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Bard College Catalogue 2009-2010
2009-2010
Mathematics
http://math.bard.edu
Faculty: Ethan D. Bloch (director), James M. Belk, Maria Belk, John Cullinan, Clíona Golden, Mark D. Halsey, Samuel K. Hsiao*, Mary Krembs, Gregory Landweber, Lauren Lynn Rose**
* on sabbatical, fall 2009
** on sabbatical, spring 2010 OverviewMathematics is at the core of human civilization and is the cornerstone of all modern science and technology. The Mathematics Program at Bard has three main functions: to provide students in the program with the opportunity to study the primary areas of contemporary mathematics, to provide physical and social science majors with the necessary mathematical tools for work in their disciplines, and to introduce all students to serious and interesting mathematical ideas and their applications.RequirementsThe program requirements are flexible enough to allow a student to prepare for graduate study in mathematics, professional schools (such as medical or law), or employment in the public or private sector. Students in the program are expected to follow the standard divisional procedure for Moderation and to fulfill the collegewide distribution and First-Year Seminar requirements.
By the time of Moderation a student in the program should have taken (or be taking) these courses or their equivalents: Mathematics 141, Calculus I; Mathematics 142, Calculus II; Mathematics 212, Calculus III; and Mathematics 261, Proofs and Fundamentals. By graduation, a student must have completed: Mathematics 242 or 331, Linear Algebra with Applications or Linear Algebra; Mathematics 332, Abstract Algebra; Mathematics 361, Real Analysis; at least two other mathematics courses numbered 300 or above; a computer science course, preferably before beginning the Senior Project; and the Senior Project. Recent Senior Projects in Mathematics: - “Combinatorial Laplacians of Simplicial Complexes”
- “Forbidden Subgraphs for Edge Domination”
- “Hilbert Sequences of Monomial Ideals”
- “On the Curvature of Polyhedra”
- “The Matching Model for Routing Permutations on Graphs”
CoursesIn addition to the core and elective courses, the Mathematics Program offers tutorials in advanced topics. A sampling of tutorials from the past few years includes Advanced Abstract Algebra, Differential Geometry, Mathematical Logic, Number Theory, Probability, and Topology.
The Mathematics of Chance
Mathematics 102
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, Nature, Science, or Scientific American. 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.
Topics in Geometrical Mathematics
Mathematics 107
Geometrical mathematics involves many topics other than traditional Euclidean geometry, including symmetry, groups, frieze and wallpaper patterns, graphs, surfaces, knots, and higher dimensions. Prerequisite: eligibility for Q courses.
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. Prerequisites: eligibility for Q courses and satisfactory performance on the precalculus entrance exam.
Exploration in Number Theory
Mathematics 131
An overview of one of the oldest areas of mathematics that 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 permission of the instructor.
Game Theory
Mathematics 135
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? A mathematical perspective can offer a quantitative analysis of issues like these and more. This course considers the advantages and disadvantages of various types of voting systems and shows that, in fact, any such system is flawed. 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 L’Hôpital’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.
String Theory
Mathematics 191
An introduction to the mathematical ideas underlying string theory, a theory of particle physics that supposes the fundamental constituents of matter and energy are not points, but rather tiny strings or loops. No prior background in physics is required. Prerequisite: Mathematics 141 or the equivalent.
Introduction to Differential Equations
Mathematics 211
cross-listed: cognitive science
Topics include the classification of differential equations; determining the 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.
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’s theorem. Prerequisites: Mathematics 141 and 142 or the equivalent.
Discrete Mathematics
Mathematics 235
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. Prerequisite: one semester of calculus or permission of the instructor.
Linear Algebra with Applications
Mathematics 242
This course covers the basics of linear algebra in n-dimensional Euclidean space, including vectors, matrices, systems of linear equations, determinants, and 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.
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.
Problem Solving
Mathematics 299
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 with a problem session in which students work together using those techniques to tackle some particularly challenging problems. Prerequisite: any 200-level mathematics course or permission of the instructor.
Numerical Analysis Lab
Mathematics 301
cross-listed: computer science
This lab starts with tutorials on the software package Mathematica and its open-source alternative, SAGE. It then discusses algorithms for finding the zeros of nonlinear functions, solving linear systems quickly, and approximating eigenvalues. The bulk of the course is devoted to curve fitting by means of polynomial interpolation, splines, Bézier curves, and least squares. Prerequisites: Mathematics 242 and any computer science course or basic programming experience.
Enumerative Combinatorics
Mathematics 302
This course develops the basic methods of enumeration, which include elementary counting techniques, the inclusion-exclusion principle, and generating functions. Students apply these counting methods to fundamental combinatorial structures such as trees and permutations.
Computational Geometry
Mathematics 303 / Computer Science 303
This class covers a variety of topics from computational geometry, with a focus on the computational complexity of the algorithms presented and appropriate data structures. Topics may include Voronoi diagrams, convex hull calculations, and line-segment intersections. Prerequisites: Mathematics 212 and 242, and some programming knowledge.
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. Topics discussed include 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. Prerequisite: Mathematics 261 or permission of the instructor.
Probability and Statistics
Mathematics 319
cross-listed: economics
Every day we make decisions based on numerical data in the face of uncertainty. We do so while reading the latest political polls, playing a card game, or analyzing a scientific experiment. Probabilistic models and statistical methods help us think through such decisions in a precise mathematical fashion. This course provides a calculus-based introduction to the techniques and applications of probability and statistics. Applications are selected from the natural and social sciences. Prerequisite: Mathematics 142 or the equivalent.
Partial Differential Equations
Mathematics 321
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, and Green’s functions.
Dynamical Systems
Mathematics 323
An introduction to the theory of discrete dynamical systems. Topics covered include iterated functions, bifurcations, chaos, fractals and fractal dimension, complex functions, Julia sets, and the Mandelbrot set. The class makes extensive use of computers to model the behavior of dynamical systems. Prerequisite: Mathematics 261 or permission of the instructor.
Linear Algebra
Mathematics 331
An introduction to the theory of abstract vector spaces, a useful concept when studying physical phenomena. Topics include linear independence and dependence, bases and dimension, linear transformations, eigenvalues, eigenvectors, diagonalization, inner product spaces, and orthogonality. Prerequisite: Mathematics 261 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 or permission of the instructor.
Point Set Topology
Mathematics 351
Topics addressed include topological spaces, metric spaces, compactness, connectedness, continuity, homomorphisms, separation criteria, and, possibly, the fundamental group. Prerequisite: Mathematics 361 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 that 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. Prerequisites: Mathematics 212 and 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 or permission of the instructor.
Complex Analysis
Mathematics 362
This course covers the basic theory of functions of one complex variable. Topics 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 other topics as time permits. Prerequisite: Mathematics 361 or permission of the instructor.
Research Seminar in Computer Science and Mathematics
Mathematics 408 / Computer Science 408
See Computer Science 408 for description.
Advanced Algebra
Mathematics 432
A continuation of Mathematics 332. The primary goal is to develop the Galois theory of fields. Students explore the theory of field extensions, including algebraic extensions, automorphisms of fields, splitting fields, and separable extensions. As time permits, students may develop some topics in advanced group theory. Prerequisites: Mathematics 331 and 332, or permission of the instructor.
Advanced Linear Algebra
Mathematics 433
This course covers the rigorous theory of abstract vector spaces over arbitrary fields. It starts with a discussion of dual spaces, direct sums, tensor products, spaces of homomorphisms and endomorphisms, inner product spaces, and quadratic forms. Also addressed are multilinear algebra, the Jordan canonical form, Hilbert spaces, elementary functional analysis, modules, algebras, symplectic linear algebra, supersymmetry, and K-theory. Prerequisites: Mathematics 261 and 332.
Topics in Algebra and Combinatorics
Mathematics 434
The course covers a selection of topics in algebraic combinatorics and computational algebra, including convex polytopes, simplicial complexes, hyperplane arrangements, Groebner bases, and multivariate splines. Prerequisite: Mathematics 331 or 332.
Representation Theory
Mathematics 436
This course covers the basic theory of representations of finite groups in characteristic zero: Schur’s lemma, Maschke’s theorem and complete reducibility, character tables and orthogonality, and direct sums and tensor products. If time permits, the theory of Brauer characters and modular representations will be introduced. Prerequisites: Mathematics 242 and 332.
Knot Theory
Mathematics 454
Knot theory is an active branch of contemporary mathematics that, as in 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. 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, and knot polynomials. Prerequisite: Mathematics 351 or 361, or permission of instructor.
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