﻿ Undergraduate Upper-Division Courses - Department of Mathematics | University of Colorado Colorado Springs

# Undergraduate Upper-Division Courses

#### MATH 3010 - Mathematics for Elementary Teachers I

Covers the whole number, integer, and rational number systems that are of prime importance to the elementary teacher. For students planning on elementary teacher certification. Approved for Compass Curriculum requirement: Explore-Physical and Natural World. This course, when taken with MATH 3020, is one of the means to satisfy the LAS and Compass Curriculum Quantitative and Qualitative Reasoning requirement.

• 3 Credits

#### MATH 3020 - Mathematics for Elementary Teachers II

Intuitive and logical development of the fundamental ideas of geometry such as parallelism, congruence, and measurement. Includes study of plane analytical geometry. For students planning on elementary teacher certification. Approved for Compass Curriculum requirement: Explore-Physical and Natural World. This course, when taken with MATH 3010, is one of the means to satisfy the LAS and Compass Curriculum Quantitative and Qualitative Reasoning requirement.

• 3 Credits

#### MATH 3100 - Statistics for the Sciences

Descriptive probability, hypothesis testing, nonparametric methods. Discrete and continuous random variables, mean and variance, confidence limits, correlation and regression. Prer., MATH 1350.

• 3 Credits

#### MATH 3110 - Theory of Numbers

A careful study, with emphasis on proofs, of the following topics associated with the set of integers: divisibility, congruences and modular arithmetic, arithmetic functions, sums of squares, and elementary results on distributions of primes. The history of various developments in the subject, along with biographies of important contributors, will be included. Prer., MATH 1360 and MATH 2150.

• 3 Credits

#### MATH 3130 - Introduction to Linear Algebra

Systems of linear equations, matrices, vector spaces, linear independence, basis, dimension, determinants, linear transformations and matrices, eigenvalues and eigenvectors. Prer., MATH 2350 with a grade of “C” or better.

• 3 Credits

#### MATH 3400 - Introduction to Differential Equations

First order differential equations, linear differential equations, the Laplace transform method, power series solutions, numerical solutions, linear systems. Prer., MATH 2350.

• 3 Credits

#### MATH 3410 - Introduction to Analysis

An introduction to proofs in analysis. Topics include completeness of the real numbers, sequences and limits, infinite series, and continuous functions. Prer., MATH 2150 and MATH 2350.

• 3 Credits

#### MATH 3480 - Functions and Modeling

Data collection and exploration of a variety of situations that can be modeled using linear, exponential, polynomial, and trigonometric functions. Use of technology in teaching, connections between various areas of mathematics, non-routine problem solving, problem-based learning, and applications for mathematics. Meets with UTLS 2040. Prer., MATH 2350.

• 3 Credits

#### MATH 3500 - Graph Theory

Standard material on the theory of both directed and undirected graphs, including the concepts of isomorphism, connectivity, trees, traversability, planar graphs, coloring problems, relations and matrices. Prer., MATH 2150.

• 3 Credits

#### MATH 3510 - Topics in Combinatorial Analysis

A survey of important areas of combinatorics. Topics may include enumeration techniques, recurrence relations, combinatorial designs, graph theory, machining and optimization. Prer., MATH 2150.

• 3 Credits

#### MATH 3650 - Advanced Computational Math

Advanced computational techniques with applications in mathematics, science and engineering. Topics include numerical linear algebra, dynamical systems and stability, calculus in the complex plane and elements of Fourier analysis, the DFT and FFT method, Monte Carlo Simulations, other applications in science and engineering. Prer., MATH 2650, MATH 3130, MATH 3400.

• 2 Credits

#### MATH 3670 - Introduction to Scientific Computation

This is the 3-credit-hour alternative to MATH 2650 (1 credit) and MATH 3650 (2 credits). Introduction to computational math (see course description for MATH 2650) and advanced computational techniques (see course description for MATH 3650). Prer., MATH 3130, MATH 3400.

• 3 Credits

#### MATH 3810 - Introduction to Probability and Statistics

The axioms of probability and conditional probability will be studied as well as the development, applications and simulation of discrete and continuous probability distributions. Also, expectation, variance, correlation, sum and joint distributions of random variables will be studied. The Law of Large Numbers and the Central Limit Theorem will be developed. Applications to statistics will include regression, confidence intervals, and hypothesis testing. Prer., MATH 2350.

• 3 Credits

#### MATH 4040 - Senior Math Seminar

This is a capstone experience for the students in the mathematics program. Students will give oral presentations on mathematical topics, and will actively participate in peer presentations. Approved for LAS Oral Communication area requirement. Approved for Compass Curriculum requirement: Summit. Prer., Senior standing.

• 1 Credits

#### MATH 4050 - Topics in Mathematics Secondary Classroom

The topics covered will vary from one offering to the next. Topics will be chosen to meet the needs of secondary mathematics teachers for additional training to teach to the Colorado Model Content Standards. Prer., One semester of calculus, or instructor approval. Meets with MATH 5050.

• 1 Credits (Minimum); 3 Credits (Maximum)

#### MATH 4100 - Technology in Mathematics Teaching and Curriculum

Methodology for using technology as a teaching/learning tool for high school and college math courses. Use of graphing calculators, computer algebra systems, computer geometry systems and the internet will be emphasized. Students are required to develop and present a portfolio of in-depth projects. Prer., MATH 1360. Meets with MATH 5100.

• 3 Credits

#### MATH 4130 - Linear Algebra I

Vector spaces, linear transformations and matrices, determinants, eigenvalues, similarity transformations, orthogonal and unitary transformations, normal matrices and quadratic forms. Prer., MATH 3130. Meets with MATH 5130.

• 3 Credits

#### MATH 4140 - Modern Algebra I

A careful study of the elementary theory of groups, rings, and fields. Mappings such as homomorphisms and isomorphisms are considered. The student will be expected to prove theorems. Prer., MATH 2150 and MATH 3130. One of MATH 3110, MATH 3500, or MATH 3510 (preferably MATH 3110) is strongly recommended.

• 3 Credits

#### MATH 4150 - Modern Algebra II

Continuation of MATH 4140. The relationship between groups and fields is explored via a thorough investigation of Galois theory. Prer., MATH 4140. Meets with MATH 4150.

• 3 Credits

#### MATH 4210 - Differential Geometry

Presents topics in geometry suitable for advanced undergraduates, such as differential geometry of curves and surfaces, geometry of manifolds, or more in-depth exploration of non-Euclidean geometries. Prer., MATH 3130, MATH 3410. Meets with MATH 5210.

• 3 Credits

#### MATH 4230 - Fractal Geometry

Introduction to iterated function systems and mathematical aspects of fractal sets. Includes metric spaces and the space fractals live in, transformations, contraction mapping and Collage Theorem, chaotic dynamics, shadowing theorem, fractal dimension, fractal interpolation, and measures on fractals. Prer., MATH 2350 and MATH 3130. Meets with MATH 5230.

• 3 Credits

#### MATH 4250 - Introduction to Chaotic Dynamical Systems

Introduction to dynamical systems or processes in motion, that are defined in discrete time by iteration of simple functions, or in continuous time by differential equations. Emphasis on understanding chaotic behavior that occurs when a simple non-linear function is iterated. Topics include orbits, graphical analysis, fixed and periodic points, bifurcations, symbolic dynamics, chaos, fractals, and Julia sets. Prer., MATH 2350. Meets with MATH 5250.

• 3 Credits

#### MATH 4310 - Modern Analysis I

Rigorous treatment of calculus. Topics include differentiation, integration, Taylor’s Theorem, uniform convergence of sequences of functions, and power series. Prer., MATH 3410.

• 3 Credits

#### MATH 4320 - Modern Analysis II

Careful theoretical study of topology of Euclidean space, metric spaces, sequences and series of functions, calculus of several variables. Prer., MATH 4310. Meets with MATH 5320.

• 3 Credits

#### MATH 4420 - Optimization

Topics selected from linear and nonlinear programming, the simplex algorithm and other approaches to linear optimization, minimax theorems, convex functions, introduction to calculus of variations. Prer., MATH 3130 and MATH 3400. Meets with MATH 5420.

• 3 Credits

#### MATH 4430 - Ordinary Differential Equations

Existence and uniqueness theorem, linear equations, linear systems with periodic coefficients (Floquet theory), stability analysis of planar systems, series solutions at regular singular points, Strum-Lioville problems. Prer., MATH 3130 and MATH 3400. Meets with MATH 5430.

• 3 Credits

#### MATH 4450 - Complex Variables

Theory of functions of one complex variable including integrals, power series, residues, conformal mapping and special functions. Prer., MATH 2350. Meets with MATH 5450.

• 3 Credits

#### MATH 4470 - Methods of Applied Mathematics

Boundary value problems for the wave, heat, and Laplace equations, separation of variables methods, eigenvalue problems, Fourier series, orthogonal systems. Prer., MATH 2350, MATH 3130 and MATH 3400. Meets with MATH 5470.

• 3 Credits

#### MATH 4480 - Mathematical Modeling

The use of diverse mathematical techniques to analyze and solve problems from science and engineering, particular problems likely to arise in nonacademic settings such as industry or government. Converting a problem to a mathematical model. Commonly encountered classes of mathematical models, including optimization problems, dynamical systems, probability models and computer simulations. Communication of results of mathematical analysis. Prer., MATH 3130, MATH 3400, and MATH 3100 or MATH 3810 or ECE 3610. MATH 2650 or adequate experience with computer programming. Meets with MATH 5480.

• 3 Credits

#### MATH 4510-Topology

An introduction to Point-set topology and elements of geometric or algebraic topology. Prer., MATH 4310. Meets with MATH 5510

• 3 credits

#### MATH 4650 - Numerical Analysis

Error analysis, root finding, numerical integration and differentiation, numerical methods for ordinary differential equations, numerical linear algebra and eigenvalue problems. Prer., MATH 3130, MATH 3400, and CS 1120 or CS 1150 or MATH 3670.

• 3 Credits

#### MATH 4670 - Scientific Computation I

Numerical solutions of initial-value problems, two-point boundary-value problems for ordinary differential equations, and applications. Numerical methods for solving linear partial differential equations, including finite difference and finite element method. Laplace, heat, and wave equation. Prer., MATH 3670 or MATH 4650/5650, and MATH 4470/5470. Meets with MATH 5670.

• 3 Credits

#### MATH 4810 - Mathematical Statistics I

Exponential, Beta, Gamma, Student, Fisher and Chi-square distributions are covered in this course, along with joint and conditional distributions, moment generating techniques, transformations of random variables and vectors. Prer., MATH 2350 and MATH 3130. Meets with MATH 5810.

• 3 Credits

#### MATH 4820 - Mathematical Statistics II

Point and confidence interval estimation, principles of maximum likelihood, sufficiency and completeness; tests of simple and composite hypotheses. Linear models and multiple regression analysis. Other topics will be included. Prer., MATH 3810 or MATH 3100. Meets with MATH 5820.

• 3 Credits

#### MATH 4830 - Linear Statistical Models

Methods and results of linear algebra are developed to formulate and study a fundamental and widely applied area of statistics. Topics include generalized inverses, multivariate normal distribution and the general linear model. Applications focus on model building, design models and computing methods. The “Statistical Analysis System” (software) is introduced as a tool for doing computation. Prer., MATH 3810 or ECE 3610, or MATH 3100 and MATH 3130. Meets with MATH 5830.

• 3 Credits

#### MATH 4850 - Stochastic Modeling

Mathematical development of continuous and discrete time Markov chains, queuing theory, reliability theory, and Brownian motion with applications to engineering and computer science. Prer., MATH 3810 or ECE 3610. MATH 2650 or adequate experience with computer programming. Meets with MATH 5850.

• 3 Credits

#### MATH 4900 - Advanced Topics Seminar

Various advanced topics in mathematics. Prer., Vary depending on course content. Consent of instructor required. Meets with MATH 5900.

• 3 Credits