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## Bard College Catalogue 2023-24

## Mathematics

### Faculty

Ethan Bloch (director), John Cullinan, Charles Doran, Mark D. Halsey, Mary C. Krembs (MAT), Caitlin Leverson, Stefan M. Mendez-Diez, Daniel Newsome, Lauren Rose, 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 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 255, *Vector Calculus* (or Physics 221, *Mathematical Methods *I); and Mathematics 261, *Proofs and Fundamentals*. By graduation, a student must also have completed: Mathematics 332, *Abstract Algebr*a; Mathematics 361,* Real Analysi*s; 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

- “Coding Theory and Supersymmetry”
- “Does Bias Have Shape? An Examination of the Feasibility of Algorithmic Detection of Unfair Bias Using Topological Data Analysis”
- “An Exploration on Condorcet-Approval-Range Voting Function with Limits”
- “Juggling: Siteswap and the Symmetric Group”

### Courses

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

**Mathematics for the Public Good**

*Mathematics 100*

DESIGNATED: OSUN COURSE

The beauty and power of mathematics is not limited to its applications in the natural sciences. Mathematics explains the mechanisms of modern democratic societies: from voting systems to gerrymandering; “fair division” of resources to biased algorithms governing the internet; and from the security of blockchain to the prying eyes of data mining AIS. This course introduces each of these topics and illustrates the mathematics behind them. No specific mathematical background is assumed, just a curiosity about what is going on “under the hood.”

**Quadrivium: Mathematics and Metaphysics**

*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**

*Mathematics 105*

CROSS-LISTED: PHILOSOPHY

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. For each model presented, the implications of the model as well as its limitations are discussed. Students are actively involved in the modeling process. While there is no particular mathematical prerequisite, students do algebraic computations from time to time and discuss deductive proofs of some of the results.

**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.

**Mathematics of Puzzles and Games**

*Mathematics 116*

DESIGNATED: ELAS COURSE

This course develops the mathematics of puzzles and games as a means to solve a puzzle or win a game, and also as a fun way to learn and develop mathematical skills. The focus is on the mathematics and strategies behind Rubik’s Cube, SET, Nim, Hex, and Sudoku. ELAS activities include guest presenters, a trip to the National Museum of Mathematics, and participation in game sessions for local K–12 students and community members.

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

**Calculus I**

*Mathematics 141*

The basic ideas of differentiation and integration of functions in 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, and series and power series.

*Prerequisite:*Mathematics 141 or the equivalent.

**Racial Disparities in Mathematics**

*Mathematics 199*

DESIGNATED: ELAS+ COURSE

In light of the recent Black Lives Matter protests, this 2-credit, P/F course begins by exploring the anti-Black narratives that exist in math textbooks and departments throughout the United States. It ultimately hopes to develop skills and strategies to dismantle the existing biases as students proceed into STEM careers. Participants write chapters and lesson plans based on math topics of their choice that incorporate ideas of racial inequality and injustices. Prerequisite: Mathematics 141 or permission of the instructor.

**Mathematics Seminar**

Mathematics 208

This 1-credit course introduces a broad range of ideas in modern mathematics through weekly talks by visiting speakers and Bard faculty. The talks may address subjects of current research interest as well as topics not typically covered in the Mathematics Program at Bard

*. Prerequisite:*Mathematics 142 or the equivalent.

**Introduction to Differential Equations**

*Mathematics 211*

The course is organized around methods for solving ordinary differential equations and incorporates many ideas from calculus. 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, including integrating factors, Laplace transforms, and power series.

*Prerequisites:*Mathematics 141 and 142, or the equivalent.

**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.

**Vector Calculus**

*Mathematics 255*

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.

**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.

**Discrete and Computational Geometry**

*Mathematics 313*

Discrete and computational geometry, which has applications in areas such as pattern recognition, image processing, computer graphics, and terrain modeling, is the study of geometric constructs in two- and three-dimensional space that arise from finite sets of points. Topics covered include convex hull, Delaunay triangulations, Voronoi diagrams, curve reconstruction, and polyhedra. Work involves both traditional proofs and implementation of algorithms via the computer system Sage, which is based upon Python.

*Prerequisites:*Mathematics 261 or Computer Science 145, and some programming experience.

**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.

*Prerequisite:*Mathematics 244 or 255, 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 261 or permission of the instructor.

**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 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.

**Point-Set Topology**

*Mathematics 351*

Topics addressed include topological spaces, metric spaces, compactness, connectedness, continuity, homomorphisms, separation criteria, an introduction to knots, and, possibly, the fundamental group.

*Prerequisite:*Mathematics 261 or permission of the instructor; one prior 300-level Mathematics course is recommended.

**Differential Geometry**

*Mathematics 352*

This course uses methods from multivariable calculus to study the geometry of curves and surfaces in three dimensions. Topics covered include curvature and torsion of curves, geometry of surfaces, geodesics, spherical and hyperbolic geometry, minimal surfaces, Gaussian curvature, and the Gauss-Bonnet theorem. Time permitting, applications to subjects such as cartography and navigation, shapes of soap bubbles, computer graphics, image processing, and general relativity are also discussed.

*Prerequisite:*Mathematics 241.

**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.

**Algebraic Topology**

*Mathematics 451*

A look at topics in algebraic topology, which is the study of how to use algebraic methods to study the shapes of spaces. Possible topics include homotopy, the fundamental group, covering spaces, simplicial complexes, simplicial homology, and knot groups.

*Prerequisite:*Mathematics 332 or permission of the instructor.