Bard College Catalogue 2012-13
Samuel K. Hsiao (director), Jules Albertini, James Belk, Maria Belk, Ethan Bloch, John Cullinan, Jennie D’Ambroise, Mark D, Halsey, Gregory Landweber*, Lauren Rose, Branden Stone
* on sabbatical, fall 2012
OverviewMathematics is at the core of human civilization and is the cornerstone of all modern science and technology. The Mathematics Program 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.
The 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, Elementary 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
- “A Structure Theorem for Plesken Lie Algebras over Finite Fields”
- “Classification of Adinkra Graphs”
- “Enumerating faces of Zonohedra”
- “Modeling Origami Folding with Thick Paper”
- “Voronoi Diagrams with Non-Linear Bisectors”
CoursesIn addition to the core and elective courses, the Mathematics Program offers tutorials in advanced topics.
Mathematics and Politics
This course considers 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. The implications of each model presented, as well as the limitations of the model, are discussed. There is no mathematical prerequisite, but the course includes some algebraic computations and discussion of deductive proofs of the main results.
Topics in Geometrical Mathematics
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.
Introduction to Mathematical Modeling
Mathematical modeling is the process of using mathematics to describe and solve problems about real-world scenarios. A mathematical model is a representation of a particular phenomenon using structures such as graphs, equations, or algorithms. This course presents the skills used in creating, interpreting, and using mathematical models to solve real-world problems. Precise writing as well as careful use of algebraic manipulations is stressed.
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.
The mathematical theory of probability is useful for quantifying the uncertainty 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. The course uses algebra as a problem-solving tool. Prerequisite: passing score on Part I of the Mathematics Diagnostic.
Communications (and Miscommunications) Using Math
This course introduces the math behind everyday communications, from mass media to cell phones. Topics covered include cryptography, as used in secure websites; and elements of sound and image analysis used in MP3 players and digital cameras. Prerequisite: Mathematics 110 or the equivalent.
Statistics for Everyday Life
Statistics is used in the stock market, weather forecasting, medical studies by insurance companies, and quality testing. This course introduces core ideas in statistical reasoning to enable students to make sense of the statistics they encounter in the media, in their classes, and in everyday life. Prerequisite: precalculus or the equivalent.
Exploration in Number Theory
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 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.
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.
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.
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.
Mathematical Models in Biology
An introduction to common approaches for dynamic modeling in biology, including difference equations, matrix algebra, and simulation. The main focus is on population and disease models, but there is some flexibility to explore other types of biological examples. Students learn the mathematical ideas involved in constructing and analyzing the models, and conduct computer simulations using Matlab. Prerequisite: one year of calculus.
Introduction to Differential Equations
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.
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.
Linear Algebra with Ordinary 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. Prerequisites: Mathematics 142 or the equivalent.
Elementary Linear Algebra
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
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.
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
cross-listed: computer science
An introduction to mathematical computation. After reviewing Taylor series and introducing algorithms for finding the zeros of nonlinear functions, solving linear systems quickly, and approximating eigenvalues, the course is devoted to curve fitting by means of polynomial interpolation, splines, bezier curves, and least squares. Other topics: matrix factorizations, the PageRank algorithm, sparse matrices, and vector processing. Corequisites: Mathematics 213 or 242, and any computer science course or basic programming experience.
Mathematics 303 / Computer Science 303
The focus of this class is 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.
This course treats the differential and integral calculus of several variables from an advanced perspective. Students are expected to be familiar with the fundamentals of multivariate calculus from Mathematics 212. Topics include curvilinear coordinates, change of variables for multiple integrals, Stokes’ theorem, divergence theorem, Fourier series and transform, and applications to probability and the physical sciences. Prerequisite: Mathematics 212 or permission of the instructor.
Data Analysis: Getting the Extra Rigor
This course provides the computational, algebraic, and statistical tools needed to understand and make contributions in empirical science. The main focus is multidimensional data—data that typically are a function of space and time. After a solid linear algebra review, topics covered comprise covariance and cross-covariance functions and matrices, spanning sets, spectral representations and truncations, discrete vs. continuous spectra and the real number continuum, singular value decomposition, and Monte Carlo techniques. Prerequisites: Mathematics 212 and 242.
Modeling Realizable Phenomena
Modeling plays a prominent role in nearly every aspect of human knowledge—conceptual, qualitative, statistical, analytic, numerical. This course explores a variety of modeling approaches and styles. Topics may include statistical modeling, time series, 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. Prerequisites: Mathematics 212 and 242.
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 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
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
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.
The study of techniques for finding optimal solutions to complex decision-making problems. The course tries to answer questions such as how to schedule classes with a limited number of classrooms on campus, how to determine a diet that is both rich in nutrients and low in calories, or how to create an investment portfolio that meets investment needs. Techniques covered include linear programming, network flows, integer/combinatorial optimization, and nonlinear programming. Prerequisites: Mathematics 212 and 242.
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.
Fourier Analysis and Wavelets
Recently, signal processing has gone through a mathematical revolution. Traditionally, it was built on the Fourier transform, a tool used to express signals as superpositions of pure sinusoidal functions. While the Fourier transform is suited to understanding physical phenomena, such as waves, it lacks the flexibility to analyze more complicated functions. A new tool, the wavelet transform, has become the staple of many signal processing tasks. This course introduces the mathematical foundations of the Fourier and the wavelet transforms, with excursions into signal processing. Prerequisites: Mathematics 212 and Mathematics 213 or Mathematics 242.
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.
Advanced Linear Algebra
This course starts with a discussion of dual spaces, direct sums, quotients, tensor products, spaces of homomorphisms and endomorphisms, inner product spaces, and quadratic forms. It then moves 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 242; corequisite: Mathematics 332.
The digital transmission of information is considered extremely reliable, although it suffers the same sorts of corruption and data loss that plague analog transmission. Digital reliability comes from sophisticated techniques that encode data so that errors can be easily detected and corrected. These error-correcting codes require surprisingly beautiful mathematics. This class introduces the basics of error-correcting codes, as well as the mathematics of data compression and encryption. Prerequisites: Mathematics 242 and either Mathematics 261 or Computer Science 145.
Point Set Topology
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.
This course uses methods from multivariable calculus to study the geometry of curves and surfaces in three dimensions. Topics include curvature and torsion of curves, geometry of surfaces, geodesics, spherical and hyperbolic geometry, minimal surfaces, Gaussian curvature, and the Gauss-Bonnet theorem. Prerequisites: Mathematics 212, 242, and 261, or permission of the instructor.
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.
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.
Computational Algebraic Geometry
This introduction to computational algebraic geometry and commutative algebra explores the idea of solving systems of polynomial equations by viewing the solutions to these systems as both algebraic and geometric objects. Students learn how these objects can be manipulated using the Groebner basis algorithm. The course includes a mixture of theory and computation as well as connections to other areas of mathematics and to computer science. Prerequisite: Mathematics 332.
Topics include first-order logic, completeness and compactness theorems, model theory, nonstandard analysis, decidability and undecidability, incompleteness, and Turing machines. Prerequisite: Mathematics 332.
Advanced Topics in Abstract Algebra
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 332 or permission of the instructor.
This course looks 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 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.