Bard College Catalogue 2017-18
John Cullinan (director), Jules Albertini, Amir Barghi, James Belk, Maria Belk, Ethan Bloch, Mark D. Halsey, Mary C. Krembs (MAT), Stefan M. Mendez-Diez, Lauren Rose, Steven Simon, Japheth Wood
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 college-wide 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 213, Linear Algebra with Ordinary Differential Equations; and Mathematics 261, Proofs and Fundamentals. By graduation, a student must have completed: Mathematics 241, Vector Calculus; Mathematics 332, Abstract Algebra; Mathematics 361, Associativity of Binary Operations on the Real Numbers; at least two other math courses numbered 300 or above; a computer science course, preferably before beginning the Senior Project; and the Senior Project.
Recent Senior Projects in Mathematics
- “Associativity of Binary Operations on the Real Numbers”
- “Cutoff for Lamplighter Markov Chains on Prisms”
- “Lose Big, Win Big, and Sum Big: Exploration of Ranked Voting Systems”
- “Orthogonal Projections of Lattice Stick Knots”
CoursesThe following descriptions represent a sampling of courses from the past four years.
An introduction to core ideas in statistics that are needed to make sense of what is found in media outlets, online surveys, and scientific journals. Most concepts are introduced in a case-study fashion; statistical software is used to analyze data and facilitate classroom discussions. Primary goals are to foster statistical reasoning and assist in making informed conclusions about topics involving data. Prerequisite: passing score on Part I of the Mathematics Diagnostic.
Data and Decisions
This course examines applications of mathematics to a number of topics related to data and decision making. Topics are chosen from three relevant areas of mathematics: voting systems, networks, and statistics, all of which involve extracting information from various types of data. No particular mathematical preparation is needed beyond basic algebra and a willingness to explore new ideas, construct convincing arguments, and use a spreadsheet. Prerequisite: passing score on Part I of the Mathematics Diagnostic.
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.
An introduction to cryptology, the science of sending, receiving, and intercepting secret messages. A variety of encryption methods are addressed, including classical substitution and transposition ciphers, as well as more modern methods such as symmetric-key algorithms and public-key cryptography. Though the focus is on the mathematical and computational aspects of encryption and code breaking, the class also discusses the history of secret codes, the role of cryptology in Internet security, and public policy issues related to secure communication and eavesdropping.
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. Prerequisite: passing score on Part I of the Mathematics Diagnostic.
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.
From the neurons in our brains to financial markets, networks are all around us. Network science helps us understand how these complex systems work. This introductory course covers topics including representations of a network as a graph or matrix, network measures, and classification of networks as small world, random, or hierarchical. The class investigates applications in biology, sociology, transportation, ecology, and epidemiology, among other disciplines. Prerequisite: Mathematics 110 or the equivalent.
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 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.
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. Prerequisite: Mathematics 142 or the equivalent.
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, Greene’s theorem, Stokes’ theorem, the divergence theorem, and power series. Prerequisite: Mathematics 213.
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.
Ordinary and Partial Differential Equations
The main focus is on first- and second-order differential equations; higher-order differential equations are also considered. Topics in ordinary differential equations include systems of equations, phase plane portraits of solutions, bifurcations, stability, and existence and uniqueness. Topics in partial differential equations include boundary conditions, and physical applications and classifications of elliptic, parabolic, and hyperbolic equations. Prerequisite: Mathematics 213.
What is a mathematical model? And how can it be used to help solve real-world problems? This course provides students with a solid foundation in modeling and simulation, advancing understanding of how to apply mathematical concepts and theory. Topics may include modeling with Markov chains, Monte Carlo simulation, discrete dynamical systems, differential equations, game theory, network science, and optimization. Prerequisite: Mathematics 213.
Game theory, a mathematical approach to modeling situations of conflict and cooperation, has applications to many fields, including economics, biology, and psychology. This course introduces game theory from a mathematical perspective; topics include mathematical models of games, two-person games, mixed strategies, and Nash equilibria. Additional topics may include continuous games, dynamic games, and stochastic games. Prerequisites: Mathematics 213 and 261.
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.
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.
This proofs-based introduction to the theory of numbers 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.
Operations research is the study of techniques for finding optimal solutions to complex decision-making problems. It 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. Prerequisite: Mathematics 213.
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.
This course samples topics from the geometry of the plane, with a primary emphasis on the synthetic approach to Euclidean geometry. Other approaches (e.g., vector methods) and types of geometry (hyperbolic or projective geometry) are also considered, time permitting. Core topics in Euclidean geometry include axioms, metrics, congruence, similarity, polygons, triangles, and circle. Prerequisites: Mathematics 213 and 261, or permission of the instructor.
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 gain practical experience using mathematical software to run probability simulations. Prerequisite: Mathematics 213.
This course is a calculus-based introduction to statistical theory and applications. Students explore the mathematical ideas underlying common statistical methods and gain experience in analyzing real data. Core topics include estimation, confidence intervals, hypothesis testing, and regression. Additional topics vary by instructor and may include bootstrapping or nonparametric methods. Statistical software is used extensively to perform simulations and data analyses. Prerequisite: Mathematics 328.
An introduction to modern abstract algebraic systems, including groups, rings, fields, and vector spaces. The course focuses primarily on a rigorous treatment of the basic theory of groups (subgroups, quotient groups, homomorphisms, isomorphisms, group actions) and vector spaces (subspaces, bases, dimension, linear maps). Prerequisites: Mathematics 213 and 261, or permission of the instructor.
Philosophy of Mathematics
Mathematics 336 / Philosophy 336
See Philosophy 336 for a full course description.
The Fundamental Theorem of Mathematics
The primary goal of the course is to develop a proof of the fundamental theorem of algebra along an approach initiated by Euler and then refined by Foncenex and Lagrange in the 18th century. Along the way, students encounter topics such as the historical development of algebra, mathematical induction in several forms, Dirichlet’s box principle, ring theory, symmetric polynomials, and Viète’s theorem. Prerequisites: Mathematics 261 and a previous course in abstract algebra, or permission of the instructor.
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 213 and 261, or permission of the instructor.
The class studies the fundamental ideas of analysis in one-dimensional Euclidean space. 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.
Numerical Real Analysis
Topics include the foundations of the real numbers, sequences, series, power series, the derivative, and the Riemann integral. The class also focuses on error estimates for numerical methods of approximating the roots, derivatives, and integrals of real analytic functions, making use of Taylor series and Taylor’s theorem. Prerequisites: Mathematics 261 and one prior 300-level mathematics course, or permission of the instructor.
Designed to help students prepare for a Senior Project in mathematics via a variety of hands-on activities related to reading, doing, and writing mathematics. There are 10 weekly meetings, each devoted to a different topic: reading a mathematics paper; searching the mathematics literature; using LaTeX for writing a Senior Project; using computer programs such as Sage and Mathematica; and expository mathematical writing. The seminar is graded pass/fail.
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.
Real Analysis II
Topics covered in this course, which continues the study of real analysis begun in Mathematics 361, include functions of several variables, metric spaces, Lebesgue measure and integration, and, time permitting, inverse and implicit function theorems, differential forms, and Stokes’s theorem. Prerequisite: Mathematics 361.