Fall 2002 Course
Descriptions
MATH 103:
Mathematical Explorations
Homer White, Beverly West. 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.
Lectures 01 & 02: Excursions in Modern Mathematics, Homer White
More info to come.
Lecture 03: Iterations and Patterns, Beverly West
Required text: ONE of the following (parallel assignments will be
given)
Peak and Frame, Chaos under Control: the art & science of complexity,
W. H. Freeman, 1994 (ISBN: 0-7167-2429-4).
OR
Stewart, Ian, Does God Play Dice? (Edition 2) Blackwell, 2002
In Lecture 3 we will use computers regularly, with easy-to-use interactive
color graphics tools. We begin by exploring patterns in the world around
us, and their mathematical aspects. Simultaneously we study iteration
— numerical, geometric, and algebraic — and the resulting
dynamical systems. During the later part of the course we will see how
these basic concepts lie behind current research in diverse fields,
including art, ecology, medicine, and finance.
The syllabus for Lecture 3 spends roughly 4 weeks each on
* iterating real-valued functions;
* geometric iteration;
* iterating real-valued systems of two variables
and spends the last two weeks on finishing special projects.
MATH 105:
Finite Mathematics For The Life and Social Sciences
Laurent Saloff-Coste. 3 credits. Prerequisite: 3 years of high
school mathematics, including trigonometry and logarithms.
This course is an introduction to linear algebra, probability, and Markov
chains which develops the parts of the theory most relevant for applications.
Specific topics include: equations of lines, the method of least squares,
solutions of linear systems, matrices; Basic concepts of probability,
permutations, combinations, binomial distribution, mean and variance,
and the normal approximation to the binomial. Examples from biology and
the social sciences are used.
MATH 111:
Calculus I
Robert Strichartz, czar. 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
Marshall Cohen, czar. 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 121:
Honors Calculus
David Henderson. 4 credits. Prerequisite: 3 years of high school
mathematics with average grade of AĠ or better, or permission of the
department.
This is a first-semester course in calculus intended for students who
have been quite successful in their previous mathematics courses. The
syllabus for the course is quite similar to that of MATH 111; however,
the approach is more theoretical and the material is covered in greater
depth.
MATH 122:
Honors Calculus
Gregory Lawler, Lawren Smithline. 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.
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. The approach is more theoretical than in MATH 112.
MATH 171:
Statistical Theory and Application In The Real World
Gene Hwang, Michael Nussbaum, Alexander Bendikov. 4 credits. Prerequisite:
High school mathematics.
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 190:
Calculus For Engineers
Alfred Schatz. 4 credits. Prerequisite: 3 years of high school
mathematics, including trigonometry and logarithms.
Course topics include: plane analytic geometry, differential and integral
calculus, and applications. This course is restricted to engineering students
who have had no previous successful experience with calculus. Students
who have had such experience but wish a first-semester calculus course
should take MATH 191.
MATH 191:
Calculus For Engineers
Robert Connelly, czar. 4 credits. Prerequisite: 3 years of high
school mathematics including trigonometry and logarithms, plus some
knowledge of calculus.
Course topics include: plane analytic geometry, differential and integral
calculus, and applications. MATH 191 covers essentially the same topics
as 190, but is designed for students with some previous successful experience
with calculus.
MATH 192:
Calculus For Engineers
Bing Cady, czar. 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
Jos¸ Ramrez. 4 credits. Prerequisite: MATH 112, 122, or 192.
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. 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.
The course may emphasize different topics in the syllabus in different
semesters.
MATH 221:
Linear Algebra and Differential Equations
Anil Nerode, José Ramírez, Edward Swartz. 4 credits.
Prerequisite: 2 semesters of calculus with high performance or
permission of the department.
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. This course
is especially recommended for students who plan to major in mathematics
or in a strongly related field.
MATH 222:
Multivariable Calculus
Allen Back, Oleg Chalykh. 4 credits. Prerequisite: MATH 221.
Course topics include: multivariable and vector differential and integral
calculus, including multiple, line, and surface integrals. This course
is especially recommended for students who plan to major in mathematics
or in a strongly related field.
MATH 223:
Theoretical Linear Algebra and Calculus
Harrison Tsai, Bent Ørsted. 4 credits. Prerequisite:
2 semesters of calculus with a grade of AĠ or better, or permission
of instructor.
Course topics include: vectors, matrices, and linear transformations;
differential calculus of functions of several variables; inverse and implicit
function theorems; quadratic forms, extrema, and manifolds; multiple and
iterated integrals. MATH 223-224 provides an integrated treatment of linear
algebra and multivariable calculus designed for students who have been
highly successful in their previous calculus courses.
MATH 293:
Engineering Mathematics
Allen Hatcher, Matthew Fickus, Jane Wang. 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
Herbert Hui. 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 321:
Manifolds and Differential Forms
Reyer Sjamaar. 4 credits. Prerequisite: Multivariable calculus
and linear algebra as taught in MATH 221-222 or 293-294.
Topics for this course include: differential forms, exterior derivative,
implicit function theorem, manifolds, orientation, boundaries, integration
of forms, generalized Stokes' theorem, Hodge star operator, Laplace operator,
basics of de Rham cohomology. We reexamine the integral theorems of vector
calculus (Green, Gauss, and Stokes) in the light of the exterior differential
calculus and apply differential forms to problems in partial differential
equations, fluid mechanics and electromagnetism.
MATH 323:
Introduction to Differential Equations
Lars B. Wahlbin. 4 credits. Prerequisite: Multivariable calculus
and linear algebra as taught in MATH 221-222 or 293-294, or permission
of instructor.
This course is intended for students who want a brief one-semester introduction
to the theory of and techniques in both ordinary and partial differential
equations. (Fuller introductions are given in MATH 427 and 428.) Topics
for ordinary differential equations may include: Initial-value and two-point
boundary value problems, the basic existence and uniqueness theorems,
continuous dependence on data, stability of fix-points, numerical methods,
special functions. Topics for partial differential equations may include:
The Poisson, heat and wave equations, boundary and initial-boundary value
problems, maximum principles, continuous dependence on data, separation
of variables, Fourier series, Green's functions, numerical methods, transform
methods.
MATH 332:
Algebra and Number Theory
Birgit Speh. 4 credits. Prerequisite: MATH 221, 223, 231 or
294.
Course covers various topics from number theory and modern algebra, usually
including most of the following: Primes and factorization, Diophantine
equations, congruences, quadratic reciprocity, continued fractions, rings
and fields, finite groups, and an introduction to the arithmetic of the
Gaussian integers and quadratic fields. Motivation and examples for the
concepts of abstract algebra are derived primarily from number theory
and geometry.
MATH 413:
Honors Introduction to Analysis I
Yulij Ilyashenko, Yuri Berest, Martin Dindos. 4 credits. Prerequisite:
A high level of performance in MATH 221-222, 223-224 or 293-294.
The sequence MATH 413-414, designed for honors students, provides an
introduction to the theory of functions of real variables, stressing a
rigorous logical development of the subject rather than applications.
Topics include: metric spaces, the real number system, continuous and
differentiable functions, uniform convergence and approximation theorems,
Fourier series, Riemann and Lebesgue integrals, calculus in several variables,
and differential forms.
MATH 420:
Differential Equations and Dynamical Systems
Rodrigo Perez. 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 425:
Numerical Solutions of Differential Equations
Alan Demlow. 4 credits. Prerequisite: MATH 221-222, 223-224,
or 293-294 and one course numbered 300 or higher in mathematics, or
permission of instructor.
Emphasis may be on numerical approximation of initial-value or two-point
boundary value problems for ordinary differential equations, or on partial
differential equations. A major component in the course is writing (or
using) computer code to illustrate the theoretical concepts introduced.
MATH 427:
Introduction to Ordinary Differential Equations
Oscar Rothaus. 4 credits. Prerequisite: MATH 221-222, 223-224,
or 293-294 and one course numbered 300 or higher in mathematics, or
permission of instructor.
Covers the basic existence, uniqueness, and stability theory together
with methods of solution and methods of approximation. Topics include:
singular points, series solutions, Sturm-Liouville theory, transform methods,
approximation methods, and application to physical problems.
MATH 431:
Linear Algebra
Peter Kahn. 4 credits. Prerequisite: MATH 221, 223, 231, or
294. Undergraduates who plan to attend graduate school in mathematics
should take MATH 433-434.
An introduction to linear algebra, including: the study of vector spaces,
linear transformations, matrices, and systems of linear equations; quadratic
forms and inner product spaces; canonical forms for various classes of
matrices and linear transformations.
MATH 433:
Honors Introduction To Algebra I
Karen Vogtmann, Dan Zaffran. 4 credits. Prerequisite: A high
level of performance in MATH 221, 223, 231, or 294.
Math 433-434 is the honors version of MATH 431-432. MATH 433-434 is more
theoretical and rigorous than 431-432 and includes additional material
such as multilinear and exterior algebra.
MATH 441:
Introduction To Combinatorics
Louis Billera. 4 credits. Prerequisite: MATH 221, 223, 231,
or 294.
Course covers enumerative combinatorics: permutation enumeration, Stirling
and Bell numbers, generating functions, exponential formula, Lagrange
inversion, recurrences, basic asymptotic methods, rational generating
functions. Also covers basic graph theory: trees and Cayley's theorem,
chromatic polynomial, eigenvalues and their application. Also considers
matching theory: equivalences, marriage theorem, flow problems, totally
unimodular matrices. Also considers Polya theory: action of a group on
a set, Burnside lemma, DeBruijn's method, applications to graphical enumeration
and algorithms.
MATH 451:
Euclidean and Spherical Geometry
David W. Henderson. 4 credits. Prerequisite: MATH 221, 223,
231, or 294, or permission of instructor.
Covers topics from Euclidean and spherical (non-Euclidean) geometry.
A nonlecture, seminar-style course organized around student participation.
This is a Writing in the Majors course.
MATH 453:
Introduction To Topology
James E. West. 4 credits. Prerequisite: MATH 311, 411 or 413,
or permission of instructor.
Course covers basic point set topology, connectedness, compactness, metric
spaces, fundamental group. Application of these concepts to surfaces such
as the torus, the Klein bottle, and the Moebius band.
MATH 471:
Basic Probability
Eugene B. Dynkin. 4 credits. Prerequisite: MATH 221, 223, 231,
or 294. May be used as a terminal course in basic probability.
Topics include: combinations, important probability laws, expectations,
moments, moment-generating functions, limit theorems. Emphasis is on diverse
applications and on development of use in statistical applications. See
also the description of MATH 671.
MATH 482:
Topics In Logic (also PHIL 432)
Harold Hodes. 4 credits. Prerequisite: 1 logic course from the
Mathematics Department at the 200 level or higher, 1 logic course from
the Philosophy Department at the 300 level or higher, or permission
of the instructor.
For description, see PHIL 432.
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 500:
College Teaching
Maria Terrell. 1 credits. Prerequisite: Graduate student standing
or permission of instructor. Meets alternate weeks.
Among the topics covered: basic topics about teaching, such as how to
plan recitations, how to prepare lesson plans for lectures, exam design
and grading, syllabus planning. Also discussed: the structure of colleges
and universities, jobs and tenure, professionalism, alternative teaching
strategies.
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.
MATH 611:
Real Analysis
Eugene B. Dynkin. 4 credits.
Measure and integration and functional analysis.
MATH 613:
Topics In Analysis
Gregory Lawler. 4 credits.
MATH 615:
Mathematical Methods In Physics
Dan M. Barbasch. 4 credits. Prerequisite: Intended for graduate
students in physics or related fields who have had a strong advanced
calculus course and at least 2 years of general physics. A knowledge
of the elements of finite dimensional vector space theory, complex variables,
separation of variables in partial differential equations, and Fourier
series will be assumed. Undergraduates will be admitted only with permission
of instructor.
Topics are designed to give a working knowledge of the principal mathematical
methods used in advanced physics. Course covers: Hilbert space, generalized
functions, Fourier transform, Sturm-Liouville problem in ODE, Green's
functions, and asymptotic expansions.
MATH 621:
Measure Theory and Lebesgue Integration
Alexander Bendikov. 4 credits.
Measure theory, integration, and Lp spaces.
MATH 631:
Algebra
Yuri Berest. 4 credits.
Finite groups, field extensions, Galois theory, rings and algebras, and
tensor and exterior algebra.
MATH 649:
Lie Algebras
Dan M. Barbasch. 4 credits.
Topics include: nilpotent, solvable and reductive Lie algebras; enveloping
algebras; root systems; Coxeter groups; and classification of simple algebras.
MATH 652:
Differentiable Manifolds I
Reyer Sjamaar. 4 credits. Prerequisite: Advanced calculus, linear
algebra (MATH 431), point-set topology (MATH 453).
This is an introduction to differential geometry and differential topology
at the level of a beginning graduate student. Topics include: smooth manifolds,
embeddings, tangent bundles, tensors, vector bundles, vector fields, and
Frobenius' theorem. Further topics chosen by instructor from other major
areas such as fibre bundles, Lie groups, connections, curvature, geodesics,
Riemannian manifolds, differential forms, and de Rham cohomology.
MATH 661:
Geometric Topology
Kai-Uwe Bux. 4 credits.
An introduction to some of the more geometric aspects of topology and
its connections with group theory. Possible topics include: surface theory,
3-manifolds, knot theory, geometric and combinatorial group theory, hyperbolic
groups, and hyperbolic manifolds.
MATH 671:
Probability Theory I
Richard Durrett. 4 credits. Prerequisite: A knowledge of Lebesgue
integration theory, at least on the real line. Students can learn this
material by taking parts of MATH 413-414 or 621.
Topics for MATH 671-672 include: properties and examples of probability
spaces; sample space, random variables, and distribution functions; expectation
and moments; independence, Borel-Cantelli lemma, zero-one law; convergence
of random variables, probability measures, and characteristic functions;
law of large numbers; selected limit theorems for sums of independent
random variables; Markov chains, recurrent events; ergodic and renewal
theorems; Martingale theory; and Brownian motion and processes with independent
increments.
MATH 722:
Topics In Complex Analysis
John H. Hubbard. 4 credits.
Selections of advanced topics from complex analysis, such as Riemann
surfaces, complex dynamics, and conformal and quasiconformal mapping.
Course content varies.
MATH 728:
Seminar In Partial Differential Equations
Irina Mitrea. 4 credits.
MATH 731:
Seminar In Algebra
R. Keith Dennis. 4 credits.
MATH 735:
Topics In Algebra
Milen Yakimov. 4 credits.
Selection of advanced topics from algebra, algebraic number theory, and
algebraic geometry. Course content varies.
MATH 751:
Seminar In Topology
Karen Vogtmann. 4 credits.
MATH 753:
Algebraic Topology
Allen Hatcher. 4 credits.
The continuation of 651. The standard topics covered in this course most
years are cohomology, cup products, Poincar¸ duality, and homotopy groups.
Other possible topics include fiber bundles, fibrations, vector bundles,
and characteristic classes. The course may sometimes be taught from a
differential forms viewpoint.
MATH 755:
Topology and Geometric Group Theory Seminar
Staff. 4 credits.
MATH 757:
Topics In Topology
James Conant. 4 credits.
Selection of advanced topics from modern algebraic, differential, and
geometric topology. Course content varies.
MATH 761:
Seminar In Geometry
Brian Smith. 4 credits.
MATH 771:
Seminar In Probability and Statistics
Staff. 4 credits.
MATH 777:
Stochastic Processes
Laurent Saloff-Coste. 4 credits.
MATH 781:
Seminar In Logic
Richard A. Shore. 4 credits.
MATH 784:
Recursion Theory
Richard A. Shore. 4 credits.
Course covers: theory of effectively computable functions; classification
of recursively enumerable sets; degrees of recursive unsolvability; applications
to logic; hierarchies; recursive functions of ordinals and higher type
objects; generalized recursion theory.
MATH 790:
Supervised Reading and Research
Staff. 1-6 credits.
Last modified:
April 7, 2003
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