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## Bard College Catalogue, 2019–20

## Mathematics

### Faculty

John Cullinan (director), Ethan Bloch, Matthew Junge, Mary C. Krembs (MAT), Stefan M. Mendez-Diez, Lauren Rose, Silvia Saccon, Steven Simon, Japheth Wood

### Overview

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.

### Requirements

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 242, *Elementary Linear Algebra*, or Mathematics 245, *Intermediate Calculus* (or Physics 221, *Mathematical Methods I*); and Mathematics 261, *Proofs and Fundamentals*. By graduation, a student must have completed: Mathematics 242, *Elementary Linear Algebra*; Mathematics 245, I*ntermediate Calculus* (or Physics 221, *Mathematical Methods I*); Mathematics 332, *Abstract Algebra*; Mathematics 361, *Real Analysis*; 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

- “Applications of Equivariant Topology in Cascading Makeev Problems”
- “Centroidal Voronoi Tessellations with Few Generator Points”
- “Exploring Sequences of Tournament Graphs with Draws”
- “The Mathematics of Cancer: Fitting the Gompertz Equation to Tumor Growth”

### Courses

*The following descriptions represent a sampling of courses from the past four years.*

**Elementary Statistics**

*Mathematics 102*

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.

**Quadrivium: Mathematics and Metaphysics in the Premodern World**

*Mathematics 103*

What does the “music of the spheres” sound like? What influence did astrology have on mathematics? Why does Newton’s rainbow have seven colors? These questions are addressed by the quadrivium, a term coined by the sixth-century philosopher Boethius for the mathematical program of the medieval university. This course explores how mathematics was seen as the structure of everything. Readings from Plato, Euclid, Boethius, Ptolemy, al Khwarizmi, Fibonacci, Oresme, Kepler, others.

*Prerequisite:*passing score on Part I of the Mathematics Diagnostic.

**Data and Decisions**

*Mathematics 104*

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.

**Time, Space, and Infinity: Mathematical Perspectives on Philosophical Paradoxes**

*Mathematics 105*

If time is composed of moments with zero duration, is change an illusion? Beginning with Zeno’s ancient paradoxes, fundamental problems on the nature of time and space—and related ones regarding infinity—have bedeviled thinkers throughout the contemporary period. This course provides a beginner-friendly tour of some of mathematics’ most profound discoveries (irrational numbers, limits, uncountability) and the concerns that arise in answering such intractable questions.

*Prerequisite:*Passing score on Part I of the Mathematics Diagnostic.

**Mathematics and Politics**

*Mathematics 106*

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.

**Chance, Skill, and Uncertainty**

*Mathematics 107*

Some of the most sought-after numbers are the probabilities of future events—their values underlie the profits of insurance companies and casinos, while incorrectly assessing them can contribute to medical misdiagnosis, wrongful incarceration, and financial crashes. This course touches on selected probability paradoxes and philosophical interpretations, along with physical theories of information, entropy, and quantum mechanics that illuminate the question of what probabilities signify.

*Prerequisite:*passing score on Part I of the Mathematics Diagnostic.

**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 also addressed.

*Prerequisite:*passing score on Part I of the Mathematics Diagnostic.

**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.

**Calculus I**

*Mathematics 141*

The basic ideas of differentiation and integration of functions of one variable are discussed. 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 reinforces the fundamental ideas of the derivative and definite integral. Topics include integration techniques, L’Hôpital’s rule, improper integrals, volumes, arc length, sequences and series, power series, continuous random variables, and separable differential equations. Prerequisite: Mathematics 141 or the equivalent.

**Mathematics Seminar**

*Mathematics 208*

Students are exposed to a broad range of ideas in modern mathematics through a series of weekly talks by visiting speakers and Bard faculty. The talks address subjects of current research interest or topics not typically covered in Bard mathematics courses. Graded pass/fail.

*Prerequisite:*at least one 200-level mathematics course.

**Problem Solving, Engagement, and the Culture of Mathematics**

*Mathematics 209*

In this 2-credit course, students investigate problem-solving techniques in mathematics, and use them to develop an activity or project that involves mathematical reasoning, analytical thinking, and open-ended exploration. The course also addresses the culture of mathematics, looking at the factors that lead to lower rates of participation by women and minorities in mathematics and other STEM fields. The class visits local schools and the National Museum of Mathematics.

*Prerequisite:*Mathematics 142 or the equivalent.

**Linear Algebra with Ordinary Differential Equations**

*Mathematics 213*

Topics 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, Green’s theorem, Stokes’s theorem, the divergence theorem, and power series.

*Prerequisite:*Mathematics 213.

**Elementary Linear Algebra**

*Mathematics 242*

The 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.

*Prerequisite:*Mathematics 142 or permission of the instructor.

**Intermediate Calculus**

*Mathematics 245*

A continuation of the study of calculus begun in Mathematics 141-142. Topics covered include double and triple integrals in curvilinear coordinates, sequences and series, power series, and an introduction to ordinary differential equations.

*Prerequisite:*Mathematics 142 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.

**Scientific Computing**

*Mathematics 301*

The course explores the theory of numerical computation, as well as how to utilize the theory to solve real problems using the computer software package MATLAB. Students learn how to use MATLAB by solving eigenvalue problems before moving on to curve fitting using least squares and polynomial interpolation, among other numerical methods for solving differential equations.

*Prerequisites:*Mathematics 213 and 241, and either Computer Science 143 or Physics 221.

**Computational Geometry**

*Mathematics 303*

Computational geometry is a branch of mathematics and computer science devoted to the study of algorithms and the appropriate data structures to solve geometric problems on (often large) data sets. This course focuses on combinatorial computational geometry, also called algorithmic geometry. Topics may include Voronoi diagrams, convex hull calculations, and line segment intersections.

*Prerequisites:*Mathematics 213, 241, and either Mathematics 261 or Computer Science 201.

**Advanced Calculus**

*Mathematics 312*

This course treats the differential and integral calculus of several variables from an advanced perspective. Topics may include the derivative as a linear transformation, change of variables for multiple integrals, parametrizations of curves and surfaces, line and surface integrals, Green’s theorem, Stokes’s theorem, the divergence theorem, manifolds, tensors, differential forms, and applications to probability and the physical sciences.

*Prerequisite:*Mathematics 241 or Physics 222, or permission of the instructor.

**Mathematical Modeling**

*Mathematics 314*

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**

*Mathematics 315*

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.

**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.

**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.

**Number Theory**

*Mathematics 318*

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.

**Differential Equations**

*Mathematics 321*

An introduction to the theory of partial differential equations, with a focus on 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, and numerical and approximation methods.

*Prerequisites:*Mathematics 213 and 241, or permission of the instructor.

**Operations Research**

*Mathematics 322*

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.

**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.

**Geometry**

*Mathematics 325*

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.

**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 gain practical experience using mathematical software to run probability simulations.

*Prerequisite:*Mathematics 213.

**Mathematical Statistics**

*Mathematics 329*

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.

**Abstract Linear Algebra**

*Mathematics 331*

The main focus of this course is the study of vector spaces and linear maps between vector spaces. Topics covered also include linear independence, bases, dimension, linear maps, isomorphisms, matrix representations of linear maps, determinants, eigenvalues, inner product spaces, and diagonalizability.

*Prerequisites:*Mathematics 213 or 242, Mathematics 261, or permission of the instructor.

**Abstract Algebra**

*Mathematics 332*

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.

**Real Analysis**

*Mathematics 361*

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.

**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.

**Galois Theory**

*Mathematics 430*

The goal of this course is to develop the Galois theory of fields, a theory that connects groups, rings, fields, and vector spaces in the study of roots of polynomials. Toward that end, the class develops the theory of field extensions, including algebraic extensions, automorphisms of fields, splitting fields, and separable extensions, with some surprising applications.

*Prerequisite:*Mathematics 332.

**Real Analysis II**

*Mathematics 461*

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.