Courses

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Mathematics 141, Calculus I

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.

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Mathematics 142, Calculus II

This course, a continuation of Calculus I, reinforces the fundamental ideas of the derivative and the definite integral. Topics covered include techniques of integration, l'Hopital's rule, improper integrals, applications of integration, functions of several variables, partial derivatives, multiple integrals. Prerequisite: Mathematics 141 or the equivalent.

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Mathematics 242, Linear Algebra

This course covers the basics of linear algebra in n-dimensional Euclidean space, including vectors, matrices, systems of linear equations, determinants, 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.

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Mathematics 255, Vector Calculus

This course investigates differentiation and integration of vector-valued functions along with related topics in multivariable calculus. Topics covered include gradient vectors, the chain rule, optimization, change of variables for multiple integrals, line and surface integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. Prerequisite: Mathematics 142 and Mathematics 242, or the equivalent.

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Mathematics 261, Proofs and Fundamentals

This course introduces students 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.

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Mathematics 331, Abstract Linear Algebra

This course is an introduction to the study of linear algebra as an abstract algebraic system.  The main focus of this course is the study of vector spaces, and linear maps between vector spaces.  Topics covered will include vector spaces, linear independence, bases, dimension, linear maps, isomorphisms, matrix representations of linear maps, determinants, eigenvalues, inner product spaces and diagonalizability.  This course satisfies the Abstract Algebra requirement of the Mathematics Program.  Prerequisites: Mathematics 242 and Mathematics 261, or permission of the instructor.

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Mathematics 332, Abstract Algebra

This course is an introduction to modern abstract algebraic systems, specifically groups, rings and fields.  The focus of the course is a rigorous treatment of the basic theory of groups (subgroups, permutation groups, quotient groups, homomorphisms, isomorphisms, group actions), and an introduction to rings and fields (ideals, polynomials, factorization). Prerequisites: Mathematics 242 and Mathematics 261, or permission of the instructor.

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Mathematics 361, Real Analysis

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 is required, and one prior 300-level mathematics course is recommended, or permission of the instructor.

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Mathematics 301, Scientific Computing

This course will explore how to solve continuous problems using numerical methods. Such problems arise in many mathematical applications. We will discuss the theory of numerical computation, as well as how to utilize the theory to solve real problems using the computer software package MATLAB. The course will begin with learning how to use MATLAB by experimenting with its use in solving eigenvalue problems. We will then study curve fitting using least squares and polynomial interpolation, among other methods. We will use these problems to focus on how to optimize our computer code for parallelization. The course will conclude by focusing on numerical methods for solving differential equations. Prerequisites: Mathematics 213 or Mathematics 242, and one of Mathematics 241, Mathematics 245, Computer Science 141, Computer Science 143 or Physics 221, or permission of the instructor.

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Mathematics 312, Advanced Calculus

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' theorem, the divergence theorem, manifolds, tensors, differential forms, and applications to probability and the physical sciences.  Prerequisite: Mathematics 241 or Mathematics 245 or Physics 222 (Mathematical Methods II), or permission of the instructor.

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Mathematics 313, Discrete and Computational Geometry

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. This class will treat fundamental topics in the field, including convex hull, Delaunay triangulations, Voronoi diagrams, curve reconstruction and polyhedra. The class will combine both theory and algorithms; the work for the class will involve both traditional proofs and implementation of algorithms using the programming language Python, which will be discussed in class. Prerequisites: Mathematics 261, or Computer Science 145 and some programming experience.

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Mathematics 314, Mathematical Modeling

What is a mathematical model? And how can it be used to help solve real world problems? This course will provide 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.

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Mathematics 316, Combinatorics

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.

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Mathematics 317, Graph Theory

Graph theory is a branch of mathematics that has applications in areas ranging from operations research to biology. This course is a survey of the theory and applications of graphs. Topics are chosen from among 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. Several applications of graph theory are discussed in depth. Prerequisite: Mathematics 261 or permission of the instructor.

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Mathematics 318, Number Theory

This is a proofs-based introduction to the theory of numbers and 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.

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Mathematics 321, Differential Equations

This course is an introduction to the theory of 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, Green's functions. Prerequisite: Mathematics 213, or Mathematics 242 and Mathematics 245.

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Mathematics 323, Dynamical Systems

An introduction to the theory of discrete dynamical systems. Topics to be covered include iterated functions, bifurcations, chaos, fractals and fractal dimension, complex functions, Julia sets, and the Mandelbrot set. We will make extensive use of computers to model the behavior of dynamical systems. Prerequisite: Mathematics 213, or Mathematics 212 and Mathematics 242.

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Mathematics 328, Probability

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 will gain practical experience using mathematical software to run probability simulations. Prerequisite: Mathematics 245 or Physics 221, or permission of the instructor.

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Mathematics 333, Abstract Algebra II

This course continues the study of abstract algebra begun in Mathematics 332. Topics are chosen by the instructor, and may include additional group theory, additional ring theory, Galois theory, modules, group representations, and commutative algebra. Prerequisite: Mathematics 332 or permission of the instructor.

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Mathematics 351, Point Set Topology

An introduction to point set topology. Topics include topological spaces, metric spaces, compactness, connectedness, continuity, homomorphisms, separation criteria, and, possibly, the fundamental group. Prerequisite: Mathematics 261 or permission of the instructor.

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Mathematics 352, Differential Geometry

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, and it 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. This course also introduces geodesics, curves of minimal length. The course culminates with the Gauss-Bonnet theorem, giving a link between the geometry and topology of surfaces. Prerequisite: Mathematics 212 and Mathematics 261, or permission of the instructor.

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Mathematics 362, Complex Analysis

This course will cover the basic theory of functions of one complex variable. Topics will 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/or other topics as time permits. Prerequisite: Mathematics 212, Mathematics 261, and one prior 300-level mathematics course is recommended, or permission of the instructor.

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Mathematics 384, Computational Algebraic Geometry

This course is an introduction to computational algebraic geometry and commutative algebra. We will explore the idea of solving systems of polynomial equations by viewing the solutions to these systems as both algebraic and geometric objects. We will also see how these objects can be manipulated using the Groebner basis algorithm. This course will include a mixture of theory and computation as well as connections to other areas of mathematics and to computer science. Prerequisite: Mathematics 332.

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Mathematics 405, Mathematical Logic

An introduction to mathematical logic. Topics include first-order logic, completeness and com-pactness theorems, model theory, nonstandard analysis, decidability and undecidability, incompleteness, and Turing machines. Prerequisite: Mathematics 332 or permission of the instructor.

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Mathematics 417, Algebraic Number Theory

In this course we will study algebraic number fields (finite extensions of the rational numbers) from an algebraic and an analytic viewpoint, motivated by the special cases of quadratic and cyclotomic fields. The goal of the course is to develop an understanding of the deep connections between algebra, analysis, and arithmetic. Topics will include: rings of integers, factorization, ideal class group, unit group, zeta and L-functions, Dirichlet's theorem. Prerequisite: Mathematics 332.

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Mathematics 453, Modern Geometry

Geometry is an ancient subject, but it has received a modern makeover in the past two centuries, where the type of geometry now studied is broader than just Euclidean geometry, and where the approach is now analytic rather than axiomatic. In this course we will look 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.

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Mathematics 461, Real Analysis II

This course continues the study of real analysis begun in Math 361. Topics include functions of several variables, metric spaces, Lebesgue measure and integration, and, time permitting, additional topics such as the Inverse and Implicit Function Theorems, differential forms and Stokes’ Theorem. Prerequisites: Mathematics 361.

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Mathematics 102, Elementary Statistics

The main focus of this course is to introduce 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 will be used to analyze data and facilitate classroom discussions. The goal of this course is to foster statistical reasoning, and to assist in making informed conclusions about topics involving data. Intended for non-math majors.  Prerequisite: passing score on Part I of the Mathematics Placement.

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Mathematics 104, Data and Decisions

This course examines applications of mathematics to a number of topics related to data and decision-making. Topics will be chosen from three relevant areas of mathematics: voting systems, networks and statistics, all of which involve extracting information from various types of data. There is no particular mathematical preparation needed for this course 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 Placement.

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Mathematics 106, Mathematics and Politics

The course examines 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, its implications and its limitations are discussed. Students are actively involved in the modeling process. There is no particular mathematical prerequisite for this course, though algebraic computations and deductive proofs of some of the main results are required. Prerequisite: passing score on Part I of the Mathematics Placement.

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Mathematics 110, Precalculus Mathematics

A course 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 given 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. Students who need to brush up on their Precalculus skills are encouraged to enroll concurrently in BLC 190. Prerequisites: passing score on Part I of the Mathematics Placement.

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Mathematics 123, Statistics for Everyday Life

Statistics is everywhere these days. It is used in the stock market, in weather forecasting, in medical studies, by insurance companies, in quality testing, and in many other areas. This course will introduce core ideas in statistical reasoning to enable you to make sense of and (in)validate the statistics you encounter in the media, in your classes, and in everyday life. Prerequisite: Precalculus or the equivalent.

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Mathematics 131, Exploration in Number Theory

This course, an overview of one of the oldest and most beautiful areas of mathematics, 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: Precalculus or the equivalent.

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Mathematics 135, Game Theory

In the 20th century the theory of games gained prominence for its application to the social sciences. 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.

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Mathematics 241, Vector Calculus

No longer offered. Replaced by Mathematics 245.

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Mathematics 299, Problem Solving Seminar

This course introduces problem solving techniques used throughout the mathematics curriculum. 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, together with a problem session where the students work together using those techniques to tackle some particularly challenging problems. Students may find this class helpful in preparing for the Putnam Exam, a national college mathematics competition given in early December. Prerequisite: Any 200-level mathematics course or permission of the instructor.