# Upper Division MATH Courses

All prerequisite courses must be passed with a grade of C- or better.

For official course descriptions, please see the current CU-Boulder Catalog.

**MATH 3001 Analysis 1**

Provides a rigorous treatment of the basic results from elementary Calculus. Topics include the topology of the real line, sequences of numbers, continuous functions, differentiable functions, and the Riemann integral.

Note: Students may not receive credit for both MATH 3001 and APPM 4440. However APPM 440 does NOT fulfill Math major/minor requirements.

Prerequisites: MATH 2001/2002 and Linear Algebra.

Generally offered: each fall and spring; sometimes offered in the summer.

**MATH 3110 Introduction to Theory of Numbers**

(a.k.a. Number Theory)

Studies the set of integers, focusing on divisibility, congruencies, arithmetic functions, sums of squares, quadratic residues and reciprocity, and elementary results on distributions of primes.

Prerequisite: MATH 2001/2002.

Generally offered: each spring.

**MATH 3120 Functions and Modeling**

Engages the students in daily projects and occasional in-class labs designed to strengthen and expand knowledge of the topics in secondary mathematics, focusing especially on topics from algebra, precalculus, and calculus. The projects and labs involve the use of multiple representations, transformations, data analysis techniques, and interconnections among ideas from geometry, algebra, probability, and calculus.

Prerequisites: Calculus I and MATH 2001/2002.

Generally offered: each fall.

**MATH 3140 Abstract Algebra 1**

(a.k.a. Abstract Algebra, Modern Algebra, Algebra I)

Studies basic properties of algebraic structures with a heavy emphasis on groups. Other topics, time permitting, may include rings and fields.

Prerequisites: MATH 2001/2002 and Linear Algebra.

Generally offered every fall and spring.

**MATH 3170 Combinatorics 1**

(a.k.a. Combinatorics)

Covers basic methods and results in combinatorial theory. Includes enumeration methods, elementary properties of functions and relations, and graph theory.

Prerequisite: MATH 2001/2002.

Generally offered every fall.

**MATH 3210 Euclidean and Non-Euclidean Geometry**

(a.k.a. Geometry, Geometry I)

Axiomatic systems; Euclid's presentation of the elements of geometry; Hilbert's axioms; neutral, Euclidean and non-Euclidean geometries and their models.

Prerequisites: MATH 2001/2002 and Linear Algebra.

Generally offered every spring.

**MATH 3430 Ordinary Differential Equations (numbered 4430 through Summer 2014)**

(a.k.a. ODEs, Diff. Eq., "Diffy Q")

Involves an elementary systematic introduction to first-order scalar differential equations, nth order linear differential equations, and n-dimensional linear systems of first-order differential equations. Additional topics are chosen from equations with regular singular points, Laplace transforms, phase plane techniques, basic existence and uniqueness, and numerical solutions.

Prerequisites: Calculus 3 and Linear Algebra.

Generally offered: every fall, spring, and summer.

**MATH 3450 Introduction to Complex Variables**

(a.k.a. Complex Analysis)

Theory of functions of one complex variable, including integrals, power series, residues, conformal mapping, and special functions.

Prerequisite: Calculus 3.

Generally offered: every spring.

**MATH 3510 Introduction to Probability and Statistics**

(a.k.a. Prob & Stats)

Introduces the basic notions of Probability: random variables, expectation, conditioning, and the standard distributions (Binomial, Poisson, Exponential, Normal). This course also covers the Law of Large Numbers and Central Limit Theorem as they apply to statistical questions: sampling from a random distribution, estimation, and hypothesis testing.

Notes: Students may not receive credit for this course and MATH 4510. This course is designed for Mathematics majors following the Secondary Education track.

Prerequisites: Calculus 2 and MATH 2001/2002.

Generally offered: every fall, spring, and summer.

**MATH 4000 Foundations of Mathematics**

(a.k.a. Foundations)

Focuses on a complete deductive framework for mathematics and and applies it to various areas. Presents Gödel's famous incompleteness theorem about the inherent limitations of mathematical systems.

Prerequisites: MATH 2001/2002 and at least one of the following: Linear Algebra, MATH 3001, 3140, or 3210

Generally offered: fall of even numbered years.

**MATH 4001 Analysis 2**

Provides a rigorous treatment of infinite series, sequences of functions, and an additional topic chosen by the instructor (for example, multivariable analysis, the Lebesgue integral, or Fourier analysis).

Prerequisites: MATH 3001 and Linear Algebra.

Generally offered: every fall.

**MATH 4120 Introduction to Operations Research**

(a.k.a. Operations Research)

Studies linear and nonlinear programming, the simplex method, duality, sensitivity, transportation; as time permits, may also include network flow problems, constrained and unconstrained optimization theory. In general, an optimization course.

Note: this course is cross-listed as APPM 4120.

Prerequisite: Linear Algebra.

Generally offered: every spring.

**MATH 4140 Abstract Algebra 2**

(a.k.a Algebra II)

Explores some topic that builds on material in 3140. Possible topics include (but are not limited to) Galois theory, representation theory, advanced linear algebra, or commutative algebra.

Prerequisite: MATH 3140.

Generally offered every spring.

**MATH 4200 Introduction to Topology**

(a.k.a. Topology)

Introduces the basic concepts of point set topology. Includes topological spaces, metric spaces, homeomorphisms, connectedness, and compactness.

Prerequisite: MATH 3001.

Generally offered: fall of odd numbered years.

**MATH 4230 Geometry of Curves and Surfaces**

(a.k.a. Differential Geometry)

Introduces the modern differential geometry of plane curves, space curves, and surfaces in 3-dimensional space. Topics include the Frenet frame, curvature and torsion for space curves; Gauss and mean curvature for surfaces; Gauss and Codazzi equations, and the Gauss-Bonnet theorem.

Prerequisites: Calculus 3, Linear Algebra, and MATH 3001

Generally offered: fall of even numbered years.

**MATH 4330 Fourier Analysis**

The notion of Fourier analysis, via series and integrals, of periodic and nonperiodic phenomena is central to many areas of mathematics. Develops the Fourier theory in depth, and considers such special topics and applications as wavelets, Fast Fourier Transforms, seismology, digital signal processing, differential equations, and Fourier optics.

Prerequisite: MATH 3001

Generally offered: every spring.

**MATH 4440 Mathematics of Coding and Cryptography**

(a.k.a. Coding & Crypt)

Gives an introduction, with proofs, to the algebra and number theory used in coding and cryptography. Basic problems of coding and cryptography are discussed.

Prerequisite: Linear Algebra. MATH 3110 and 3140 are recommended.

Generally offered: every fall.

**MATH 4470 Partial Differential Equations**

(a.k.a. PDEs)

Studies initial and boundary value problems for the wave, heat, and Laplace equations. Also highlights separation of variables method, eigenvalue problems, Fourier series, and orthogonal systems.

Prerequisite: MATH 3430

Generally offered: every fall and spring.

**MATH 4510 Introduction to Probability Theory**

(a.k.a. Probability)

Studies axioms, combinatorial analysis, independence and conditional probability, discrete and absolutely continuous distributions, expectation and distribution of functions of random variables, laws of large numbers, central limit theorems, and simple Markov chains if time permits.

Note: Students may not receive credit for this course and MATH 3510, APPM 3570, or ECEN 3810. However, APPM 3570 and ECEN 3810 do NOT fulfill Math major/minor requirements.

Prerequisites: Calculus 3 and Linear Algebra.

Generally offered: every fall and spring, and summer.

**MATH 4520 Introduction to Mathematical Statistics**

(a.k.a. Statistics, Math Stats)

Topics include point and confidence interval estimation. Examines principles of maximum likelihood, sufficiency, and completeness, as well as tests of simple and composite hypotheses, linear models, and multiple regression analysis if time permits. Analyzes various distribution-free methods.

Note: This course is cross-listed as APPM 4520.

Prerequisite: MATH 4510.

Generally offered: every fall and spring; sometimes offered in the summer.

**MATH 4540 Introduction to Time Series**

(a.k.a. Time Series)

Studies basic properties, trend-based models, seasonal models, modeling and forecasting with ARIMA models, spectral analysis, and frequency filtration.

Note: This course is cross-listed as APPM 4540.

Prerequisites: MATH 4510 and MATH 4520.

Generally offered: every spring.

**MATH 4650 Intermediate Numerical Analysis 1**

(a.k.a. Numerical Analysis)

Focuses on numerical solution of nonlinear equations, interpolation, methods in numerical integration, numerical solution of linear systems, and matrix eigenvalue problems. Stresses significant computer applications and software.

Note: This course is cross-listed as APPM 4650. Although the title includes the word "Intermediate", MATH 4650 is the introductory course in numerical analysis.

Prerequisites: Linear Algebra, MATH 3430 and knowledge of a programming language

Generally offered: every fall, spring, and summer.

**MATH 4660 Intermediate Numerical Analysis 2**

(a.k.a. Numerical II)

Continuation of MATH 4650. Examines numerical solution of initial-value problems and two-point boundary-value problems for ordinary differential equations. Also looks at numerical methods for solving partial differential equations

Note: This course is cross-listed as APPM 4660.

Prerequisite: MATH 4650.

Generally offered: every spring.

**MATH 4730 Set Theory**

Studies in detail the theory of cardinal and ordinal numbers, definition by recursion, the statement of the continuum hypothesis, simple cardinal arithmetic, and other topics chosen by the instructor.

Prerequisites: MATH 2001/2002 plus one upper division MATH course.

Generally offered: fall of odd numbered years.

**MATH 4820 History of Mathematical Ideas**

(a.k.a. History of Math)

Examines the evolution of a few mathematical concepts (e.g., number, geometric continuum, or proof), with an emphasis on the controversies surrounding these concepts. Begins with Ancient Greek mathematics and traces the development of mathematical concepts through the middle ages into the present.

Prerequisites: MATH 2001/2002 plus one upper division MATH course; prior completion of upper-division Written Communication is strongly recommended.

Generally offered: every fall and spring.