This Web document is updated twice a year, on or about the first day of registration for Fall and Spring semesters.

1030  Introduction to Quantitative Reasoning (3) Prerequisite: Math ACT score of 23 or grade of C or better in MATH 1010. Fulfills Quantitative Reasoning (Math).
   In this course we analyze quantitative information about change and growth through specific case studies. The mathematics covered includes topics from financial mathematics, linear and exponential growth, geometric measurements and scaling. This course is primarily for undergraduates who will not take any further mathematics except for statistics.

1040  Introduction to Statistics and Probability (3) Prerequisite: Math ACT score of 23 or grade of C or better in MATH 1010. Fulfills Quantitative Reasoning (Statistics/Logic).
   Topics covered: how to collect, organize, analyze, display, and interpret data; deviation, variance and standard deviation, Empirical rule; basic concepts of probability and counting, conditional probability, multiplication and addition rule; probability distributions, binomial distributions; standard normal (bell-shaped) distributions; correlation and linear regression. 

1050  College Algebra (4) Prerequisite: Math ACT score of 23 or grade of C or better in MATH 1010. Fulfills Quantitative Reasoning (Math).
   Functions, inverses and graphs; polynomial, rational, radical, exponential and logarithmic functions; systems of equations and matrices; applications; arithmetic and geometric sequences and series.

1060  Trigonometry (3) Prerequisite: Math ACT score of 23 or grade of C or better in MATH 1010. Fulfills Quantitative Reasoning (Math).
   Trigonometric functions, inverses, equations and identities with applications; introduction to vectors.

1070  Introduction to Statistical Inference (3) Prerequisite: Math ACT score of 23 or grade of C or better in MATH 1010. Fulfills Quantitative Reasoning (Statistics/Logic).
   The important topics used in making inferences from data will be presented and illustrated. As well as material on descriptive statistics, estimation of the mean, or of the proportion, in one or two populations, simple linear regression, and one-way analysis of variance are covered.

1090  College Algebra for Business and Social Sciences (3) Prerequisite: Math ACT score of 23 or grade of C or better in MATH 1010. Fulfills Quantitative Reasoning (Math).
   Functions and graphs, linear and quadratic functions, matrices, Gaussian elimination, Leontieff models, exponential and logarithmic functions, growth, periodic and continuously compounded interest, arithmetic and geometric series, annuities and loans.

1100  Quantitative Analysis (3) Prerequisite: Math ACT score of 28 or C or better in MATH 1090. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Not for students who have completed more than one semester of calculus. Differentiation, maximization and minimization of functions, marginal analysis and the optimization of constrained functions, integration and applications.

1170  Calculus for Biologists I (4) Prerequisite: Math ACT score of 28 or grade of C or better in MATH 1050 and 1060. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Derivation of dynamical models of biological systems and their analysis with differential and integral calculus. Discrete-time dynamical systems for growth, breathing, selection, the heart, etc. Differentiation and its applications to stability, approximation of functions, maximization, and limits. Differential equations describing growth, diffusion, and selection, and their solution with integral calculus. Computer lab using Maple.

1180  Calculus for Biologists II (4) Prerequisite: A grade of C or better in MATH 1170 or consent of instructor. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Higher dimensional and probabilistic models of biological systems and their analysis. Phase plane analysis of interacting populations and the neuron. Derivation and analysis of stochastic dynamical systems describing growth, diffusion, and selection. Introduction to probability theory including distributions, random variables, and probability density functions. Applications of binomial, exponential, Poisson, and normal distributions. Introduction to statistics including parameter estimation, maximum likelihood, hypothesis testing and regression. Computer lab using Maple.

1210  Calculus I (4) Prerequisite: Math ACT score of 28 or grade of C or better in MATH 1050 AND 1060. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Functions and their graphs, differentiation of polynomial, rational and trigonometric functions. Velocity and acceleration. Geometric applications of the derivative, minimization and maximization problems, the indefinite integral, and an introduction to differential equations. The definite integral and the Fundamental Theorem of Calculus.

1220  Calculus II (4) Prerequisite: A grade of C or better in MATH 1210. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Geometric applications of the integral, logarithmic, and exponential functions, techniques of integration, conic sections, improper integrals, numerical approximation techniques, infinite series and power series expansions, differential equations (continued).

1250  Calculus for AP Students I (4) Prerequisite: Math AP AB test score of 4 or 5. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   MATH 1250 and MATH 1260 together replace the three semester sequence MATH 1210, MATH 1220, MATH 2210. Review of introductory calculus, applications of differential and integral calculus, introduction to differential equations, conic sections and polar coordinates, numerical approximation, sequences and series, power series.

1260  Calculus for AP Students II (4) Prerequisite:  A grade of C or better in MATH 1250.  Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Completion of MATH 1260 is equivalent to completing the entire three-semester Calculus I, II, III sequence. Vectors in the plane and in 3-space, differential calculus in several variables, integration and its applications in several variables, vector fields, and line, surface and volume integrals. Green's and Stokes' Theorems.

1270  Accelerated Engineering Calculus I (4) Prerequisite: Math AP AB score of 3 or 4 or Math ACT score of 30.
   Math 1270 and 1280 together are equivalent to the three semester sequence Math 1210, Math 1220, and Math 2210. This sequence is intended for engineering majors. Review of introductory calculus, applications of differential and integral calculus, introduction to differential equations, conic sections and polar coordinates, numerical approximation, sequences and series, power series.

1280  Accelerated Engineering Calculus II (4) Prerequisite: A grade of C or better in MATH 1270.
   Completion of Math 1280 is equivalent to completing the entire three semester Calculus I, II, II sequence. Vectors in the plane and in 3-space, differential calculus in several variables, integration and its applications in several variables, vector fields, and line, surface and volume integrals, Green's and Stokes Theorems.

1900  Topics in Mathematics (1 to 4) Prerequisite: Instructor's consent.
   Various special topics in mathematics to be treated at the appropriate level.

2160  Introduction to Scientific Computing Using C (3) Prerequisite: A grade of C or better in MATH 1210 or consent of instructor. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   A short introduction to those aspects of C and C++ essential for mathematics, followed by extensive work with mathematics problems in which computation plays an important role.

2200  Introduction to Discrete Mathematics (3) Prerequisite: A grade of C or better in MATH 1220.
   Fundamentals of logic, set theory, order, relations, functions. Elementary number theory, modular arithmetic. Combinatorics; counting permutations, generating functions, matrix operations. Basic algebraic structures; groups, rings. Discrete probability. Introduction to graphy theory, trees, search optimization problems. Boolean algebra.

2210  Calculus III (3) Prerequisite: A grade of C or better in MATH 1220. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Vectors in the plane and in 3-space, differential calculus in several variables, integration and its applications in several variables, vector fields and line, surface, and volume integrals. Green's and Stokes' theorems.

2250  Differential Equations and Linear Algebra (4) Prerequisite: A grade of C or better in MATH 1210/1220 and MATH 2210 or PHYS 2210 or 3210, MATH 1250/1260; 1270/1280. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   This is a hybrid course which teaches the allied subjects of linear algebra and differential equations. These topics underpin the mathematics required for most students in the Colleges of Science, Engineering, Mines & Earth Science.

2270  Linear Algebra (4) Prerequisite: A grade of C or better in MATH 1220 or 1260. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Euclidean space, linear systems, Gaussian elimination, determinants, inverses, vector spaces, linear transformations, quadratic forms, least squares and linear programming, eigenvalues and eigenvectors, diagonalization. Includes theoretical and computer lab components.

2280  Introduction to Differential Equations (4) Prerequisite: A grade of C or better in MATH 2270. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Linear and nonlinear differential equations and systems of equations, with applications. Matrix exponential, fundamental solution matrix, phase-space and portraits, stability, initial- and boundary-value problems, introduction to partial differential equations. Requires familiarity with linear algebra. Includes theoretical and computer lab components.

2900  Honors Seminar in Mathematics (2) Co-requisite: A grade of C or better in MATH 1210 or 1250.
   Fostering the ability to understand theorems and their purpose by studying selected groups of theorems in contexts that are new to the students and not part of the regular introductory courses.

3000  Undergraduate Colloquium (1)
   Colloquium of presentations of topics of contemporary mathematical interest.

3010  Topics in the History of Mathematics (3) Prerequisite: A grade of C or better in MATH 1210 or equivalent. Fulfills Comm/Wrtg & Phys/Life Sci Exploration.
   A brief look at the history of mathematics, focusing on the principal ideas of importance in the development of the subject.

3070  Applied Statistics I (4) Prerequisite: A grade of C or better in MATH 1220 or 1250. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   An introduction to basic probability theory, sampling from normal populations, large-sample problems, sampling from one or two populations, estimation, and testing. SAS is used to perform statistical analyses. There are three lectures and one 1 1/2 hour lab per week.

3080  Applied Statistics II (3) Prerequisite: A grade of C or better in MATH 3070. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Introduction to analysis of variance, regression analysis, correlation analysis, and nonparametric techniques. Continued use of SAS programming language. There are two lectures and one 1 1/2 hour lab per week.

3090  Design of Experiments (3) Prerequisite: A grade of C or better in MATH 3070.
   Introduction to the design of experiments, multiple regression, factorial and nested designs. SAS is used for computations.

3100  Foundations of Geometry (3) Prerequisite: A grade of C or better in MATH 2210 and MATH 2200. Co-requisite:  MATH 4030. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Modern axiomatic development of Euclidean geometry and of trigonometry, also incidence theorems, projective invariants, straight-edge and compass constructions, spherical and hyperbolic geometries. Mathematics teaching majors should take the accompanying practicum, MATH 3105, the following semester.

3105  Geometry Practicum (1) Prerequisite: A grade of C or better in MATH 3100. Co-Requisite: MATH 4035 and 4090
   Application of the geometry studied in MATH 3100 to the secondary- school geometry classroom including appropriate use of technology, analysis of textbooks, interviews with students and opportunities to teach.  A significant portion of this course is field work. (Second session course).

3150  Partial Differential Equations for Engineering Students (2) Prerequisite: A grade of C or better in MATH 2250 and either MATH 1260 or MATH 2210.
   Fourier series and boundary-value problems for the wave, heat, and Laplace equations, separation of variables, Sturm-Liouville problems and orthogonal expansions, Bessel functions and Legendre polynomials. Fourier transform.

3160  Applied Complex Variables (2) Prerequisite: A grade of C or better in MATH 2210 or 1260 or 1280 or consent of instructor.
   Analytic functions, complex integration, Cauchy integral theorem, Taylor and Laurent series, residues and contour integrals, conformal mappings with applications to electrostatics, heat, and fluid flow.

3170  SAS Lab I (1)
   For students who wish to participate only in the Lab component of MATH 3070.

3180  SAS Lab II (1)
   For students who wish to participate only in the Lab component of MATH 3080.

3190  SAS Lab III (1)
   For students who wish to participate only in the Lab component of MATH 3090.

3210  Foundations of Analysis I (4) Prerequisite: A grade of C or better in MATH 2210. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Logic, methods of proof and mathematical argument in mathematical analysis. Rigorous reconsideration of the real-number system, infinite series and of continuity, differentiation and integration for functions of one variable. The emphasis is on improving the student's ability to understand and explain concepts in a logical and complete manner.

3220  Foundations of Analysis II (4) Prerequisite: A grade of C or better in MATH 2270, 3210. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Advanced multivariable calculus. Topics include continuity, compactness, differentiation and affine approximation, chain rule, Taylor series, extremization, error estimation, inverse and implicit function theorems, Riemann integration, Fubini's Theorem, change of variables formula. The emphasis is on further developing the student's ability to understand more abstract concepts and to write an effective and rigorous mathematical argument.

3300  Laboratory in Computational Science (3) Prerequisite: A grade of C or better in MATH 1210-1220 or MATH 1170-1180, and MATH 2250 or 2270.
   Mathematical and computational experimentation to understand principles of physical, chemical, and biological processes. Develop hypotheses, and use Matlab to test, questions regarding diffusion, molecular interactions, populations dynamics, epidemics, and ranking of sports teams. Emphasis throughout on using mathematical reasoning to understand computer simulation results.

3900  Topics in Mathematics (1 to 4) Prerequisite: Instructor's consent.
   Various special topics in mathematics to be treated at the appropriate level.

3910  Supervised Reading (1 to 6) Prerequisite: Instructor's consent.
   A course of independent study overseen by a faculty member.

4010  Mathematics for Elementary School Teachers I (4) Prerequisite: A grade of C or better in MATH 1050. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   This is the first course in a two-course sequence for prospective elementary school teachers. This is a content course that provides teachers with a deeper understanding of the real number system and arithmetic operations for whole numbers, fractions, and decimals. This provides the conceptual framework that allows teachers to analyze and correct common student misunderstandings in Grades K-6. See the Utah State Core Curriculum at www.usoe.k12.ut.us. Teaching methods pertaining to this material are discussed in TL 5360.

4020  Mathematics for Elementary School Teachers II (4) Prerequisite: A grade of C or better in MATH 4010. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   This is a continuation of MATH 4010, but deals with a different part of the mathematics curriculum. It is a content course that gives teachers a better understanding of topics in geometry appropriate to Grades K-6, including measurement, symmetry, geometric shapes, congruence and similarity. The course presents ideas from an intuitive perspective that prepares teachers to discuss geometry with children, and from a computational perspective to enable teachers to work with students to calculate distance, area and volume in both customary and metric units, measure angles, construct figures, and more. A brief discussion of topics in statistics and probability for Grades K-6 is also included. See the Utah State Core Curriculum at www.usoe.k12.ut.us. Teaching methods pertaining to this material are discussed in TL 5360.

4030  Foundations of Algebra (3) Prerequisite: A grade of C or better in MATH 2210 and MATH 2200. Co-Requisite:  MATH 3100. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   The integers, unique factorization, and modular arithmetic. Polynomial rings. Introduction to abstract algebraic systems. Mathematics teaching majors should also take the accompanying practicum, MATH 4035, the following semester..

4035  Algebra Practicum (1) Prerequisite: A grade of C or better in MATH 4030. Co-Requisite: MATH 4090 and 3105.
   Application of the material studied in MATH 4030 to the secondary- school algebra classroom including appropriate use of technology, analysis of textbooks, interviews with students and opportunities to teach. A significant portion of this course is field work. (Second session course.)

4090  Teaching of Secondary School Mathematics (3) Prerequisite: A grade of C or better in MATH 3100 and MATH 4030 Fulfills Quantitative Reasoning (Math & Stat/Logic). Co-Requisite: MATH 3105 and 4035
   In this course we discuss teaching methods for secondary classroom, including appropriate use of technology. Important components are developing topics across the curriculum, connections between mathematical concepts, as well as other important issues in running one's classroom: setting norms, goals, assessing students work and knowledge, providing practice and direction, managing diverse populations. This is a hands on class with ample opportunities for teaching. (First session course).

4190 Secondary Mathematics Teaching Practicum (3) Prerequisite:  A grade of C or better in MATH 3220 and either 4090, 4035 or 3105.
   Attend Intermediate Algebra course, assist in grading, teach a weekly review session, meet weekly to discuss pedagogy and evaluations of teaching.

4200  Introduction to Complex Variables (3) Prerequisite: A grade of C or better in MATH 3220. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Complex functions and their differentiability, complex integrals, power series, the Cauchy theorem and formulas, residues and applications to evaluating integrals, conformal mappings and applications. Graduate students who need this course should consult the instructor.

4400  Introduction to Number Theory (3) Prerequisite: A grade of C or better in MATH 2250 or MATH 2270. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   An overview of algebraic number theory, covering factorization and primes, modular arithmetic, quadratic residues, continued fractions, quadratic forms, and diophantine equations.

4510  Introduction to Topology (3) Prerequisite: A grade of C or better in MATH 3220. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Introduction to the ideas of topologies, compactness, connectedness, separation axioms, metric spaces. Graduate students who need this course should consult the instructor.

4530  Curves and Surfaces in Euclidean Space (3) Prerequisite: A grade of C or better in MATH 3220. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Frenet theory, Gaussian and mean curvatures, Gauss-Bonnet theorem, minimal surfaces, and surfaces of constant curvature. Graduate students who need this course should consult the instructor.

4600 Mathematics in Physiology and Medicine (4) Prerequisite:  A grade of C or better in MATH 2250.
   The goals of the class are (i) to introduce the students to a range of modern mathematical tools; (ii) to teach the students the skill of building tractable mathematical models of biological processes; (iii) to show how to combine the mathematical knowledge, the numerical simulations (in Matlab) and biological intuition to derive new insights into the functioning of living systems. Mathematical topics include introduction to linear algebra, complex numbers, geometric dynamical systems, bifurcation theory, probability, Markov chain, partial differential equations. Biological topics may include modeling heart and circulation, kidneys, circadian clocks, brain rhythms, HIV, antibiotic resistance in bacteria, regulation of gene expression, biological pattern formation.

4750  Elementary Mathematical Fluid Dynamics (3) Prerequisite: A grade of C or better in MATH 2250 and 3150 or consent of instructor.
   An elementary introduction to fluid dynamics for the advanced undergraduate sciences student. The course will discuss the mathematical description of a variety of interesting fluid phenomena.

4800  Undergraduate Research in Mathematics (3) Prerequisite: Permission of instructor required.
   Explore a topic of significant mathematical interest, or an application of mathematics to a significant problem in science, engineering, or business. Students help to present the material or the results of their own investigations, and write a report on their findings. Prerequisites vary depending on the topic.

4910  Internship in Mathematics (1 to 4)
   Mathematics-related work in industry, business, or government.

4950  Special Research Projects (2 to 6)
   Projects to be completed as part of the requirements for the Departmental Honors program in mathematics.

4999  Honors Thesis/Project (3)
   Restricted to students in the Honors Program working on their University Honors degree.

5000  Undergraduate Problem Seminar (1) Prerequisite: A grade of C or better in MATH 1210.
   Difficult problems presented for solution sharpen skills and develops problem-solving techniques. Prepares students for Putnam Examination (given annually by the Mathematical Association of America).

5010  Introduction to Probability (3) Prerequisite: A grade of C or better in MATH 2210 or 1260. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6805. Combinatorial problems, random variables, distributions, independence and dependence, conditional probability, expected value and moments, law of large numbers, and central-limit theorems.

5030  Actuarial Mathematics (3) Prerequisite: A grade of C or better in MATH 5010. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Basic introduction to actuarial and insurance mathematics. Prepares students for the actuarial exam.

5040  Stochastic Processes and Simulation I (3) Prerequisite: A grade of C or better in MATH 5010. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6810. An introductory course in the theory and practice of random processes with special emphasis on problem solving and simulation analysis.

5050  Stochastic Processes and Simulation II (3) Prerequisite: A grade of C or better in MATH 5040. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6815. Second half of the course described under the listing for MATH 5040.

5075  Time Series Analysis (3) Prerequisite: A grade of C or better in MATH 5010.
   Meets with MATH 6820. An introduction to the basic topics: difference equations and lag operators, stationary autoregressive moving average processes, forecasting, estimation of parameters, spectral analysis, Kalman filter, introduction to nonlinear time series, processes with deterministic trends, processes with unit roots, cointegration, time series models for heteroskedasticity, and time series with changes in regime.

5080  Statistical Inference I (3) Prerequisite: A grade of C or better in MATH 5010. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Sampling, sampling distributions, Central Limit Theorem, transformation of data, complete and sufficient statistics, point estimation, optimality.

5090  Statistical Inference II (3) Prerequisite: A grade of C or better in MATH 5080. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Interval estimation, hypothesis testing, likelihood method, errors, optimality, order statistics, nonparametric methods, rank statistics.

5110  Mathematical Biology I (3) Cross listed as BIOL 5011. Prerequisite: A grade of C or better in MATH 2250 or 2280 or consent of instructor. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6830. Mathematical modeling in the biological and medical sciences. Topics will include continuous and discrete dynamical systems describing interacting and structured populations, resource management, biological control, reaction kinetics, biological oscillators and switches, and the dynamics of infectious diseases.

5120  Mathematical Biology II (3) Cross listed as BIOL 5012. Prerequisite: A grade of C or better in MATH 5110. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6835. Mathematical models of spatial processes in biology including pattern formation in the embryo and during tissue differentiation, applications of traveling waves to population dynamics, epidemiology, and chemical reactions, and models for neural patterns.

5210  Introduction to Real Analysis (4) Prerequisite: A grade of C or better in MATH 3220. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Metric spaces, fixed-point theorems and applications, Lebesgue integral, normed linear spaces, approximation, the Fundamental Theorem of Calculus.

5215  Applied Fourier Analysis (3) Prerequisite: A grade of C or better in MATH 2270 and MATH 2280.
   Fourier series and integrals, uncertainty principle, approximation and convergance theorems, discrete Fouier transforms and the Fast Fourier transform, signal and image processing. A project/paper will be required.

5250  Matrix Analysis (3) Prerequisite: A grade of C or better in MATH 2270.
   Linear transformations and their eigenvalues, factorizations and canonical forms, vector and matrix norms, special matrix types, matrix-valued functions, generalized inverses, matrix groups.

5310  Introduction to Modern Algebra I (3) Prerequisite: A grade of C or better in MATH 2250 or 2270 and MATH 3210 or 2900. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   An introduction to groups, rings, and fields.

5320  Introduction to Modern Algebra II (3) Prerequisite: A grade of C or better in MATH 5310. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Second half of the course described under the listing for MATH 5310.

5405  Cryptography, Codes, and Computational Number Theory (3) Prerequisite: A grade of C or better in MATH 4400 and MATH 5310.
   Classic and modern methods of encryption, applications to public-key ciphers (RSA, El Gamnal, etc.), random number generators, attacks on encryption systems, error correcting codes; computational number theory. A project/paper will be required.

5410  Introduction to Ordinary Differential Equations (4) Prerequisite: A grade of C or better in MATH 3220. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6840. Linear ordinary differential equations: initial-value problems and behavior of solutions. Nonlinear equations: existence, uniqueness, perturbations, extension to the boundary. Introduction to dynamical systems and their relation to differential equations.

5420  Ordinary Differential Equations and Dynamical Systems (3) Prerequisite: A grade of C or better in MATH 5410 or consent of instructor. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6845. Continuation of the study of dynamical systems, through a discussion of stability and its absence, concrete examples. Sturm-Liouville theory, including the existence of complete orthormal systems of eigenfunctions.

5440  Introduction to Partial Differential Equations (3) Prerequisite: A grade of C or better in MATH 3220. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6850. Classical wave, Laplace, and heat equations. Fourier analysis, Green's functions. Methods of characteristics.

5470  Chaos and Nonlinear Systems (3) Prerequisite: A grade of C or better in MATH 2250 or 2280. Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Meets with MATH 6440. Introduction to chaotic motions, strange attractors, fractal geometry. Models from fluid dynamics and mechanical and electrical oscillators.

5520  Introduction to Algebraic/Geometric Topology (3) Prerequisite: A grade of C or better in MATH 4510 and 5310 or equivalent. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Introduction to algebraic topology via the fundamental group of a topological space. Includes selected topics in geometric topology.

5600  Survey of Numerical Analysis (4) Prerequisite: A grade of C or better in MATH 2210, either MATH 2250 or 2270 and computing experience. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6855. Numerical linear algebra, interpolation, integration, differentiation, approximation (including discrete and continuous least squares, Fourier analysis, and wavelets), initial- and boundary-value problems of ordinary and partial differential equations.

5610  Introduction to Numerical Analysis I (4) Prerequisite: A grade of C or better in MATH 2210, either MATH 2250 or 2270 and computing experience. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6860. Numerical linear algebra, interpolation, integration, differentiation, approximation (including discrete and continuous least squares, Fourier analysis, and wavelets).

5620  Introduction to Numerical Analysis II (4) Prerequisite: A grade of C or better in MATH 5610. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6865. Numerical solution of initial and boundary value problems of ordinary and partial differential equations.

5700  Capstone Course in Mathematics (3) Prerequisite: A grade of C or better in MATH 4030 and 3100
   The Capstone Course in Mathematics examines secondary school mathematics from an advanced point of view. The topics covered are drawn from Abstract Algebra, Analysis, and Geometry and are rooted in the core secondary curriculum of number and operations, algebra, geometry, and functions. Students learn to generalize definitions and theorems that help to unite and explain mathematics. As they explore familiar secondary mathematic problems from a higher perspective, they draw connections between ideas taught separately in different courses. Through their work in the course, they improve their ability to promote their students' understanding of mathematics and to make better decisions regarding the direction of their lessons and curriculum.

5710  Introduction to Applied Mathematics I (3) Prerequisite: A grade of C or better in MATH 2250 and 3150 and 3160. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Symmetric linear systems, positive definite matrices, eigenvalue problems, equilibrium equations for discrete and continuous systems, boundary value problems in ODEs and PDEs, boundary integrals.

5720  Introduction to Applied Mathematics II (3) Prerequisite: A grade of C or better in MATH 5710. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Fourier methods, initial value problems in ODEs and PDEs, conservation laws, network flows and combinatorics, optimization.

5740  Mathematical Modeling (3) Prerequisite: A grade of C or better in MATH 5600 or CP SC 5220. Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6870. Development of mathematical models for physical, biological, engineering, and industrial phenomena and problems, using mainly ordinary and partial differential equations. Involvement of analytical and numerical tools suitable for analysis and visualization of the solutions of these problems, including packages such as LINPACK, EISPACK, Maple and Matlab.

5750  Topics in Applied Mathematics (3) Fulfills Quant Reason(Math & Stat/Log) & Quant Intensive BS.
   Meets with MATH 6880. Consult Math Department for specific offering. Possible topics include integral equations, calculus of variations, control theory, continuum mechanics, applied matrix theory, vector and tensor analysis, applications of probability and statistics. Will be offered occasionally on the basis of need or interest.

5760  Introduction to Mathematical Finance I (3) A grade of C or better in MATH 2280, 5010 or equivalent.
   Meets with MATH 6890. A basic introduction to the theory of financial derivative pricing. Topics include no arbitrage principle, risk-neutral measure, Black-Scholes theory, numerical model implementation and parameter calibration.

5765  Introduction to Mathematical Finance II (3) A grade of C or better in MATH 2280, 5010 or equivalent.
   Meets with MATH 6895. Topics include interest rate models, credit derivatives, and Monte Carlo simulations.

5900  Topics in Mathematics (1 to 4) Prerequisite: Instructor's consent.
   Various topics in mathematics to be treated at the appropriate level.

5910  Supervised Reading (1 to 6) Fulfills Quantitative Reasoning (Math & Stat/Logic).

5950  Senior Seminar in Mathematics (3) Prerequisite: Instructor's consent.
   A seminar on advanced topics in mathematics, centering around senior theses and projects.

5960  Undergraduate Special Projects (4) Fulfills Quantitative Reasoning (Math & Stat/Logic).
   Special computer project to serve as a senior thesis for students in scientific-computing emphasis.

5969  Special Topics in Statistics (1 to 6) Cross listed as OIS 5969, ED PS 5969, FP MD 5969, ECON 5969, FCS 5969, PSY 5969, SOC 5969, STAT 5969.
   Topics vary. Taught by members of the University Statistics Committee. Check current class schedule for cross-listings.

6010  Linear Models (3) Prerequisite: A grade of C or better in MATH 5010 and 5080 and 5090 and 2270.
   Univariate linear models with applications to regression and ANOVA.

6020  Multilinear Models (3) Prerequisite: A grade of C or better in MATH 6010.
   Multivariate linear models with applications to regression and ANOVA.

6040  Mathematical Probability (3) Prerequisite: A grade of C or better in MATH 6210.
   Analytical approach to probability theory, random variables and their distributions, limit theorems for sums of independent random variables.

6070  Mathematical Statistics (3) Prerequisite: A grade of C or better in MATH 2270 and 5080.
   Topics from distribution theory, estimation, and hypothesis testing.

6130  Introduction to Algebraic Geometry I (3) Prerequisite: A grade of C or better in MATH 6310 and 6320.
   Affine and projective varieties, tangent spaces and singularities, curve theory.

6140  Introduction to Algebraic Geometry II (3) Prerequisite: A grade of C or better in MATH 6130.
   Surfaces, intersection theory, special varieties, introduction to schemes.

6150  Complex Manifolds (3) Prerequisite: A grade of C or better in MATH 6220.
   Material will be selected from Riemann surfaces and algebraic curves, Kaehler geometry, Stein manifold theory, compact surfaces, etc.

6170  Introduction to Riemannian Geometry (3) Prerequisite: A grade of C or better in MATH 6520.
   Riemannian metrics, connections, geodesics, normal coordinates, completeness, spaces of constant curvature, submanifolds, Bonnet's and Meyer's theorem, Cartan-Hadamard theorem, Alexandrov and Topogonov comparison theorems, closed geodesics, cut locus, sphere theorem.

6210  Real Analysis (3) Prerequisite: A grade of C or better in MATH 5210.
   Measures and integrals, Lp-spaces, Hilbert spaces, Banach spaces, Fourier series.

6220  Complex Analysis (3) Prerequisite: A grade of C or better in MATH 4200 and 6210.
   Analytic functions, complex integration, conformal mapping, families of analytic functions, zeros of analytic functions, analytic continuation.

6240  Lie Groups/Lie Algebras I (3) Prerequisite: A grade of C or better in MATH 6220.
   Basic theory of Lie groups and Lie algebras.

6250  Lie Groups/Lie Algebras II (3) Prerequisite: A grade of C or better in MATH 6240.
   Structure theory, classification, and finite dimensional representations of Lie groups. Compact Lie groups.

6310  Modern Algebra I (3) Prerequisite: A grade of C or better in MATH 5320.
   Groups, rings, modules, homological algebra, fields, and Galois theory.

6320  Modern Algebra II (3) Prerequisite: A grade of C or better in MATH 6310.
   Second half of the course described under the listing for MATH 6310.

6350  Commutative Algebra (3) Prerequisite: A grade of C or better in MATH 6320.
   Various topics in commutative algebra. The course will be offered on the basis of need or interest. May be repeated for credit when the topics vary.

6410  Ordinary Differential Equations (3) Prerequisite: A grade of C or better in MATH 5210.
   Existence, uniqueness theory; stability theory; invariant sets and manifolds; periodic and quasiperiodic motions; boundary value problems; ODEs in Banach spaces; applications.

6420  Partial Differential Equations (3) Prerequisite: A grade of C or better in MATH 5210.
   First-order equations: characteristics, transport equations, shocks, Hamilton-Jacobi theory. Boundary value problems for the Laplace equation: maximum principles, Green's functions, Hilbert space methods. Cauchy and initial-boundary value problems for the heat equation and wave equation: existence and basic properties.

6430  Advanced Partial Differential Equations (3) Prerequisite: A grade of C or better in MATH 6420.
   Advanced topics from among the following: systems of conservations laws, nonlinear elliptic and parabolic equations, viscosity solutions of evolution problems, free boundary problems, applications.

6440  Advanced Dynamical Systems (3)
   Meets with MATH 5470. Basic abstract dynamics; stable, unstable, center manifold theory; index theories; KAM theory; chaos; dimensions of attractors; forced oscillations; applications.

6510  Differentiable Manifolds (3) Prerequisite: A grade of C or better in MATH 4510 and 5520.
   Manifolds, tangent spaces, orientation, Whitney's embedding theorem, transversality, Sard's theorem, partitions of unity, tubular neighborhoods, fiber bundles, degree theory, vector fields, flows, Lie derivatives, Frobenius' integrability theorem, differential forms, DeRham cohomology.

6520  Introduction to Algebraic Topology (3) Prerequisite: A grade of C or better in MATH 5520 and 6510.
   Simplicial and cell complexes, homology and cohomology with coefficients, excision, Mayer-Vietoris sequence, cup and cap products, DeRham theorem, Euler characteristic, Poincare-Hopf theorem, higher homotopy groups, long exact sequence of a fiber bundle, elementary homotopy theory.

6550  Algebraic Topology (3) Prerequisite: A grade of C or better in MATH 6510 and 6520.
   Topics depend on the instructor. Possibilities include: Morse theory, Lefschetz fixed-point theorem, simple-homotopy theory, obstruction theory, vector bundles, characteristic classes, spectral sequences, duality theorems, rational homotopy theory, topological K-theory.

6570  Geometric Topology (3) Prerequisite: A grade of C or better in MATH 6510 and 6520.
   Topics depend on the instructor. Possibilities include: low-dimensional topology (geometric structures on surfaces, Nielsen-Thurston theory of surface homeomorphisms, topology and geometry of 3-manifolds, topology of 4-manifolds), surgery and the classification of high-dimensional manifolds.

6610  Analysis of Numerical Methods I (3) Prerequisite: A grade of C or better in MATH 5620.
   Mathematical analysis of numerical methods in linear algebra, interpolation, integration, differentiation, approximation (including least squares, Fourier analysis, and wavelets), initial- and boundary-value problems of ordinary and partial differential equations.

6620  Analysis of Numerical Methods II (3) Prerequisite: A grade of C or better in MATH 6610.
   Second half of the course described under the listing for MATH 6210.

6630  Numerical Solutions of Partial Differential Equations (3) Prerequisite: A grade of C or better in MATH 6610 and 6620 and 6420.
   Analysis and implementation of numerical methods for solving partial differential equations. Issues of stability and accuracy. Linear and nonlinear problems.

6710  Applied Linear Operator and Spectral Methods (3) Prerequisite: A grade of C or better in MATH 5210 and 5410.
   The theory of linear operators applied to matrix, differential and integral equations, the Fredholm alternative, spectral theory, inverse and pseudo-inverse operators, Hilbert-Schmidt theory and eigenfunction expansions.

6720  Applied Complex Variables and Asymptotic Methods (3) Prerequisite: A grade of C or better in MATH 5210 and 5410.
   Cauchy-Riemann equations, Cauchy integral formulas, Taylor and Laurent series, multivalued functions, branch points and cuts, analytic continuation, Jordan's lemma, evaluation of real integrals; potential theory, stream functions, conformal mapping, special functions, Fourier, Laplace, Hilbert, and Z transforms, scattering theory, asymptopic analysis of integrals, Laplace's method, Watson's lemma, method of steepest descents.

6730  Asymptotic and Perturbation Methods (3) Prerequisite: A grade of C or better in MATH 6720.
   Asymptotic analysis, initial-value problems, multiscale analysis and the averaging theorem, homogenization theory, boundary- and transition-layer problems, matched asymptotic expansions, relaxation oscillations and propagating transition layers. Applications to problems from the physical and life sciences.

6740  Bifurcation Theory (3) Prerequisite: A grade of C or better in MATH 6210 and 6220.
   Degree theories; method of Liapunov and Schmidt; local and global bifurcation theory; Hopf bifurcation; Liusternik-Shnirelman theory; applications.

6750  Fluid Dynamics (3) Prerequisite: A grade of C or better in MATH 5440 or 6420.
   Derivation of equations of fluid dynamics, Euler and Navier-Stokes equations, Bernoulli's theorem, Kelvin's circulation theorem, potential flow, exact solutions, hydrodynamic paradoxes, vorticity, compressibility, thermal convection, waves in fluids, fluid instabilities, turbulence.

6760  Continuum Mechanics: Solids (3)
   Linear and nonlinear elasticity theory, transport phenomena, electromagnetic and elastic wave propogation and variational principles. Additional possible topics include piezoelectricity, thermoelectricity, viscoelasticity, magnetic materials, the Hall effect, quasiconvexity and phase transitions, shape memory and composite materials.

6770  Mathematical Biology I (3)
   Topics will alternate between (a) ecology and population biology and (b) physiology and cell biology.

6780  Mathematical Biology II (3)
   Second half of the course described under the listing for MATH 6770, of which it is the continuation.

6790  Case Studies in Computational Engineering and Science (3) Prerequisite: A grade of C or better in MATH 5740.
   Two to five faculty members from various disciplines will describe in detail a project in which they are engaged that involves all ingredients of computational engineering and science: a scientific or engineering problem, a mathematical problem leading to mathematical questions, and the solution and interpretation of these questions obtained by the use of modern computing techniques. Participating faculty will vary from year to year. To be offered on the basis of need.

6795  Seminar in Computational Engineering and Science (1 to 5) Prerequisite: A grade of C or better in MATH 6790.
   Students in the final semester of the Computational Engineering and Science Program will present their own CES-related research. To be offered on the basis of need.

6805  Introduction to Probability (3) Prerequisite: A grade of C or better in MATH 2210.
   Meets with MATH 5010. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6810  Stochastic Processes and Simulation I (3) Prerequisite: A grade of C or better in MATH 5010.
   Meets with MATH 5040. For Ph.d. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6815  Stochastic Processes and Simulation II (3) Prerequisite: A grade of C or better in MATH 5010, MATH 5040.
   Meets with MATH 5050. For Ph.d. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6820  Time Series Analysis (3) Prerequisite: A grade of C or better in MATH 5010.
   Meets with MATH 5075. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6830  Mathematical Biology I (3) Prerequisite: A grade of C or better in MATH 2280 or MATH 3150 or consent of instructor.
   Meets with MATH 5110. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6835  Mathematical Biology II (3) Prerequisite: A grade of C or better in MATH 5110.
   Meets with MATH 5120. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6840  Introduction to Ordinary Differential (4) Prerequisite: A grade of C or better in MATH 3220.
   Meets with MATH 5410. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6845  Ordinary Differential Equations and Dynamical Systems (3) Prerequisite: A grade of C or better in MATH 5410 or consent of instructor.
   Meets with MATH 5420. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6850  Introduction to Partial Differential Equations (3) Prerequisite: A grade of C or better in MATH 3220.
   Meets with MATH 5440. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6855  Survey of Numerical Analysis (4) Prerequisite: A grade of C or better in MATH 2210, 2250, 2280.
   Meets with MATH 5600. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6860  Introduction to Numerical Analysis I (4) Prerequisite: A grade of C or better in MATH 2210, either MATH 2250 or 2270, and MATH 2280 and computing.
   Meets with MATH 5610. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6865  Introduction to Numerical Analysis II (4) Prerequisite: A grade of C or better in MATH 5610.
   Meets with MATH 5620. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6870  Mathematical Model (3) Prerequisite: A grade of C or better in MATH 5600 or CP SC 5220.
   Meets with MATH 5740. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6875  Methods of Optimization (3) Prerequisite: A grade of C or better in MATH 3150.
   Meets with MATH 5450. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6880  Topics in Applied Mathematics (3)
   Meets with MATH 5750. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6890  Introduction to Mathematical Finance I (3) A grade of C or better in MATH 2280, 5010 or equivalent.
   Meets with MATH 5760. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6895  Introduction to Mathematical Finance II (3) A grade of C or better in MATH 2280, 5010 or equivalent.
   Meets with MATH 5765. For Ph.D. students only. Extra work is required; this should be arranged with the instructor before the end of the second week of classes.

6910  Supervised Reading (1 to 6)

6960  Special Projects (1 to 6)

6970  Thesis Research: Master's (1 to 9)

6980  Faculty Consultation (3)
   Usually only one sequence from each decade (e.g., 711 to 719) is offered in any year.

7210  Representations of Lie Groups I (3) Prerequisite: A grade of C or better in MATH 6210 and 6220.
   Basic theory of unitary representations of Lie groups.

7220  Representations of Lie Groups II (3) Prerequisite: A grade of C or better in MATH 7210.
   Infinite dimensional representations of semi-simple Lie groups. Theory of Harish-Chandra modules.

7240  Several Complex Variables I (3) Prerequisite: A grade of C or better in MATH 6220.
   Local theory of functions of several complex variables.

7250  Several Complex Variables II (3) Prerequisite: A grade of C or better in MATH 7240.
   Global theory of functions of several complex variables.

7280  Operator Theory (3) Prerequisite: A grade of C or better in MATH 6210 and 6220.
   A study of linear operators, primarily in Hilbert spaces.

7710  Optimization (3)
   Discusses modern problems in calculus of variations and optimal control, especially in the structural optimizations, as well as the foundations of these disciplines. Offered on the basis of need or interest.

7730  Nonlinear Oscillations (3)
   Offered on the basis of need or interest.

7740  Nonlinear Waves (3)
   Offered on the basis of need or interest.

7760  Mathematics of Materials (3)
   Offered on the basis of need or interest.

7800  Topics in Algebraic Geometry (3)
   Various topics in the area of algebraic geometry, offered on the basis of need or interest.

7805  Seminar in Algebraic Geometry (1 to 3)

7810  Topics in Riemannian Geometry (1 to 3)
   Various topics in the area of Riemannian geometry, offered on the basis of need or interest.

7815  Seminar in Differential Geometry (1 to 3)

7825  Seminar in Analysis (1 to 3)

7830  Topics in Commutative Algebra (3)
   Various topics in the area of commutative algebra, offered on the basis of need or interest.

7835  Seminar in Algebra (1 to 3)

7840  Topics in Differential Equations (3)
   Various topics in the area of differential equations, offered on the basis of need or interest.

7845  Seminar in Differential Equations (1 to 3)

7850  Topics in Algebraic Topology (3)
   Various topics in algebraic topology, offered on the basis of need or interest.

7853  Topics in Geometric Topology (3)
   Various topics in the area of geometric topology, offered on the basis of need or interest.

7855  Seminar in Topology (1 to 3)

7860  Topics in Numerical Analysis (3)
   Various topics in the area of numerical analysis, offered on the basis of need or interest.

7870  Topics in Applied Mathematics (1)
   Various topics in applied mathematics, offered on the basis of need or interest.

7875  Seminar in Applied Mathematics (1 to 3)

7880  Topics in Probability (3)
   Various topics in the area of probability, offered on the basis of need or interest.

7885  Seminar in Probability and Statistics (1 to 3)

7890  Topics in Representation Theory (3)
   Various topics in representation theory, to be offered on the basis of need or interest. May be repeated for credit when the topics vary.

7895  Seminar in Representation Theory (1 to 3)
   No description.

7970  Thesis Research: Ph.D. (1 to 9)

7980  Faculty Consultation (3)

7990  Continuing Registration: Ph.D. (0)