Undergraduate Course Descriptions

Also visit the graduate course descriptions page.

21-100–199   Freshman level
21-200–299   Sophomore level
21-300–399   Junior level
21-400–599   Junior/Senior level
21-600–900   Graduate level

21-101 Freshman Mathematics Seminar
Fall: 3 units

This course is offered in the second half of the Fall semester for first semester Freshmen interested in majoring in mathematics. Topics vary from year to year. Recent topics have been finite difference equations, convexity, and fractals. 3 hrs. lec.

21-105 Pre Calculus (For H&SS students only)
Fall: 9 units

Review of basic concepts, logarithms, functions and graphs, inequalities, polynomial functions, complex numbers, and trigonometric functions and identities. 3 hrs. lec., 1hr. rec.

21-106 Topics in Precalculus
Fall, first mini: 5 units

A review of precalculus and its use in solving problems similar to those to be encountered in calculus. 3 hrs. lec., 2 hrs. rec. Placement based upon Placement Test.

21-110 Problem Solving in Recreational Mathematics (For H&SS students only)
Offered intermittently: 9 units

The emphasis is on learning to solve problems in elementary mathematics. Topics may vary among offerings of the course, but typically include puzzles, algebraic problems, number theory, and graph theory. 3 hrs. lec.

21-111 Calculus for Humanities Students I (For H&SS students only)
Fall or Spring: 10 units

Review of basic algebra, functions, limits, derivatives of algebraic, exponential and logarithmic functions, curve sketching, applications with emphasis on economic models. 3 hrs. lec., 2 hrs. rec.

21-112 Calculus for Humanities Students II (For H&SS students only)
Fall or Spring: 10 units

Indefinite integral, definite integral and applications, techniques of integration, trigonometric functions, functions of several variables, partial derivatives, maximum-minimum problems, Lagrange multipliers, geometric series, Newton's method, applications. 3 hrs. lec., 2 hrs. rec.

21-115 Differential Calculus
Offered for AP credit only: 5 units

Functions, limits, derivatives, curve sketching, Mean Value Theorem, trigonometric functions, related rates, linear and quadratic approximations, maximum-minimum problems. 3 hrs. lec., 2 hrs. rec.

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21-120 Differential and Integral Calculus
Fall and Spring: 10 units

Functions, limits, derivatives, logarithmic, exponential, and trigonometric functions, inverse functions; L'Hospital's Rule, curve sketching, Mean Value Theorem, related rates, linear and quadratic approximations, maximum-minimum problems, inverse functions, definite and indefinite integrals, and hyperbolic functions; applications of integration, integration by substitution and by parts. 3 hrs. lec., 2 hrs. rec.

A lecture of 21-120 will be offered each Fall for first year students in Business Administration and Economics.

21-121 Integration and Differential Equations
Fall and Spring: 10 units

Differentiation review, maximum-minimum problems. Definite and indefinite integrals; hyperbolic functions; applications of integration, integration by substitution and by parts. Integration by trigonometric substitution and partial fractions; arclength; improper integrals; Simpson's and Trapezoidal Rules for numerical integration; separable differential equations, first order linear differential equations, homogeneous second order linear differential equations with constant coefficients. 3 hrs. lec., 2 hrs. rec., Prerequisite: 21-115 or equivalent.

21-122 Integration, Differential Equations and Approximation
Fall and Spring: 10 units

Integration by trigonometric substitution and partial fractions; arclength; improper integrals; Simpson's and Trapezoidal Rules for numerical integration; separable differential equations, first order linear differential equations, homogeneous second order linear differential equations with constant coefficients; series solution, Newton's method, Taylor's Theorem including a discussion of the remainder, sequences, series, power series;. 3 hrs. lec., 2 hrs. rec., Prerequisite: 21-120 or 21-111/112.

21-123 Calculus of Approximation
Fall and Spring, first minis: 5 units

Newton's method, Taylor's Theorem including a discussion of the remainder, sequences, series, power series. 3 hrs. lec., 2 hrs. rec., Prerequisite: 21-121.

21-124 Modeling with Differential Equations
Fall or Spring: 3 units

An introduction to mathematical modeling using Matlab. This course is designed to accompany 21-260 Differential Equations. 1 hr. lec.

21-125 Maple Lab
Fall and Spring: 3 units

An introduction to the symbolic programming package Maple using topics chosen from calculus and matrix algebra. Recommended to accompany any calculus course beyond 21-120 Differential and Integral Calculus. 1 hr. lec.

21-126 Introduction to Mathematical Software
Fall and Spring: 3 units

This course provides an introduction to the use of several software packages which are useful to mathematics students. Among the packages are Maple and Mathematica for symbolic computing, TeX and LaTeX for mathematical documents, and Matlab for numerical computing. The course will also introduce the mathematical facilities built into spreadsheets such as Excel. The aim of the course is to provide the student with some basic skills in the use of this software without attempting complete coverage. A deeper knowledge of the software will be easy to obtain after completing this course. There are no prerequisites for the course, other than basic computer literacy and a knowledge of elementary mathematics. It is suggested that the course should be taken during the first two years of undergraduate studies.

21-127 Concepts of Mathematics
Fall and Spring: 9 units

This course introduces the basic concepts, ideas and tools involved in doing mathematics. As such, its main focus is on presenting informal logic, and the methods of mathematical proof. These subjects are closely related to the application of mathematics in many areas, particularly computer science. Topics discussed include a basic introduction to elementary number theory, induction, the algebra of sets, relations, equivalence relations, congruences, partitions, and functions, including injections, surjections, and bijections. 3 hrs. lec., 1 hr. rec. No prerequisite. A prerequisite for 15-211.

21-131 Analysis I
Fall: 10 units

An enriched first course in calculus which includes a greater concentration on the foundations of the subject. Recommended for students with some prior background in calculus and who seek a deeper calculus course. Functions, limits, continuity; the Intermediate Value Theorem; the Riemann integral; the Fundamental Theorem of Calculus; integrability of continuous functions; the derivative and its significance; product rule, quotient rule, chain rule; Mean Value Theorem; inverse functions. 3 hrs. lec., 2 hrs. rec.

21-132 Analysis II
Spring: 10 units

A continuation of Analysis I. 'Hopital's rule; trigonometric, logarithmic, and exponential functions; techniques of integration; approximation by polynomials, Taylor's theorem; sequences, series, power series; introduction to linear differential equations. 3 hrs. lec., 2 hrs. rec. Prerequisite: 21-131, or consent of the instructor.

21-170 Introduction to Mathematical Finance
Spring: 9 units

This course introduces the mathematical ideas which underlie the recent rapid expansion of financial markets. Topics include: an introduction to financial instruments and markets, simple models for the random variation of prices in financial markets, the fundamental concept of arbitrage, and the use of arbitrage-free models for valuation of securities and for risk management. 3 hrs. lec.

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21-201 Undergraduate Colloquium
Fall and Spring: 1 unit

All mathematics majors meet for one hour each week to hear discussions on current research by faculty or students, presentations on mathematics from mathematicians outside academia, and expository talks on selected mathematical topics not part of the usual curricula. Also will include topics of special interest to undergraduates such as preparation for graduate school.

21-228 Discrete Mathematics
Fall and Spring: 9 units

The techniques of discrete mathematics arise in every application of mathematics which is not purely continuous, for example in computer science, economics, and general problems of optimization. This course introduces two of the fundamental areas of discrete mathematics: enumeration and graph theory. The introduction to enumeration includes permutations, combinations, and topics such as discrete probability, combinatorial distributions, recurrence relations, generating functions, Ramsey's Theorem, and the principle of inclusion and exclusion. The introduction to graph theory includes topics such as paths, walks, connectivity, Eulerian and Hamilton cycles, planar graphs, Euler's Theorem, graph coloring, matchings, networks, and trees. 3 hrs. lec., 1 hr. rec. Prerequisite: 21-127.

21-229 Set Theory
Spring: 9 units

Set theory was invented about 110 years ago by George Cantor as an instrument to understand infinite objects and to compare different sizes of infinite sets. Since then set theory has come to play an important role in several branches of modern mathematics, and serves as a foundation of mathematics. Contents: Basic properties of natural numbers, countable and uncountable sets, construction of the real numbers, some basic facts about the topology of the real line, cardinal numbers and cardinal arithmetic, the continuum hypothesis, well ordered sets, ordinal numbers and transfinite induction, the axiom of choice, Zorn's lemma. Optional topics if time permits: Infinitary combinatorics, filters and large cardinals, Borel and analytic sets of reals. Prerequisite: 21-127.

21-235/6 Mathematical Studies I & II
Fall and Spring: 20 units each

A unified and intensive presentation of algebra, analysis, and geometry by a team of instructors. For capable and dedicated students who can allot a substantial amount of time to the study of mathematics. Aims at providing a modern background in mathematics for a career in pure or applied mathematics, science, or engineering. Topics covered include analysis in IR, analysis in Euclidean spaces, linear algebra, geometry, algebraic theories, multi-linear algebra, ordinary differential equations. Covers and extends the material taken up in the courses Calculus in Three Dimensions, Advanced Calculus I, Algebraic Structures, Linear Algebra I and II. Normally taken by students in the third and fourth semesters. On completion a number of options are open, among them the Honors Degree Program. Participation by invitation. Interested Freshmen should contact the Department in March. Prerequisites: 21-131 and 21-132 or the equivalent. 6 hrs. lec., 2 hrs. rec.

21-241 Matrix Algebra
Fall or Spring: 9 units

Vectors and matrices, the solution of linear systems of equations, vector spaces and subspaces, orthogonality, determinants, real and complex eigenvalues and eigenvectors, linear transformations. 3 hrs. lec. Prerequisite: 21-127 recommended.

21-256 Multivariate Analysis and Approximations
Fall or Spring: 9 units

Taylor's Theorem; geometric sequences and series and their applications in finance; vectors and matrices, lines, and planes; partial derivatives, directional derivatives, gradient, chain rule, maximum-minimum problems, Lagrange multipliers and the Kuhn-Tucker Theorem. 3 hrs. lec., 2 hrs. rec. Prerequisite: 21-120.

21-257 Models and Methods for Optimization
Fall or Spring: 9 units

Introduces basic methods of operations research and is intended primarily for Business Administration and Economics majors. Review of linear systems; linear programming, including the simplex algorithm, duality, and sensitivity analysis; the transportation problem; other structured optimization problems. 3 hrs. lec., 1 hr. rec. Prerequisite: 21-256.

21-259 Calculus in Three Dimensions
Fall or Spring: 9 units

Vectors, lines, planes, quadratic surfaces, polar, cylindrical and spherical coordinates, partial derivatives, directional derivatives, gradient, divergence, curl, chain rule, maximum-minimum problems, multiple integrals, parametric surfaces and curves, line integrals, surface integrals, Green-Gauss theorems. 3 hrs. lec. 2 hrs. rec. Prerequisite: 21-120

21-260 Differential Equations
Fall and Spring: 9 units

Ordinary differential equations: first and second order equations, applications, Laplace transforms; partial differential equations: partial derivatives, separation of variables, Fourier series; systems of ordinary differential equations; applications. 3 hrs. lec., 1 hr. rec. Prerequisite: 21-122 or 21-123

21-292 Operations Research I
Spring: 9 units

Operations research offers a scientific approach to decision making, most commonly involving the allocation of scarce resources. This course develops some of the fundamental methods used. Linear programming: the simplex method and its linear algebra foundations, duality, post-optimality and sensitivity analysis; the transportation problem; the critical path method; non-linear programming methods. 3 hrs. lec. 1 hr. rec. Prerequisite: 21-122 or 21-123, and 21-241.

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21-300 Basic Logic
Fall: 9 units

Propositional and predicate logic: Syntax, proof theory and semantics up to completeness theorem, Lowenheim Skolem theorems, and applications of the compactness theorem. Prerequisite: 21-127 and one of 21-228, 21-484, 21-373, 21-132.

21-301 Combinatorics Analysis
Fall: 9 units

A major part of the course concentrates on algebraic methods, which are relevant in the study of error correcting codes, and other areas. Topics covered in depth include permutations and combinations, generating functions, recurrence relations, the principle of inclusion and exclusion, and the Fibonacci sequence and the harmonic series. Additional topics may include existence proofs, partitions, finite calculus, generating combinatorial objects, Polya theory, codes, probabilistic methods. 3 hrs. lec., 1 hr. rec. Prerequisite: 21-122 or 21-122 or 21-123, 21-127 or permission of instructor.

21-320 Symbolic Programming Methods
Fall or Spring: 9 units

The objective of this course is to learn to program in Maple, a powerful symbolic mathematics package available on many platforms at Carnegie Mellon. After learning what Maple can do with the commands provided with the package, students will learn to develop their own Maple functions to accomplish extended mathematical computations. Grades in the course will be based mostly on project work. Projects may come from any relevant field and may be graphical, numerical, or symbolic or all three. The course will involve online demonstrations in most classes. 3 hrs. lec.

21-341 Linear Algebra I
Fall or Spring: 9 units

Vector spaces: subspaces and linear independence, basis and dimension, row equivalence of matrices, general theorems about vector spaces, systems of linear equations, linear manifolds. Linear transformations: addition and multiplication, matrices of linear transformations. Inner products: angle and orthogonality, Gram-Schmidt orthogonalization, orthogonal transformations. Determinants: existence and uniqueness, multiplication theorem. 3 hrs. lec. Prerequisite: None, but 21-127 Concepts of Mathematics is highly recommended.

21-342 Linear Algebra II
Spring: 9 units

Theory of a single linear transformation: basic concepts, invariant subspaces, triangular form theorem, rational and Jordan canonical forms. Dual vector spaces and multilinear algebra: quotient spaces and dual spaces, bilinear forms and duality, direct sums and tensor products, elementary divisor theorem. Orthogonal and unitary transformations: structure of orthogonal transformations, principal axis theorem, unitary transformations, spectral theorem for self-adjoint and normal transformations. Applications to differential equations. 3 hrs. lec. Prerequisite: 21-341.

21-350 History of Mathematics
Fall or Spring: 9 units

Mathematics has a long and interesting history, and there is much insight into both mathematics and history to be gained from its study. The emphasis here will be on learning the mathematics with the added value of appreciating it in historical context. Selected topics may range from early number systems, the development of geometry, the emergence of the ideas of analysis, through to the origins of modern set theory.

21-355 Advanced Calculus I
Fall or Spring: 9 units

This course expands on topics introduced in the calculus sequence and considers them at a higher mathematical level. Infinite series and sequences, completeness of the real numbers, continuous and differentiable functions. Riemann integral. 3 hrs. lec. Prerequisite: 21-122 and 21-127.

21-356 Advanced Calculus II
Spring: 9 units

The calculus of functions of several variables: limits and continuity; differentiability, inverse and implicit function theorems; integrability, Fubini's Theorem, change of variables in multiple integrals; vector analysis, including Stokes' Theorem. 3 hrs. lec. Prerequisites: 21-241, 21-259, 21-355.

21-357 Sequences and Series of Functions
Fall: 9 units

This course serves as a sequel to Advanced Calculus I. The course begins with a thorough coverage of uniform and pointwise convergence of sequences and series of functions. This is followed by application to power series and Fourier series. Additional topics may include (at the discretion of the instructor and as time permits) the Weierstrass approximation theorem, metric spaces, contraction mapping, existence of solutions to ODEs, the Arzela-Ascoli theorem, and wavelets. 3 hrs. lec. Prerequisite: 21-241, 21-259, 21-355.

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21-365 Projects in Applied Mathematics
Fall or Spring: 9 units

This course provides students with an opportunity to solve problems posed by area companies. It is also designed to provide experience working as part of a team to solve problems for a client. The background needed might include linear programming, simulation, data analysis, scheduling, numerical techniques, etc. Prerequisite: Permission of the instructor; technical background necessary varies with the project.

21-369 Numerical Methods
Fall or Spring: 9 units

This course provides an introduction to the use of computers to solve scientific problems. Methods for the computational solution of linear algebra systems, nonlinear equations, the interpolation and approximation of functions, differentiation and integration, and ordinary differential equations. Analysis of roundoff and discretization errors and programming techniques. 3 hrs. lec. Prerequisite: 21-259.

21-370 Discrete-Time Finance
Fall: 9 units

This course treats the multi-period binomial model for derivative security pricing. Options, futures, exotic options, bonds, and interest rate swaps are treated, and prices of some of these are computed by backward recursion and by Monte Carlo simulation. The Black-Scholes equation will be obtained as a limiting case of the binomial model. 3 hrs. lec. Prerequisite: 36-225 and 21-257 or 21-292.

21-371 Functions of a Complex Variable
Fall or Spring: 9 units

This course provides an introduction to one of the basic topics of both pure and applied mathematics and is suitable for those with both practical and theoretical interests. Algebra and geometry of complex numbers; complex differentiation and integration. Cauchys theorem and applications; conformal mapping; applications. 3 hrs. lec. Prerequisites: 21-259 and 21-260.

21-372 Partial Differential Equations
Spring: 9 units

This course provides an introduction to partial differential equations and is recommended for majors in mathematics, physical science, or engineering. Boundary value problems on an interval, Fourier series, uniform convergence, the heat, wave, and potential equations on bounded domains, general theory of eigenfunction expansion, the Fourier integral applied to problems on unbounded domains, introduction to numerical methods. 3 hrs. lec. Prerequisites: 21-259 and 21-260.

21-373 Algebraic Structures
Fall or Spring: 9 units

This course introduces the basic concepts of algebra, preparing the student to understand abstract concepts, and thus to go on to other courses. Algebraic systems, groups, rings, fields, integral domains, fields, polynomials, unique factorization domains, rings and ideals, applications to computer science and coding theory. 3 hrs. lec. Prerequisite: 21-341.

21-374 Field Theory
Spring: 9 units

The purpose of this course is to provide a successor to Algebraic Structures, with an emphasis on applications of groups and rings within algebra to some major classical problems. These include constructions with a ruler and compass, and the solvability or unsolvability of equations by radicals. It also offers an opportunity to see group theory and basic ring theory "in action", and introduces several powerful number theoretic techniques. The basic ideas and methods required to study finite fields will also be introduced. These ideas have recently been applied in a number of areas of theoretical computer science including primality testing and cryptography. 3 hrs. lec. Prerequisite: 21-241, 21-373.

21-380 Introduction to Mathematical Modeling
9 units

This course shall examine mathematical models which may be used to describe natural phenomena. Examples which have been studied include: continuum description of highway traffic, discrete velocity models of a monotonic gas, chemotactic behavior in biological systems, European options pricing, and cellular-automata. Systems such as the first four are described by partial differential equations; the last involves discrete-time and discrete-phase dynamical systems which have been used to successfully represent both physical and biological systems. The course will develop these models and then examine the behavior of the underlying systems, both analytically and numerically. The mathematical tools required will be developed in the course. Prerequisites: 21-260 Differential Equations, 21-241 Matrix Algebra, Computer literacy and experience with at least one programming language.

21-393 Operations Research II
Fall: 9 units

An important goal of this course is for the student to gain experience with the process of working in a group to solve a problem. Much of the course is devoted to a group project based upon case studies and methods presented. Topics may include combinatorial optimization, game theory, integer programming, heuristic methods. 3 hrs. lec. Prerequisite: 21-292. 21-228 recommended.

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21-420 Continuous-Time Finance
Spring: 9 units

Transition from discrete- to continuous-time models. Review of partial differential equations emphasizing the heat equation. Formal approach to Ito's lemma. Use of Ito's lemma in financial modeling: applications to options pricing. Derivation of Black-Scholes equation. Solution by analytical and numerical methods. Modifications for early exercise. Modeling and solutions for Asian and other path dependent options. Introduction to the classical single-factor interest rate models. Basics of pricing bond options. 3 hrs. lec. Prerequisites: 21-260, 21-370.

21-440 Selected Topics in Algebra
Fall or Spring: 9 units

Typical of courses which are offered from time to time are Boolean algebras, algebraic theory of semigroups, rings and ideals, number theory, inequalities. Prerequisites: Variable.

21-441 Number Theory
Fall: 9 units

Number theory deals with the integers, the most basic structures of mathematics. It is one of the most ancient, beautiful, and well-studied branches of mathematics, and has recently found surprising new applications in communications and cryptography. Course contents: Structure of the integers, greatest common divisiors, prime factorization. Modular arithmetic, Fermat's Theorem, Chinese Remainder Theorem. Number theoretic functions, e.g. Euler's function, Mobius functions, and identities. Diophantine equations, Pell's Equation, continued fractions. Modular polynomial equations, quadratic reciprocity. 3 hrs. lec. Prerequisite: 21-127.

21-450 Selected Topics in Geometry
Fall or Spring: 9 units

Typical of courses which are offered from time to time are convex sets, differential geometry, projective geometry, and classical geometry. Prerequisites: Variable.

21-460 Topology
Spring: 9 units

This course introduces the topological concepts that underlie analysis. Included are metric spaces, topological spaces, separation, compactness, convergence, and connectedness. Also included are constructions and concepts in topological spaces that parallel those found elsewhere in mathematics such as quotients, products, sums, factorization of mappings, and isomorphisms. Other topics included as time permits according to the interests of the instructor. 3 hrs. lec. Prerequisite: 21-355.

21-470 Selected Topics in Analysis
Fall or Spring: 9 units

Typical of courses which are offered from time to time are finite difference equations, calculus of variations, and applied control theory. Prerequisites: Variable.

21-476 Ordinary Differential Equations
Fall: 9 units

Review of solution techniques, modeling techniques, existence and uniqueness, numerical procedures, linear equations and systems, special functions, autonomous non-linear systems, qualitative techniques. 3 hrs. lec. Prerequisites: 21-241 and 21-260.

21-484 Graph Theory
Spring: 9 units

Graph theory uses basic concepts to approach a diversity of problems and nontrivial applications in operations research, computer science and other disciplines. It is one of the very few mathematical areas where one is always close to interesting unsolved problems. Topics include graphs and subgraphs, trees, connectivity, Euler tours and Hamilton cycles, matchings, graph colorings, planar graphs and Euler's Formula, directed graphs, network flows, counting arguments, and graph algorithms. 3 hours lecture. Prerequisite: 21-127 or permission of instructor.

21-499 Undergraduate Research Topic
Fall or Spring: units vary

Students conduct research on problems suggested within a specific area under the supervision of a faculty member. Work would be done individually or in teams as appropriate. The course prerequisites, topic and enrollment limitation vary with the instructer and topic.

Times to be arranged

21-599 Undergraduate Reading and Research
Fall or Spring: units vary

Individual reading courses or projects in mathematics and its applications. Prerequisites and units to be negotiated with individual instructors.

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The following graduate courses are available to properly prepared undergraduates:

21-600 Mathematical Logic I
21-602 Introduction to Set Theory
21-603 Introduction to Model Theory
21-605 Teaching Mathematics
21-606 Mathematics Course Design
21-610 Algebra I
21-611 Topics in Algebra
21-620 Real Analysis
21-621 Introduction to Lebesgue Integration
21-622 Finite-Dimensional Analysis
21-623 Complex Analysis I
21-624 Topics in Analysis
21-630 Ordinary Differential Equations
21-633 Mathematical Techniques in Engineering
21-640 Functional Analysis
21-651 General Topology
21-660 Introduction to Numerical Analysis I
21-661 Numerical Solution of Partial Differential Equations I
21-662 Numerical Solution of Partial Differential Equations II
21-679 Foundations of Probability
21-680 Topics in Probability
21-681 Stochastic Calculus in Finance
21-690 Fundamentals of Optimization
21-700 Mathematical Logic II

Visit the graduate course descriptions page for upper level graduate courses.

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