mathematics

Designed for students capable of independent and original research. May be repeated for a maximum of three (3) units. Enrollment Requirement: Twelve (12) units of upper-division mathematics. Enrollment restricted to students who have obtained consent of instructor.

MATH 505 (3) Readings from Original Sources

Mathematics studied through the reading, analysis, and discussion of original papers. May be repeated once for credit with consent of instructor. Enrollment requirements to be determined by instructor.

Prerequisite for undergraduate students and enrollment requirement for graduate students: MATH 470 with a grade of C (2.0) or better.

MATH 523 (3) Cryptography and Computational Number Theory

Algorithms for factorization and primality testing: pseudo-primes, quadratic sieve, Lucas Test, continued fractions, factorization using elliptic curves, and public key cryptosystems such as RSA, which is widely used for secure transfer of data on the internet. Additional background material (such as the rudiments of elliptic curves) will be introduced as needed. Combines theoretical ideas with computer lab experimentation and implementation. Some familiarity with a computer language is useful but not required. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 350 or 370 with a grade of C (2.0) or better.

MATH 510 (3) Mathematical Communication

MATH 528 (3)

Selected topics in advanced mathematics chosen to demonstrate appropriate use of technology and effective organization and presentation of mathematics in oral and written form. Includes three aspects of mathematical writing: writing expository mathematics, writing formal mathematics, and writing as a tool to learn; preparation of mathematical lectures; and development software modules/notebooks. Prerequisite for undergraduates and enrollment

Advanced Linear Algebra

requirement for graduate students: MATH 350. Additional enrollment requirement for all students: At least nine (9) other units of upper-division mathematics.

MATH 520 (3) Algebra

Review and continuation of the study of algebra begun in MATH 470. Covers some of the following: the theory of finite group theory including the Sylow Theorems, polynomial ring, unique factorization, number fields, and finite fields. The latter half of the course will cover field extensions and Galois Theory, including the classic theorems on the unsolvability of the general quintic and the impossibility of certain ruler and compass constructions, such as trisecting an angle. Prerequisite for undergraduate students and enrollment requirement for graduate students: MATH 470 with a grade of C (2.0) or better.

Vector spaces; dual spaces; linear transformations; bilinear forms and their matrix representations; Jordan and other canonical forms; finite-dimensional spectral theory; and connections to other branches of mathematics. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 with a grade of C (2.0) or better.

MATH 530 (3) Measure Theory

Lebesgue measure, measurable functions, the Lebesgue integral, Fubini’s Theorem, Lp-spaces, and differentiation. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 430.

MATH 532 (3) Ordinary Differential Equations

Theory and applications of ordinary differential equations. Existence and uniqueness of solutions, methods for solving equations, linear differential equations, singularities, qualitative analysis of solutions, and systems of equations. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 and 430.

MATH 534 (3)

MATH 542 (3)

Partial Differential Equations

Advanced Graph Theory

Theory and applications of partial differential equations. Cauchy problems, boundary problems, the Cauchy-Kovalevsky Theorem, Fourier Series, harmonic functions, elliptic equations, and hyperbolic equations. Enrollment Requirement: MATH 260 and 362. Prerequisite for

Graphs and digraphs; traversability; factorization; planarity and embedding; coloring; graph Ramsey theory; probabilistic methods; extremal graph theory; and algebraic graph theory.

undergraduates and enrollment requirement for graduate students: MATH 374 and 430.

Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 350 or 370 or 470 or 472 or 474.

MATH 544 (3)

MATH 535 (3)

Advanced Combinatorics

Multivariable Advanced Calculus

Enumeration; combinatorial set systems; combinatorial designs; Ramsey theory; combinatorial optimization; matroids; and axiomatic social choice. Prerequisite for undergraduates and enrollment requirement

Analysis in several variables including multivariable derivatives and integrals, inverse function theorem, implicit function theorem, and generalizations of the fundamental theorem of calculus (e.g., Stokes’ Theorem). Some of these topics may be presented from the point of view of differential forms. Enrollment Requirement: MATH 260. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 and 430.

MATH 536 (3) Complex Analysis

Study of functions of a complex variable, including analytic functions, contour integrals, Cauchy’s Theorem, poles and residues, Liouville’s Theorem, Laurent Series, the Residue Theorem, analytic continuation, and conformal mappings. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 430.

MATH 537 (3) Calculus of Variations

Study of the theory of maximum and minimum values of functions defined on spaces of infinite dimension. Includes topics such as Euler’s equation, geodesics, the isoperimetric problem, optimization constrained by subsidiary conditions, and the Weierstrass-Erdman corner conditions. Emphasis to be on both theory and application. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 362, 374, and 430.

MATH 538 (3) Applicable Analysis

Foundations of functional analysis; linear and metric spaces; different modes of convergence; Hilbert Space; and applications. May include topics such as calculus of variations, fixed point theorems, and operator theory. Prerequisites for undergraduates and enrollment requirement for graduate students: MATH 362, 374, and 430.

MATH 540 (3) Concrete Mathematics

Blend of continuous and discrete topics including sums, recurrences, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 350 or 370 or 470 or 472 or 474.

MATH 541 (3) Structural Graph Theory

Material covered will be selected from a subset of the following subjects: trees and cycles; independence and matching; graph partitioning, packing, and covering; tournaments; flows; algorithmic aspects; topological graph theory; and facility location. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 350 or 370 or 470 or 472 or 474.

for graduate students: MATH 350 or 370 or 470 or 472 or 474.

MATH 550 (3) Geometry

Geometric ideas selected from the following fields: euclidean geometry, hyperbolic geometry, projective geometry, introductory algebraic geometry, and computational geometry. Combines theoretical ideas with hands-on laboratory experience. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 and 470.

MATH 552 (3) Introduction to Differential Topology and Geometry

Introduction to curves, surfaces, and possibly higher dimensional manifolds from the point of view of differential topology and/or differential geometry. Includes some of the following: Curves (e.g., Frenet-Serret Theorem and its consequences, isoperimetric inequality, four-vertex theorem, line integrals, Fenchel’s Theorem); the topological classification of surfaces, vector fields, and curvature on surfaces (leading up to some of the following: geodesics, minimal surfaces, Gauss’s theorema egregium, and the Gauss-Bonnet Theorem); and introduction to higher dimensional manifolds, differential forms, and integration (possibly including Stokes’ Theorem and global invariants such as the Euler characteristic and de Rham cohomology). Enrollment Requirement: MATH 260. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 and 430.

MATH 555 (3) General Topology

Topological spaces, open and closed sets, metric spaces, continuity, compactness, and connectedness. Other subjects may include separation axioms, fundamental groups, classification of surfaces, and completion of metric spaces. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 430.

MATH 561 (3) Computational Linear Algebra

Provides a thorough background in the formulation and analysis of algorithms for numerical linear algebra. Includes fundamentals of scientific computation, subspaces, rank-revealing matrix factorizations, numerical solutions of linear systems, linear least squares, regularization, perturbation theory, and iterative methods. Combines theoretical ideas with laboratory experience. Knowledge of computer language is required. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374.

MATH 563 (3)

MATH 699 (3)

Numerical Solution of Ordinary Differential Equations

Thesis

Survey of numerical methods for the solution of ordinary differential equations including Runge-Kutta, Taylor’s, Voterra, and multistep methods. Analysis of convergence and implementation of various methods using numerical software. Covers the stability of differential equations and stability regions for numerical schemes. Subjects include the method of lines, two-point boundary value problems, and Volterra integral equations. Prerequisite: MATH 362

Preparation of a thesis for the master’s degree. Graded Credit/ No

and MATH 374 with a grade of C (2.0) or better.

MATH 564 (3) Nonlinear Programming

Theory and techniques for solving constrained and unconstrained nonlinear programming problems. Techniques include Quasi-Newton Secant Methods, Broyden’s Method, conjugate gradient methods, and line search methods. Theoretical aspects include convexity, Lagrangian Multipliers, optimality conditions, convergence, primal problem, duality, saddle points, and line searches. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 374 or 480 or CS 480.

MATH 570 (3) Introduction to Stochastic Processes

Elements of stochastic processes, discrete-time and continuous-time Markov chains, random walks, branching processes, birth and death processes, and Poisson point processes. Applications to queues and stochastic networks, resource management, biology, and physics. May include optimal stopping, hidden Markov models, renewal processes, martingales, Brownian motion, and Gaussian processes. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 430 and 440.

MATH 571 (3) Probability and Random Processes

Framework for probability theory: probability spaces as measure spaces, random variables, expectation and conditional probability. Major results such as limit theorems for sums of random variables, zero-one laws, and ergodic theorems. Applications may include branching processes, Markov Chains, Markov Random Fields, martingales, percolation, Poisson Processes, queuing theory, random walks, and renewal processes. Combines theoretical ideas with hands-on laboratory experience using appropriate computer software packages. Prerequisite for undergraduates and enrollment requirement for graduate students: MATH 430 or 440.

MATH 620 (3) Seminar in Advanced Mathematics

Advanced mathematics chosen from areas represented in the program faculty and intended to build on 500-level material. Covers the following: algebra and number theory, analysis, combinatorics and graph theory, computational mathematics, geometry, and probability. May be repeated for a maximum of twelve (12) units of credit for MATH 620 and 621. Enrollment restricted to students who have obtained consent of instructor.

MATH 621 (3) Seminar in Advanced Mathematics with Lab

Advanced mathematics chosen from areas represented in the program faculty and intended to build on 500-level material. Covers the following: algebra and number theory, analysis, combinatorics and graph theory, computational mathematics, geometry, and probability. May be repeated for a maximum of twelve (12) units of credit for MATH 620 and 621. This course meets for four hours per week. Enrollment restricted to students who have obtained consent of instructor.

MATH 697 (1) Workshop in the Teaching of Mathematics

Discussion of syllabus construction, lecture preparation, assignment and grading of homework, construction and grading of exams, and resolution of classroom problems. May be repeated, but credit will not be counted toward the Master of Science degree. Graded Credit/No Credit. Enrollment restricted to students with graduate standing in mathematics.

Credit. May be repeated for a total of six (6) units, but students may enroll in only one section per semester. Enrollment requires approval of the graduate coordinator.