Core Courses
(offered every year)

• Mathematics 141, Calculus I
• Mathematics 142, Calculus II
• Mathematics 161, Discrete Mathematics
• Mathematics 211, Ordinary Differential Equations
• Mathematics 212, Calculus III
• Mathematics 242, Elementary Linear Algebra
• Mathematics 261, Proofs and Fundamentals
• Mathematics 331, Linear Algebra
• Mathematics 332, Abstract Algebra
• Mathematics 361, Real Analysis

Elective Courses
(offered at least once every three years or by tutorial)

• Mathematics 351, Point Set Topology
• Mathematics 372, Combinatorics
• Mathematics 373, Graph Theory
• Mathematics 405, Mathematical Logic
• Mathematics 434, Advanced Topics in Algebra and Combinatorics

Tutorials
The Mathematics Program offers tutorials in advanced topics not covered by courses taught in the current semester. The following is a sampling of mathematics tutorials from the past few years.

• Advanced Abstract Algebra
• Mathematical Logic
• Number Theory
• Topology
• Differential Geometry
• Probability

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Course Descriptions

The Mathematics of Chance
Mathematics 102
Students and the instructor choose serious applications of probability and statistics as the focus of the course. Concepts in probability and statistics are developed to the extent necessary to understand the applications. Most -topics are introduced in a case-study fashion, usually by reading an article in a current periodical such as the New York Times. Other examples are drawn from journals such as Chance, Nature, Science, and Scientific American. Primary reading is supplemented by readings on basic probability and statistics. The goal is to enable the student to make critical judgments and come to informed conclusions about current issues involving chance. Prerequisite: eligibility for Q courses.

Mathematics and Politics
Mathematics 106
The course examines applications of mathe-matics to political science. Five major topics are covered: a model of escalatory behavior, game-theoretic models of international conflict, yes-no voting systems, political power, and social choice. For each model presented, its implications and its limitations are discussed. Students are actively involved in the modeling process. There is no particular mathematical prerequisite for this course, though algebraic computations and deductive proofs of some of the main results are required. Prerequisite: eligibility for Q courses.

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Topics in Geometrical Mathematics
Mathematics 107
Geometrical mathematics involves many topics other than traditional Euclidean geometry. This course explores various topics from semester to semester and may include some, but not all, of the following: symmetry, groups, frieze and wallpaper patterns, graphs, surfaces, knots, and higher dimensions. Prerequisites: eligibility for Q courses and a willingness to explore new ideas and construct convincing arguments.

Precalculus Mathematics
Mathematics 110
For students who intend to take calculus and need to acquire the necessary skills in algebra and trigonometry. The concept of function is stressed, with particular attention to linear, quadratic, general polynomial, trigonometric, exponential, and logarithmic functions. Graphing in the Cartesian plane and developing the trigonometric functions as circular functions are included. This class makes extensive use of the TI-82 graphing calculator. Prerequisites: eligibility for Q courses and satisfactory performance on precalculus entrance exam.

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Statistics and Probability
Mathematics 125
Focuses on mathematical methods used in designing and analyzing results from experiments in the sciences, particularly biology and chemistry. Among topics covered are elementary probability and statistics, fitting and hypothesis testing, characteristics of frequency distributions, regression analysis, and propagation of uncertainties. Learning computer methods of statistical analysis is a central aim of the course. Prerequisite: eligibility for Q courses.

Exploration in Number Theory
Mathematics 131
This course, an overview of one of the oldest and most beautiful areas of mathematics, is designed for any student who wants a taste of mathematics outside the calculus sequence. Topics include number puzzles, prime numbers, congruences, quadratic reciprocity, sums of squares, Diophantine equations, cryptography, coding theory, and continued fractions. Prerequisite: Mathematics 110 or permis-sion of the instructor.

Game Theory
Mathematics 135
In the 20th century the theory of games gained prominence for its application to the social sciences. Game theory is a mathematical approach to modeling situations of conflict, whether real or theoretical. Using algebra and some analytical geometry, students explore the mathematical foundations of game theory. At the same time, they encounter a wide range of applications of the theory of games. Topics include zero-sum games, nonzero-sum games, pure and mixed strategies, von Neumann's minimax theorem, Nash equilibria, and cooperative games. Prerequisite: Mathematics 110 or permission of the instructor.

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Calculus I
Mathematics 141
An introduction to the basic ideas of differentiation and integration of functions of one variable. Topics include limits, techniques of differentiation, definite integrals, the fundamental theorem of calculus, and applications. Prerequisite: Mathematics 110 or the equivalent.

Calculus II
Mathematics 142
This course, a continuation of Calculus I, reinforces the fundamental ideas of the derivative and the definite integral. Topics covered include L'Hopital's rule, integration techniques, improper integrals, volumes, arc length, sequences and series, power series, continuous random variables, and separable differential equations. Prerequisite: Mathematics 141 or the equivalent.

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Discrete Mathematics
Mathematics 161
As a complement to calculus, which models continuous phenomena, this course studies discrete processes and structures. Discrete mathematics provides the mathematical foundation for many areas of computation and can be applied to such diverse problems as designing an optimal phone-switching network and designing a computer circuit. Five core areas are covered: enumeration and recurrence relations; fundamentals of logic; sets, relations, and functions; recursion and induction; and basic graph theory. Other topics may include elementary probability theory, generating functions, discrete optimization, modular arithmetic and finite groups, and Boolean algebra. A brief introduction to the methods of mathematical proof is taught. Prerequisite: one semester of calculus or permission of the instructor.

Ordinary Differential Equations
Mathematics 211
This course is organized around methods for solving ordinary differential equations and incorporates many ideas from calculus. Topics include the classification of differential equations; determining existence and uniqueness of ordinary differential equations; and solving first- and second-order differential equations using a variety of mathematical tools, such as integrating -factors, Laplace transforms, and power series. Prerequisites: Mathematics 141 and 142 or permission of the instructor

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Course III
Mathematics 212
This course investigates differentiation and integration of multivariable functions. Topics covered include vectors, coordinate systems, vector valued functions, partial derivatives, gradients, Lagrange multipliers, multiple integrals, change of variables, line integrals, Green's theorem, and Stoke's theorem. Prerequisite: Mathematics 141 and 142 or the equivalent.

Elementary Linear Algebra
Mathematics 242
This course covers the basics of linear algebra in n-dimensional Euclidean space, including vectors, matrices, systems of linear equations, determinants, eigenvalues and eigenvectors, as well as applications of these concepts to the natural, physical, and social sciences. Equal time is given to computational, applied, and theoretical aspects of the course material. Prerequisite: Mathematics 141 or permission of the instructor.

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Proofs and Fundamentals
Mathematics 261
An introduction to the methodology of the mathematical proof. The logic of compound and quantified statements; mathematical induction; and basic set theory, including functions and cardinality, are covered. Topics from foundational mathematics are developed to provide students with an opportunity to apply proof techniques. Prerequisite: Mathematics 142 or permission of the instructor.

Operations Research
Mathematics 322<
Operations research is a scientific approach to decision making that seeks to determine how best to design and operate a system, usually under conditions requiring the allocation of scarce resources. This course introduces students to the branch of operations research known as deterministic optimization, which tackles problems such as how to schedule classes with a limited number of classrooms on campus, determine a diet that is both rich in nutrients and low in calories, and create an investment portfolio that meets your investment needs. Techniques covered include linear programming, network flows, integer/combinatorial optimization, and non-linear programming. Emphasis is on the importance of problem formulation as well as how to apply algorithms to real-world problems. Prerequisites: working knowledge of multivariable calculus and basic linear algebra.

Linear Algebra
Mathematics 331
An introduction to the theory of abstract vector spaces. The concept of a vector space is often useful when studying physical phenomena. Topics include linear independence and dependence, bases and dimension, linear transformations, eigenvalues, eigenvectors, diagonalization, inner product spaces, and orthogon-ality. Prerequisite: Mathematics 261 or permission of the instructor.

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Abstract Algebra
Mathematics 332
An introduction to modern abstract algebraic systems. The structures of groups, rings, and fields are studied together with the homo-morphisms of these objects. Topics include equiva-lence relations, finite groups, group actions, integral domains, polynomial rings, and finite fields. Prerequisite: Mathematics 261 or permission of the instructor.

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Point Set Topology
Mathematics 351
An introduction to point set topology. Topics include topological spaces, metric spaces, compactness, connectedness, continuity, homomorphisms, separation criteria, and, possibly, the fundamental group. Prerequisite: Mathematics 361 or permission of the instructor.

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Real Analysis
Mathematics 361
The fundamental ideas of analysis in one-dimensional Euclidean space are studied. Topics covered include the completeness of real numbers, sequences, Cauchy sequences, continuity, uniform continuity, the derivative, and the Riemann integral. As time permits, other topics may be considered, such as infinite series of functions or metric spaces. Prerequisite: Mathematics 261 or permission of the instructor.

Combinatorics
Mathematics 372
Combinatorial mathematics is the study of how to combine objects into finite arrangements. Topics covered in this course are chosen from enumeration and generating functions, graph theory, matching and optimization theory, combinatorial designs, ordered sets, and coding theory. Prerequisite: Mathematics 261 or permission of the instructor.

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Graph Theory
Mathematics 373
Graph theory is a branch of mathematics that has applications in areas ranging from operations research to biology. This course is a survey of the theory and applications of graphs. Topics are chosen from among connectivity, trees, Hamiltonian and Eulerian paths and cycles; isomorphism and reconstructability; planarity, coloring, color-critical graphs, and the four-color theorem; intersection graphs and vertex and edge domination; matchings and network flows; matroids and their relationship with optimization; and random graphs. Applications of graph theory are discussed in depth. Prerequisite: Mathematics 261 or permission of the instructor.

Mathematical Logic
Mathematics 405
An introduction to mathematical logic. Topics include first-order logic, completeness and com-pactness theorems, model theory, nonstandard analysis, decidability and undecidability, incompleteness, and Turing machines. Prerequisite: Mathematics 332 or permission of the instructor.

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Advanced Topics in Algebra and Combinatorics
Mathematics 434
Covers a selection of topics in algebraic combinatorics and computational algebra. Possible topics include convex polytopes, simplicial complexes, multivariate splines, hyperplane arrangements, and Groebner bases. Prerequisite: Mathematics 331 or 332.

Advanced Topics in Topology
Mathematics 451
The course covers topics in algebraic, geometric, and differential topology chosen according to student interest and background. Possible topics include the fundamental group, covering spaces, simplicial homology, classification of compact connected surfaces, topological and smooth manifolds, critical points of smooth maps, and vector fields. Prerequisite: Mathematics 351 or permission of the instructor.