Mathematics Program at Bard College
Greg Landweber
Mathematics Chair
Phone: (845) 758-7093
Email: gregland@bard.edu

Mathematics Program
Bard College
P.O. Box 5000
Annandale, NY 12504

COURSES

Placement

Which mathematics course should I take?

Core Courses
(offered every year)

Please note that our sequence of core courses has changed recently. Mathematics 213 is now the sequel to Calculus II, and is followed by Mathematics 241. The courses Mathematics 211, Mathematics 212, and Mathematics 242 are no longer offered.

Elective Courses
(offered every two or three years, or by tutorial)

Tutorials
The Mathematics Program also offers tutorials in advanced topics not covered by courses taught in the current semester.


Course Descriptions


100 Level Courses

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: passing score on Part I of the Mathematics Diagnostic.

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: passing score on Part I of the Mathematics Diagnostic, and a willingness to explore new ideas and construct convincing arguments.

Precalculus Mathematics Mathematics 110
A course 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 given 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. Students who need to brush up on their Precalculus skills are encouraged to enroll concurrently in BLC 190. Prerequisites: passing score on Part I of the Mathematics Diagnostic.

Chance Mathematics 119
The mathematical theory of probability is useful for quantifying the uncertainty that we face in everyday life. This course introduces basic ideas in discrete probability and explores a wide range of practical applications such as evaluating medical diagnostic tests, courtroom evidence, and data from surveys. We will use algebra as a problem-solving tool throughout this course. Prerequisite: passing score on Part I of the Mathematics Diagnostic.

Communications (and Miscommunications) using Mathematics Mathematics 122
This course introduces the mathematics behind everyday communications, from the mass media to cellphones. Topics covered include cryptography used in secure web sites, elements of sound and image analysis used in MP3 players and digital cameras, and accurately understanding media reporting on topics such as health, science, and politics. Prerequisite: Precalculus or the equivalent.

Statistics for Everyday Life Mathematics 123
Statistics is everywhere these days. It is used in the stock market, in weather forecasting, in medical studies, by insurance companies, in quality testing, and in many other areas. This course will introduce core ideas in statistical reasoning to enable you to make sense of and (in)validate the statistics you encounter in the media, in your classes, and in everyday life. Prerequisite: Precalculus or the equivalent.

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: passing score on Part I of the Mathematics Diagnostic.

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: Precalculus or the equivalent.

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.

Voting Theory Mathematics 136
Who should have won the 2000 Presidential Election? Do any two senators really have equal power in passing legislation? How can marital assets be divided fairly? While these questions are of interest to many social scientists, a mathematical perspective can offer a quantitative analysis of issues like these and more. In this course, we will discuss the advantages and disadvantages of various types of voting systems and show that, in fact, any such system is flawed. We will also examine a quantitative definition of power and the principles behind fair division. Prerequisite: Mathematics 110 or the equivalent.

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 techniques of integration, l'Hopital's rule, improper integrals, applications of integration, functions of several variables, partial derivatives, multiple integrals. Prerequisite: Mathematics 141 or the equivalent.

The Mathematics of String Theory Mathematics 191
This class will introduce the mathematical ideas underlying string theory, a theory of particle physics which supposes that the fundamental constituents of matter and energy are not points, but rather tiny strings or loops. No prior background in physics is required. We will read and discuss several popular books on string theory as a means to motivate concepts from the entire mathematical curriculum. These ideas include: complex numbers, quaternions, and their generalizations; Taylor series and Fourier series; spaces of dimensions 4 and higher; the geometry and topology of two dimensional surfaces; non-commutative algebra; and supersymmetry. Prerequisites: Mathematics 141 or the equivalent.


200 Level Courses

Introduction to Differential Equations Mathematics 211
No longer offered. Replaced by Mathematics 213.

Calculus III Mathematics 212
No longer offered. Replaced by Mathematics 241.

Linear Algebra with Ordinary Differential Equations Mathematics 213
This course is an introduction to two fields of mathematics, linear algebra and ordinary differential equations, that are of fundamental importance throughout mathematics and its applications, and that are related by the important use of linear algebra in the study of systems of linear differential equations. Topics in linear algebra include n-dimensional Euclidean space, vectors, matrices, systems of linear equations, determinants, eigenvalues and eigenvectors; topics in ordinary differential equations include graphical methods, separable differential equations, higher order linear differential equations, systems of linear differential equations and applications. Prerequisite: Mathematics 142 or the equivalent.

Vector Calculus Mathematics 241
This course investigates differentiation and integration of vector-valued functions, and related topics in calculus. Topics covered include vector-valued functions, equations for lines and planes, gradients, the chain rule, change of variables for multiple integrals, line integrals, Greenís theorem, Stokesí theorem, the Divergence theorem, and power series. Prerequisite: Mathematics 142 and either Mathematics 213 or Mathematics 242, or the equivalent.

Elementary Linear Algebra Mathematics 242
No longer offered. Replaced by Mathematics 213.

Proofs and Fundamentals Mathematics 261
This course introduces students 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.

Problem Solving Seminar Mathematics 299
This course introduces problem solving techniques used throughout the mathematics curriculum. The course focuses on solving difficult problems stated in terms of elementary combinatorics, geometry, algebra, and calculus. Each class combines a lecture describing the common tricks and techniques used in a particular field, together with a problem session where the students work together using those techniques to tackle some particularly challenging problems. Students may find this class helpful in preparing for the Putnam Exam, a national college mathematics competition given in early December. Prerequisite: Any 200-level mathematics course or permission of the instructor.


300 Level Courses

Numerical Analysis Lab Mathematics 301
This lab is an introduction to mathematical computation. It will start with tutorials on the two software packages that we will be using: Mathematica and its new open source alternative, Sage. The course will then discuss algorithms for finding the zeros of non-linear functions, solving linear systems quickly, and approximating eigenvalues. The bulk of the course will be devoted to curve fitting by means of polynomial interpolation, splines, bezier curves, and least squares. Corequisites: Mathematics 242 and any Computer Science course or basic programming experience.

Computational Geometry Mathematics 303
This class will cover a variety of topics from computational geometry. Focus will be given to computational complexity of the algorithms presented as well as appropriate data structures. Topics may include Voronoi Diagrams, convex hull calculations, line segment intersections and more. Prerequisites: Mathematics 212, Mathematics 242 and some programming knowledge.

Advanced Calculus Mathematics 312
This is a proofs-based introduction to the theory of numbers and covers the fundamentals of quadratic number fields. Topics include factorization, class group, unit group, Diophantine approximation, zeta functions, and applications to cryptography. Prerequisite: Mathematics 212 or permission of the instructor.

Modeling Realizable Phenomena Mathematics 314
In nearly every aspect of human knowledge--conceptual, qualitative, statistical, analytic, numerical--modeling plays a prominent role. Modeling strategies vary widely depending on the phenomenon being studied, the nature and type of the salient state variables, the problem's dimensions, and additional simplifying constraints. In this course, we will learn some modeling approaches and styles, considering both situations when the governing equations are unknown, and when the governing equations are known but are imperfect. Topics may include statistical modeling, timeseries, spatial modeling and kriging, modeling in terms of special complete sets, auto- and cross-correlation functions, Markov chains, differential equations, data assimilation and combining data with models. Prerequisite: Mathematics 212 and Mathematics 242.

Combinatorics Mathematics 316
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.

Graph Theory Mathematics 317
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. Several applications of graph theory are discussed in depth. Prerequisite: Mathematics 261 or permission of the instructor.

Number Theory Mathematics 318
This is a proofs-based introduction to the theory of numbers and covers the fundamentals of quadratic number fields. Topics include factorization, class group, unit group, Diophantine approximation, zeta functions, and applications to cryptography. Prerequisite: Mathematics 261.

Partial Differential Equations Mathematics 321
This course is an introduction to the theory of partial differential equations. The primary focus is the derivation and solutions of the main examples in the subject rather than on the existence and uniqueness theorems and higher analysis. Topics include hyperbolic and elliptic equations in several variables, Dirichlet problems, the Fourier and Laplace transform, Green's functions. Prerequisite: Mathematics 211.

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.

Dynamical Systems Mathematics 323
An introduction to the theory of discrete dynamical systems. Topics to be covered include iterated functions, bifurcations, chaos, fractals and fractal dimension, complex functions, Julia sets, and the Mandelbrot set. We will make extensive use of computers to model the behavior of dynamical systems. Prerequisite: Mathematics 213, or Mathematics 212 and Mathematics 242.

Probability Mathematics 328
A calculus-based introduction to probability with an emphasis on computation and applications. Topics include continuous and discrete random variables, combinatorial methods, conditional probability, joint distributions, expectation, variance, covariance, laws of large numbers, and the Central Limit Theorem. Students will gain practical experience using mathematical software to run probability simulations. Prerequisite: Mathematics 212 or Mathematics 241, or permission of the instructor.

Abstract Algebra Mathematics 332
An introduction to modern abstract algebraic systems. The structures of groups, rings, and fields are studied together with the homomorphisms of these objects. Topics include equivalence relations, finite groups, group actions, integral domains, polynomial rings, and finite fields. Prerequisite: Mathematics 261, and Mathematics 213 or Mathematics 242, or permission of the instructor.

Abstract Algebra II Mathematics 333
This course is a continuation of Mathematics 332. After a review of group theory, we will cover selected topics from ring, field and module theory. The second half of the course will include applications to Representation Theory and/or Galois Theory, as time permits. Prerequisite: Mathematics 332.

Advanced Linear Algebra Mathematics 335
This course covers the advanced theory of abstract vector spaces over arbitrary fields. It will start with a discussion of dual spaces, direct sums, quotients, tensor products, spaces of homomorphisms and endomorphisms, inner product spaces, and quadratic forms. It will then move on to multilinear algebra, discussing symmetric and exterior powers, before turning to the Jordan canonical form and related topics. Other more advanced topics may include Hilbert spaces, modules, algebras, and matrix Lie groups. Prerequisite: Mathematics 342. Co-requisite: Mathematics 332.

Coding Theory Mathematics 340
he digital transmission of information is considered to be extremely reliable, and yet it suffers from the same sorts of interference, corruption, and data loss that plague analog transmission. The reliability of digital transmission comes from the use of sophisticated techniques that encode the digital data so that errors can be easily detected and corrected. This theory of error correcting codes, while having broad applications ranging from CDs to the internet to high definition television, requires some surprisingly beautiful mathematics. We will interpret strings of bits as vectors in an abstract vector space, which allows us to manipulate binary data using linear algebra over finite fields. This class will introduce students to the basics of error correcting codes, as well as touching on the mathematics of data compression and encryption. If time permits, we will also discuss quantum error correction. Although this course will mention encryption, the emphasis will NOT be on cryptography. This course will not involve any programming. Prerequisites: Mathematics 242 and either Mathematics 261 or CMSC 242 (Discrete Mathematics).

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 261 or permission of the instructor.

Differential Geometry Mathematics 352
This course explores the mathematics of curved spaces, particularly curved surfaces embedded in three-dimensional Euclidean space. Originally developed to study the surface of the earth, differential geometry is an active area of research, and it is fundamental to physics, particularly general relativity. The basic issue is to determine whether a given space is indeed curved, and if so, to quantitatively measure its curvature using multivariable calculus. This course also introduces geodesics, curves of minimal length. The course culminates with the Gauss-Bonnet theorem, giving a link between the geometry and topology of surfaces. Prerequisite: Mathematics 212 and Mathematics 261, or permission of the instructor.

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 and one prior 300-level mathematics course is recommended, or permission of the instructor.

Complex Analysis Mathematics 362
This course will cover the basic theory of functions of one complex variable. Topics will include the geometry of complex numbers, holomorphic and harmonic functions, Cauchy's theorem and its consequences, Taylor and Laurent series, singularities, residues, elliptic functions and/or other topics as time permits. Prerequisite: Mathematics 212, Mathematics 261, and one prior 300-level mathematics course is recommended, or permission of the instructor.

Computational Algebraic Geometry Mathematics 384
This course is an introduction to computational algebraic geometry and commutative algebra. We will explore the idea of solving systems of polynomial equations by viewing the solutions to these systems as both algebraic and geometric objects. We will also see how these objects can be manipulated using the Groebner basis algorithm. This course will include a mixture of theory and computation as well as connections to other areas of mathematics and to computer science. Prerequisite: Mathematics 332.


400 Level Courses

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.

Algebraic Number Theory Mathematics 417
In this course we will study algebraic number fields (finite extensions of the rational numbers) from an algebraic and an analytic viewpoint, motivated by the special cases of quadratic and cyclotomic fields. The goal of the course is to develop an understanding of the deep connections between algebra, analysis, and arithmetic. Topics will include: rings of integers, factorization, ideal class group, unit group, zeta and L-functions, Dirichlet's theorem. Prerequisite: Mathematics 332.

Advanced Topics in Abstract Algbra Mathematics 432
This course continues the study of abstract algebra begun in Mathematics 332. Topics are chosen by the instructor, and may include some additional group theory, Galois theory, modules, group representations, and commutative algebra. Prerequisite: Mathematics 332 or permission of the instructor.

Representation Theory Mathematics 436
This course will cover the basic theory of representations of finite groups in characteristic zero: Schur's Lemma, Mashcke's Theorem and complete reducibility, character tables and orthogonality, direct sums and tensor products. The major goals of the course will be to understand the representation theory of the symmetric group and the general linear group over a finite field. If time permits, the theory of Brauer characters and modular representations will be introduced. Corequisites: Mathematics 242 and Mathematics 332.

Modern Geometry Mathematics 453
Geometry is an ancient subject, but it has received a modern makeover in the past two centuries, where the type of geometry now studied is broader than just Euclidean geometry, and where the approach is now analytic rather than axiomatic. In this course we will look at Euclidean, non-Euclidean (hyperbolic and elliptic) and projective geometries, making use of tools from linear algebra and abstract algebra. Prerequisites: Mathematics 242 and Mathematics 332 (which can be taken simultaneously with this course), or permission of instructor.

Knot Theory Mathematics 454
Knot theory is an active branch of contemporary mathematics that, similarly to number theory, involves many problems that are easy to state but difficult to solve; unlike number theory, knot theory involves a lot of visual reasoning. Knot theory is a branch of topology, but it also has applications to aspects of biology, chemistry and physics (though for lack of time these applications will not be discussed in class). This course is an introduction to the theory of knots and links. Topics include methods of knot tabulation, knot diagrams, Reidemeister moves, invariants of knots, knots and surfaces, and knot polynomials. Prerequisites: Mathematics 351 or Mathematics 361, or permission of instructor.

Real Analysis II Mathematics 461
This course continues the study of real analysis begun in Math 361. Topics include functions of several variables, metric spaces, Lebesgue measure and integration, and, time permitting, additional topics such as the Inverse and Implicit Function Theorems, differential forms and Stokesí Theorem. Prerequisites: Mathematics 361.