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Core Courses
(offered every year)
Elective
Courses
(offered at least once every three years or by tutorial)
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.
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Mathematics
and Politics
Mathematics 106
The course examines applications of mathematics 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.
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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.
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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.
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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.
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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.
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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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Calculus
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 Stokes' theorem. Prerequisites:
Mathematics 141 and 142 or the equivalent.
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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.
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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.
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.
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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.
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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.
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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.
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