Spring 2004
Course Descriptions
MATH 103:
Mathematical Explorations
Staff. 3 credits.
This course is for students who wish to experience how mathematical ideas
naturally evolve. The homework consists of the students actively investigating
mathematical ideas. The course emphasizes ideas and imagination as opposed
to techniques and calculations. Topics vary depending on the instructor
and are announced (www.math.cornell.edu) several weeks before the semester
begins. Some assessment is done through writing assignments.
MATH 106:
Calculus for the Life and Social Sciences
Reyer Sjamaar. 3 credits. Prerequisite: Readiness for calculus,
such as can be obtained from 3 years of high school mathematics (including
trigonometry and logarithms) or from MATH 109 or EDUC 115. MATH 111,
rather than 106, is recommended for those planning to take 112.
Course serves as an introduction to differential and integral calculus,
partial derivatives, elementary differential equations. Examples from
biology and the social sciences are used.
MATH 111:
Calculus I
Staff. 4 credits. Prerequisite: MATH 109 or 3 years of high
school mathematics, including trigonometry and logarithms.
Course topics include: functions and graphs, limits and continuity, differentiation
and integration of algebraic, trigonometric, inverse trig, logarithmic,
and exponential functions; applications of differentiation, including
graphing, max-min problems, tangent line approximation, implicit differentiation,
and applications to the sciences; the mean value theorem; and antiderivatives,
definite and indefinite integrals, the fundamental theorem of calculus,
substitution in integration, the area under a curve. Graphing calculators
are used, and their pitfalls are discussed, as applicable to the above
topics. MATH 111 can serve as a one-semester introduction to calculus
or as part of a two-semester sequence in which it is followed by MATH
112 or 122.
MATH 112:
Calculus II
Staff. 4 credits. Prerequisite: MATH 111 with a grade of C or
better or excellent performance in MATH 106. Those who do well in MATH
111 and expect to major in mathematics or a strongly mathematics-related
field should take 122 instead of 112.
Course focus is on integration: applications, including volumes and arc
length; techniques of integration, approximate integration with error
estimates, improper integrals, differential equations (separation of variables,
initial conditions, systems, some applications). Also covered are infinite
sequences and series: definition and tests for convergence, power series,
Taylor series with remainder, and parametric equations.
MATH 122:
Honors Calculus II
Mark McClure. 4 credits. Prerequisite: 1 semester of alculus
with a high performance or permission of the department. Students planning
to continue with MATH 213 are advised to take 112 instead of this course.
The approach of this course to calculus is more theoretical than that
in MATH 112. Topics covered include: differentiation and integration of
elementary transcendental functions, techniques of integration, applications,
polar coordinates, infinite series, and complex numbers, as well as an
introduction to proving theorems.
MATH 135:
The Art of Secret Writing
Lawren Smithline. 3 credits. Prerequisite: 3 years of high school
mathematics.
The course examines classical and modern methods of message encryption,
decryption, and cryptoanalysis. We develop mathematical tools to describe
these methods (modular arithmetic, probability, matrix arithmetic, number
theory) and become acquainted with some of the fascinating history of
the methods and people involved.
MATH 171:
Statistical Theory and Application In The Real World
Staff. 4 credits. Prerequisite: High school mathematics. No
credit if taken after ECON 319, 320 or 321.
This introductory statistics course discusses techniques for analyzing
data occurring in the real world and the mathematical and philosophical
justification for these techniques. Topics include: population and sample
distributions, central limit theorem, statistical theories of point estimation,
confidence intervals, testing hypotheses, the linear model, and the least
squares estimator. The course concludes with a discussion of tests and
estimates for regression and analysis of variance (if time permits). The
computer is used to demonstrate some aspects of the theory, such as sampling
distributions and the Central Limit Theorem. In the lab portion of the
course, students learn and use computer-based methods for implementing
the statistical methodology presented in the lectures. (No previous familiarity
with computers is presumed.)
MATH 191:
Calculus For Engineers
Kasso Okoudjou. 4 credits. Prerequisite: 3 years of high school
mathematics including trigonometry and logarithms, plus some knowledge
of calculus.
MATH 191 covers essentially the same topics as 190, but is designed for
students with some previous successful experience with calculus. Course
topics include: plane analytic geometry, differential and integral calculus,
and applications.
MATH 192:
Calculus For Engineers
Staff. 4 credits. Prerequisite: MATH 190 or 191.
Course topics include: polar coordinates, infinite series, and power
series. Also covered are: vectors and calculus of functions of several
variables through double and triple integrals.
MATH 213:
Calculus III
Paul Jung. 4 credits. Prerequisite: MATH 112, 122, or 192.
This course is designed for students who wish to master the basic techniques
of calculus, but whose major will not require a substantial amount of
mathematics. Course topics include: vectors and vector-valued functions;
multivariable and vector calculus including multiple and line integrals;
first- and second-order differential equations with applications; systems
of differential equations; and elementary partial differential equations.
The course may emphasize different topics in the syllabus in different
semesters.
MATH 221:
Linear Algebra and Differential Equations
Staff. 4 credits. Prerequisite: 2 semesters of calculus with
high performance or permission of the department.
This course is recommended for students who plan to major in mathematics
or in a related field. Course covers linear algebra and differential equations.
Topics include: vector algebra, linear transformations, matrices, and
linear differential equations, as well as an introduction to proving theorems.
MATH 222:
Multivariable Calculus
Staff. 4 credits. Prerequisite: MATH 221.
This course is recommended for students who plan to major in mathematics
or in a related field. It covers differential and integral calculus of
functions in several variables, line and surface integrals as well as
the theorems of Green, Stokes and Gauss.
MATH 224:
Theoretical Linear Algebra and Calculus
Ravi Ramakrishna. 4 credits. Prerequisite: MATH 223.
Course topics include: vector fields; line integrals; differential forms
and exterior derivative; work, flux, and density forms; integration of
forms over parametrized domains; and Green's, Stoke's, and divergence
theorems.
MATH 231:
Linear Algebra
Oleg Chalykh. 3 credits. Prerequisite: MATH 111 or equivalent.
Students interested in the mathematics major should take MATH 221 or
294.
This course is an introduction to linear algebra. A wide range of applications
are discussed and computer software may be used. The main topics are systems
of linear equations, matrices, determinants, vector spaces, orthogonality,
and eigenvalues. Typical applications are population models, input/output
models, least squares, and difference equations.
MATH 275:
Elementary Probability for Applications
Richard Durrett. 3 credits. Prerequisite: 1 semester of calculus
An introduction to probability emphasizing applications. The course begins
with basics: combinatorial probability, mean and variance, independence,
conditional probability and Bayes formula. The law of large numbers and
central limit theorem are stated and their implications for statistics
are discussed. The course concludes with a discussion of Markov chains
and their applications.
MATH 293:
Engineering Mathematics
Staff. 4 credits. Prerequisite: MATH 192.
The conclusion of vector calculus, including line integrals, vector fields,
Green's theorem, Stokes' theorem, and the divergence theorem; followed
by an introduction to ordinary and partial differential equations, including
Fourier series and boundary value problems. May include computer use in
solving problems.
MATH 294:
Engineering Mathematics
Staff. 4 credits. Prerequisite: MATH 192.
Linear algebra and its applications. Topics include matrices, determinants,
vector spaces, eigenvalues and eigenvectors, orthogonality and inner product
spaces; applications include brief introductions to difference equations,
Markov chains, and systems of linear ordinary differential equations.
May include computer use in solving problems.
MATH 311:
Introduction to Analysis
Staff. 4 credits. Prerequisite: MATH 221-222, 223-224 or 293-294.
Provides a transition from calculus to real analysis. Topics include:
rigorous treatment of fundamental concepts in calculus: including limits
and convergence of sequences and series, compact sets; continuity, uniform
continuity and differentiability of functions. Emphasis will be placed
upon understanding and constructing mathematical proofs.
MATH
335: Introduction to Cryptology
Edward B. Swartz. 4 credits. Prerequisite: COM S 100 and MATH
222 or 294.
MATH 335 is an introduction to the algorithmic and mathematical concepts
in modern cryptanalysis. Topics will include elementary probability, Euclidean
algorithm, Chinese remainder theorem, finite fields, quadratic residues,
primality tests, factoring algorithms, cryptographic hash functions, secret
and public key methods.
MATH 336:
Applicable Algebra
Staff. 4 credits. Prerequisite: MATH 221, 223, 231 or 294.
An introduction to the concepts and methods of abstract algebra and number
theory that are of interest in applications. Course covers the basic theory
of groups, rings and fields and their applications to such areas as public-key
cryptography, error-correcting codes, parallel computing, and experimental
designs. Applications include the RSA cryptosystem and use of finite fields
to construct error-correcting codes and Latin squares. Topics include
elementary number theory, Euclidean algorithm, prime factorization, congruences,
theorems of Fermat and Euler, elementary group theory, Chinese remainder
theorem, factorization in the ring of polynomials, and classification
of finite fields.
MATH 356:
Groups and Geometry
James E. West. 4 credits. Prerequisite: MATH 221, 223, 231 or
294.
Groups were introduced in the nineteenth century as the sets of symmetries
of algebraic or geometric objects. This viewpoint has become central in
modern mathematics. This course studies the geometry of the plane and
of patterns in the plane in terms of the group of symmetries ("isometries")
of the plane. Prior knowledge of group theory is not a prerequisite. The
purpose of the course is to prepare students for the 400-level courses
in several ways. On one hand, the course offers experience in modern algebra
and geometry (including the geometry of complex numbers). It presents
some very beautiful and important topics and a sense of the unity of mathematics.
On the other hand, special care is taken to initiate the student into
the writing of proofs and the language of mathematics. Topics include:
Symmetries, groups of transformations, subgroups and cosets. Homomorphisms
and isomorphisms. Orbits and fixed points. Frieze groups, wallpaper groups
("2-dimensional crystallographic groups") and the associated tesselations
of the Euclidean plane.
MATH 362:
Dynamic Models in Biology (also BIOEE 362)
John Guckenheimer and Stephen Ellner. 3 credits. Prerequisite:
Two semesters of introductory biology (BIO G 101-102, 105-106, 107-108,
109-110, or equivalent) and completion of the mathematics requirement
for the Biological Sciences major or equivalent.
For description, see BIOEE 362.
MATH 401:
Honors Seminar: Topics In Modern Mathematics
Ravi Ramakrishna. 4 credits. Prerequisite: 2 courses in mathematics
numbered 300 or higher or permission of instructor.
This course is a participatory seminar primarily aimed at introducing
senior and junior mathematics majors to some of the challenging problems
and areas of modern mathematics. The seminar helps students develop research
and expository skills in mathematics, which is important for careers in
any field that makes significant use of the mathematical sciences (i.e.,
pure or applied mathematics, physical or biological sciences, business
and industry, medicine). The content varies from year to year.
MATH 403:
History of Mathematics
Robert S. Strichartz. 4 credits. Prerequisite: 2 courses in
mathematics above 300, or permission of instructor.
Survey of the development of mathematics from antiquity to the present,
with an emphasis on the achievements, problems, and mathematical viewpoints
of each historical period and the evolution of such basic concepts as
number, geometry, construction, and proof. Readings from original sources
in translation. Students are required to give oral and written reports.
MATH 408:
Mathematics In Perspective
Karen Vogtmann. 4 credits. Prerequisite: Consent of instructor.
The purpose of this course is for students to step back and form an overview
of the mathematics they have learned. The course is intended for junior
and senior mathematics majors and other undergraduates with strong backgrounds
in mathematics. In spring 2004, subjects will be chosen with a view towards
their interest for prospective high school teachers
MATH 413:
Honors Introduction to Analysis I
Martin Dindos. 4 credits. Prerequisite: a high level of performance
in MATH 221-222, 223-224, or 293-294 and a familiarity with proofs.
Students who do not intend to take MATH 414 are encouraged to take MATH
413 in the spring.
This course provides an introduction to the rigorous theory underlying
calculus, covering the real number system and functions of one variable.
The course is entirely based on proofs, and the student is expected to
know how to read and, to some extent, construct proofs before taking this
course. Topics typically include: Construction of the real number system,
properties of the real number system, continuous functions, differential
and integral calculus of functions of one variable, sequences and series
of functions.
MATH 414:
Honors Introduction to Analysis II
John Smillie. 4 credits. Prerequisite: MATH 413.
This is a proof-based introduction to further topics in analysis. Topics
may include: the Lebesgue measure and integration, functions of several
variables, differential Calculus, implicit function theorem, infinite
dimensional normed and metric spaces, Fourier series, ordinary differential
equations.
MATH 418:
Introduction To The Theory of Functions of One Complex Variable
Irina Mitrea. 4 credits. Prerequisite: MATH 223-224, 311, 411
or 413, or permission of instructor.
A theoretical and rigorous introduction to complex variable theory. Topics
include: complex numbers, differential and integral calculus for functions
of a complex variable including Cauchy's theorem and the calculus of residues,
elements of conformal mapping. Students interested in the applications
of complex analysis should consider MATH 422.
MATH 420:
Differential Equations and Dynamical Systems
Hsiao-Bing Cheng. 4 credits. Prerequisite: High level of performance
in MATH 293-294, 221-222, 223-224, or permission of instructor.
Course covers ordinary differential equations in one and higher dimensions:
qualitative, analytic, and numerical methods. Emphasis is on differential
equations as models and the implications of the theory for the behavior
of the system being modeled and includes an introduction to bifurcations.
MATH 422:
Applied Complex Analysis
Leonard Gross. 4 credits. Prerequisite: MATH 221-222, 223-224,
293-294, or 213 and 231.
Course covers complex variables, Fourier transforms, Laplace transforms
and applications to partial differential equations. Additional topics
may include an introduction to generalized functions.
MATH 424:
Wavelets and Fourier Series
Kasso Okoudjou. 4 credits. Prerequisite: MATH 221-222, 223-224,
293-294, or permission of instructor.
Both Fourier series and wavelets provide methods to represent or approximate
general functions in terms of simple building blocks. Such representations
have important consequences, both for pure mathematics and for applications.
Fourier series use natural sinusoidal building blocks and may be used
to help solve differential equations. Wavelets use artificial building
blocks that have the advantage of localization in space. A full understanding
of both topics requires a background involving Lebesgue integration theory
and functional analysis. This course presents as much as possible on both
topics without such formidable prerequisites. The emphasis is on clear
statements of results and key ideas of proofs, working out examples, and
applications. Related topics that may be included in the course: Fourier
transforms, Heisenberg uncertainty principle, Shannon sampling theorem,
and Poisson summation formula.
MATH 428:
Introduction To Partial Differential Equations
Lars B. Wahlbin. 4 credits. Prerequisite: MATH 221-222, 223-224,
or 293-294 or permission of instructor.
Topics will be selected from first-order quasilinear equations, classification
of second-order equations, with emphasis on maximum principles, existence,
uniqueness, stability, Fourier series methods, approximation methods.
MATH 432:
Introduction to Algebra
Stephen U. Chase. 4 credits. Prerequisite: MATH 332, 336, 431
or 433, or permission of instructor. Undergraduates who plan to attend
graduate school in mathematics should take MATH 433-434.
An introduction to various topics in abstract algebra, including groups,
rings, fields, factorization of polynomials and integers, congruences,
and the structure of finitely generated abelian groups. Optional topics
are modules over Euclidean domains and Sylow theorems.
MATH 434:
Honors Introduction to Algebra
R. Keith Dennis. 4 credits. Prerequisite: MATH 332, 336, 431,
or 433, or permission of instructor.
This is the honors version of a course in abstract algebra, which treats
the subject from an abstract and axiomatic viewpoint, including universal
mapping properties. Topics include groups, groups acting on sets, Sylow
theorems; rings, factorization: Euclidean rings, principal ideal domains,
the structure of finitely generated modules over a principal ideal domain,
fields, and Galois theory. The course emphasizes understanding the theory
with proofs in both homework and exams. An optional computational component
using the computer language GAP is available. A less theoretical course
that covers similar subject matter is MATH 432.
MATH 442:
Introduction to Combinatorics II
Louis Billera. 4 credits. Prerequisite: MATH 221, 223, 231,
or 294.
This is a continuation of the first semester, although formally independent
of the material covered there. The emphasis here is the study of certain
combinatorial structures, such as Latin squares and combinatorial designs
(which are of use in statistical experimental design), classical finite
geometries and combinatorial geometries (also known as matroids, which
arise in many areas from algebra and geometry through discrete optimization
theory). There is an introduction to partially ordered sets and lattices,
including general Möbius inversion and its application, as well as the
Polya theory of counting in the presence of symmetries.
MATH 452:
Classical Geometries
Indira Chatterji. 4 credits. Prerequisite: MATH 221, 223, 231,
or 294, or permission of instructor.
This is an introduction to hyperbolic and projective geometry — the classical
geometries that developed as Euclidean geometry was better understood.
For example, the historical problem of the independence of Euclid's fifth
postulate is understood when the existence of the hyperbolic plane is
realized. Straightedge (and compass) constructions and stereographic projection
in Euclidean geometry can be understood within the structure of projective
geometry. Topics in hyperbolic geometry include: models of the hyperbolic
plane and relations to spherical geometry. Topics in projective geometry
include: homogeneous coordinates and the classical theorems about conics
and configurations of points and lines. Optional topics include: principles
of perspective drawing, finite projective planes, orthogonal Latin squares,
and the cross ratio.
MATH 454:
Introduction to Differential Geometry
David W. Henderson. 4 credits. Prerequisite: MATH 221-222, 223-224,
or 293-294, plus at least one mathematics course numbered 300 or above.
MATH 453 is not a prerequisite.
Course covers differential geometry of curves and surfaces. Also covers
curvature, geodesics, and differential forms. Serves as an introduction
to n-dimensional Riemannian manifolds. This material provides some background
for the study of general relativity; connections with the latter are indicated.
MATH 472:
Statistics
Michael Nussbaum. 4 credits. Prerequisite: MATH 471 and knowledge
of linear algebra such as taught in MATH 221. Some knowledge of multivariable
calculus helpful but not necessary.
Statistics have proved to be an important research tool in nearly all
of the physical, biological, and social sciences. This course will serve
as an introduction to statistics for students who already have some background
in calculus, linear algebra, and probability theory. Topics covered in
the course will include parameter estimation, hypothesis testing, and
linear regression. The course will emphasize both the mathematical theory
of statistics as well as techniques for data analysis that are useful
in solving scientific problems.
MATH 486:
Applied Logic (also COM S 486)
Anil Nerode. 4 credits. Prerequisite: MATH 221-222, 223-224,
or 293-294; COM S 280 or equivalent (such as MATH 332, 336, 432, 434,
436, or 481); and some additional course in mathematics or theoretical
computer science.
Course covers: propositional and predicate logic; compactness and completeness
by tableaux, natural deduction, and resolution. Other possible topics
include: equational logic; Herbrand Universes and unification; rewrite
rules and equational logic, Knuth-Bendix method and the congruence-closure
algorithm and lambda-calculus reduction strategies; topics in Prolog,
LISP, ML, or Nuprl; and applications to expert systems and program verification.
MATH 490:
Supervised Reading and Research
Staff. 1-6 credits.
Supervised reading and research by arrangement with individual professors.
Not for material currently available in regularly scheduled courses.
MATH 505:
Educational Issues In Undergraduate Mathematics
David W. Henderson. 4 credits. Prerequisite: Graduate standing
or permission of the instructor.
This course examines various educational issues in undergraduate mathematics
and the relationship of these issues to the mathematics itself. The precise
choice of topics varies, but the intent is that a balance of different
views be presented and discussed. There are extensive readings in the
course and occasional guest lectures. Possible topics include: nature
of proof and how and when to teach it, calculus "reform," teaching mathematics
to school teachers, using writing, using history, alternative assessments,
alternatives to lecturing, equity issues, effective uses of technology,
what is mathematical understanding and how do we recognize it, what should
every mathematics major know, and research in undergraduate mathematics.
MATH 507:
Teaching Secondary Mathematics: Theory and Practices
Avery Solomon. 4 credits.
This course provides direct experience of new approaches, curricula and
standards in mathematics education. Discussion of articles, activities
for the secondary classroom and videotape of classroom teaching is tied
to in-class exploration of math problems. Experience in the computer lab,
examining software environments and their use in the mathematics classroom
is included. Participants are expected to write short papers, share ideas
in class and present their opinions on issues.
MATH 508:
Mathematics For Secondary School Teachers
Avery Solomon. 1-6 credits. Prerequisite: Secondary school mathematics
teacher or permission of instructor.
An examination of the principles underlying the content of the secondary
school mathematics curriculum, including connections with the history
of mathematics and current mathematics research.
Graduate courses will be announced
early in the fall semester.
MATH 790:
Supervised Reading and Research
Staff. 1-6 credits.
Last modified:
January 9, 2004
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