Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. MATH 115 is an introductory course designed for students who have not seen calculus before.

Units: 1

Max Enrollment: 24

Prerequisites: Not open to students who have completed MATH 116, MATH 120, MATH 205 or the equivalent. Not open to students whose placement is MATH 205 or MATH 206.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Summer; Spring; Fall

Semesters Offered this Academic Year: Spring; Fall

Notes:

Introduction to differential and integral calculus for functions of one variable. The heart of calculus is the study of rates of change. Differential calculus concerns the process of finding the rate at which a quantity is changing (the derivative). Integral calculus reverses this process. Information is given about the derivative, and the process of integration finds the "integral," which measures accumulated change. This course aims to develop a thorough understanding of the concepts of differentiation and integration, and covers techniques and applications of differentiation and integration of algebraic, trigonometric, logarithmic, and exponential functions. In addition to the material from Math 115, this course spends additional time strengthening students' precalculus skills, covering topics such as proportions and percents, linear and exponential growth, and logarithms. MATH 115P is an introductory course designed for students who have not seen calculus before and who would benefit from extra academic support on precalculus topics.

Units: 1.25

Max Enrollment: 30

Prerequisites: Permission of the instructor.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Fall

Notes:

This class will offer a treatment of first-semester calculus aimed at students interested in the biological and social sciences. The course material is motivated by real-life problems in laboratory and data-driven studies. Students will be expected to work in groups both in and out of class, give presentations at the chalkboard, and submit work in both problem set and project formats. Topics include: functions, limits, continuity, differentiation and an introduction to integration.

Units: 1

Max Enrollment: 24

Prerequisites: Not open to students who have completed MATH 116, MATH 120, MATH 205 or the equivalent. Not open to students whose placement is MATH 205 or MATH 206.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Semesters Offered this Academic Year: Not Offered

Notes:

The course begins with applications and techniques of integration. It probes notions of limit and convergence and adds techniques for finding limits. Half of the course covers infinite sequences and series, where the basic question is, What meaning can we attach to a sum with infinitely many terms and why might we care? The course can help students improve their ability to reason abstractly and also teaches important computational techniques. Topics include integration techniques, l'Hôpital's rule, improper integrals, geometric and other applications of integration, infinite series, power series, and Taylor series. MATH 116 is the appropriate first course for many students who have had AB calculus in high school.

Units: 1

Max Enrollment: 24

Prerequisites: MATH 115 or Math 115Z, or the equivalent. Not open to students who have completed MATH 120 or MATH 205. Not open to students whose math placement is MATH 206.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Summer; Spring; Fall

Semesters Offered this Academic Year: Spring; Fall

Notes:

This course is a variant of MATH 116 for students who have a thorough knowledge of the techniques of differentiation and integration, and familiarity with inverse trigonometric functions and the logarithmic and exponential functions. It includes a rigorous and careful treatment of limits, sequences and series, Taylor's theorem, approximations and numerical methods, Riemann sums, improper integrals, l'Hôpital's rule, and applications of integration.

Units: 1

Max Enrollment: 24

Prerequisites: Open by permission of the department to students who have completed a year of high school calculus. Students who have studied Taylor series should elect MATH 205. Not open to students who have completed MATH 116, MATH 205 or the equivalent.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Fall

Notes:

How can a candidate in a political race win the majority of votes yet lose the election? How can two competing candidates interpret the same statistic as being in their favor? How can the geometry of the voting district disenfranchise entire groups of voters? Can we quantify the power the President of the United States has? In this course, we will look at the mathematics behind these and related questions that arise in politics. We will study topics such as fairness, voting paradoxes, social choice, game theory, apportionment, gerrymandering, and data interpretation. The goal of the class will be to illustrate the importance of rigorous reasoning in various social and political processes while providing an introduction to some fascinating mathematics.

Units: 1

Max Enrollment: 24

Crosslisted Courses: PEAC 123,MATH 123

Prerequisites: None.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Fall

Notes:

This course is a first-year seminar for students in the Wellesley Plus program. It will introduce students to important basic mathematical concepts as set theory, proof techniques, propositional and predicate calculus, graph theory, combinatorics, probability, and recursion.

Units: 1

Max Enrollment: 20

Prerequisites: None. Open to First-Years only.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Other Categories: FYS - First Year Seminar

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Fall

Notes: Mandatory Credit/Non Credit.

This seminar surveys the work of Leonhard Euler (1707-1783), one of the most influential and prolific mathematicians of all time. It is geared toward students who would like a broad overview of what advanced mathematics (beyond calculus) is about, and how it got that way. Topics are drawn from a wide range of areas in pure and applied mathematics, such as algebra, number theory, analysis, and geometry. Highlights include the Basel problem, complex exponentials, the calculus of variations, the Euler line, and the bridges of Königsberg. The seminar is discussion-based: students retrace Euler's steps by making definitions, proposing conjectures, generating examples, and crafting and critiquing proofs, ever attentive to the balance between intuitive ingenuity and rigorous argument.

Units: 1

Max Enrollment: 15

Prerequisites: None. Open to First-Years only.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Other Categories: FYS - First Year Seminar

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Spring

Notes:

This course is intended for students who are interested in mathematics and its applications in economics and finance. The following topics will be covered: mathematical models in economics, market equilibrium, first and second order recurrences, the cobweb model, profit maximization, derivatives in economics, elements of finance, constrained optimization, Lagrangians and the consumer, microeconomic applications, business cycles, European and American options, call and put options, Black-Scholes analysis.

Units: 1

Max Enrollment: 24

Prerequisites: MATH 116 or the equivalent.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Semesters Offered this Academic Year: Not Offered

Notes:

Most real-world systems that one may want to model, whether in the natural or in the social sciences, have many interdependent parameters. To apply calculus to these systems, we need to extend the ideas and techniques of single-variable Calculus to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, partial derivatives, gradients and directional derivatives, Lagrange multipliers, multiple integrals, vector calculus: line integrals, surface integrals, divergence, curl, Green's Theorem, Divergence Theorem, and Stokes’ Theorem.

Units: 1

Max Enrollment: 24

Prerequisites: MATH 116 or MATH 120, or the equivalent.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Spring; Fall

Notes:

Linear algebra is one of the most beautiful subjects in the undergraduate mathematics curriculum. It is also one of the most important with many possible applications. In this course, students learn computational techniques that have widespread applications in the natural and social sciences as well as in industry, finance, and management. There is also a focus on learning how to understand and write mathematical proofs and an emphasis on improving mathematical style and sophistication. Topics include vector spaces, subspaces, linear independence, bases, dimension, inner products, linear transformations, matrix representations, range and null spaces, inverses, and eigenvalues.

Units: 1

Max Enrollment: 24

Prerequisites: MATH 205 or MATH 215; or, with permission of the instructor, MATH 116, MATH 120, or the equivalent.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Spring; Fall

Notes:

This course is designed to examine the degree to which a function can be determined by an algebraic relationship it has with its derivative(s) --- a so-called ordinary differential equation (ODE). For instance, can one completely catalog all functions which equal their own derivative? In service of developing techniques for solving certain classes of differential equations, some fundamental notions from linear algebra and complex numbers are presented.  Differential equation topics include modeling with and solving first- and second-order ODEs, separable ODEs, and a discussion of higher order and non-linear ODEs; linear algebra topics include solving systems via elementary row operations, bases, dimension, determinants, column space, and eigenvalues/vectors.

Units: 1

Max Enrollment: 24

Prerequisites: MATH 116 or MATH 120, or the equivalent.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes:

Probability is the mathematics of uncertainty.  We begin by developing the basic tools of probability theory, including counting techniques, conditional probability, and Bayes's Theorem.  We then survey several of the most common discrete and continuous probability distributions (binomial, Poisson, uniform, normal, and exponential, among others) and discuss mathematical modeling using these distributions. Often we cannot calculate probabilities exactly, and we need to approximate them.  A powerful tool here is the Central Limit Theorem, which provides the link between probability and statistics.  Another strategy when exact results are unavailable is simulation.  We examine Markov chain Monte Carlo methods, which offer a means of simulating from complicated distributions.  

Units: 1

Max Enrollment: 24

Crosslisted Courses: STAT 220

Prerequisites: MATH 205

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Semesters Offered this Academic Year: Fall

Notes:

Number theory is the study of the most basic mathematical objects: the natural numbers (1, 2, 3, etc.). It begins by investigating simple patterns: for instance, which numbers can be written as sums of two squares? Do the primes go on forever? How can we be sure? The patterns and structures that emerge from studying the properties of numbers are so elegant, complex, and important that number theory has been called "the Queen of Mathematics." Once studied only for its intrinsic beauty, number theory has practical applications in cryptography and computer science. Topics include the Euclidean algorithm, modular arithmetic, Fermat's and Euler's Theorems, public-key cryptography, quadratic reciprocity. MATH 223 has a focus on learning to understand and write mathematical proofs; it can serve as valuable preparation for MATH 305.

Units: 1

Max Enrollment: 24

Prerequisites: One of the following - MATH 116, MATH 120, or CS 230 with permission of the instructor.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring

Semesters Offered this Academic Year: Spring

Notes:

Combinatorics is the art of counting possibilities: for instance, how many different ways are there to distribute 20 apples to 10 kids? Graph theory is the study of connected networks of objects. Both have important applications to many areas of mathematics and computer science. The course will be taught emphasizing creative problem-solving as well as methods of proof, such as proof by contradiction and induction. Topics include: selections and arrangements, generating functions, recurrence relations, graph coloring, Hamiltonian and Eulerian circuits, and trees.

Units: 1

Max Enrollment: 24

Prerequisites: MATH 116 or MATH 120, or the equivalent; or CS 230 together with permission of the instructor.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes:

Units: 1

Max Enrollment: 25

Prerequisites:

Semesters Offered this Academic Year: Spring; Fall

Notes:

Units: 0.5

Max Enrollment: 25

Prerequisites:

Semesters Offered this Academic Year: Spring; Fall

Notes:

Real analysis is the study of the rigorous theory of the real numbers, Euclidean space, and calculus. The goal is to thoroughly understand the familiar concepts of continuity, limits, and sequences. Topics include compactness, completeness, and connectedness; continuous functions; differentiation and integration; limits and sequences; and interchange of limit operations as time permits.

Units: 1

Max Enrollment: 24

Prerequisites: MATH 205 and MATH 206.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes:

This course is a continuation of MATH 302 with a treatment of real analysis focusing on the theory of integration. We will see why Riemann integration is supplanted by Lebesgue integration, and how different metrics on function space yield different topological structures. Topics in this course will include the convergence of series of functions, summation tests, the Monotone Convergence Theorem, the Dominated Convergence Theorem, and Fatou's Lemma. If there is time, we will also study the theory of Fourier transforms.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring

Semesters Offered this Academic Year: Spring

Notes: This is a topics course and can be taken more than once for credit as long as the topic is different each time.

In this course, students examine the structural similarities between familiar mathematical objects such as number systems, matrix sets, function spaces, general vector spaces, and mod n arithmetic. Topics include groups, rings, fields, homomorphisms, normal subgroups, quotient spaces, isomorphism theorems, divisibility, and factorization. Many concepts generalize number theoretic notions such as Fermat's little theorem and the Euclidean algorithm. Optional subjects include group actions and applications to combinatorics.

Units: 1

Max Enrollment: 24

Prerequisites: MATH 206

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes:

Topic for Spring 24: Galois Theory

This course offers a continued study of the algebraic structures introduced in MATH 305, culminating in the Fundamental Theorem of Galois Theory, a beautiful result that depicts the circle of ideas surrounding field extensions, polynomial rings, and automorphism groups. Applications of Galois theory include the unsolvability of the quintic by radicals and geometric impossibility proofs, such as the trisection of angles and duplication of cubes. Cyclotomic extensions and Sylow theory may be included in the syllabus.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 305

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Semesters Offered this Academic Year: Spring

Notes: This is a topics course and can be taken more than once for credit as long as the topic is different each time.

This course covers some basic notions of point-set topology, such as topological spaces, metric spaces, connectedness and compactness, Heine-Borel Theorem, quotient spaces, topological groups, groups acting on spaces, homotopy equivalences, separation axioms, Euler characteristic, and classification of surfaces. Additional topics include the study of the fundamental group (time permitting).

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302. Co-requisite - MATH 305.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall

Notes:

This course will introduce students to aspects of set theory and formal logic. The notion of set is one of the fundamental notions of modern mathematics. In fact, other mathematical notions, such as function, relation, number, etc., can be represented in terms of purely set theoretical notions, and their basic properties can be proved using purely set theoretic axioms. The course will include the Zermelo-Fraenkel axioms for set theory, the Axiom of Choice, transfinite arithmetic, ordinal numbers, and cardinal numbers.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302 or MATH 305.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Not Offered

Notes:

Complex analysis is the study of the differential and integral calculus of functions of a complex variable. Complex functions have a rich and tightly constrained structure: for example, in contrast with real functions, a complex function that has one derivative has derivatives of all orders and even a convergent power series. This course develops the theory of complex functions, leading up to Cauchy's theorem and its consequences, including the theory of residues. While the primary viewpoint is calculus, many of the essential insights come from geometry and topology, and can be used to prove results such as the Fundamental Theorem of Algebra.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Semesters Offered this Academic Year: Fall

Notes:

This course examines the number 𝝅 from various points of view in pure and applied mathematics. Topics may include: (1) Geometry: Archimedes’ estimates; volume and surface area of spheres in arbitrary dimensions; Buffon’s needle (and noodle); Galperin’s colliding balls; the isoperimetric inequality; triangles in spherical and hyperbolic geometry; Descartes’s theorem on total angular defect (discrete Gauss-Bonnet). (2) Digit hunting: Viète’s infinite product; Wallis’s product and related ideas (the Gaussian integral and its multidimensional extension, saddle point approximation, Stirling’s approximation); the Leibniz-Gregory formula and Machin-type formulae; spigot algorithms and the Bailey-Borwein-Plouffe formula; elliptic integrals, the arithmetic-geometric mean, and the Brent-Salamin algorithm. (3) Analysis: complex exponentials; Fourier series; the Riemann zeta function, dilogarithms, Bernoulli numbers, and applications to number theory (means of arithmetic functions). (4) Algebra: the irrationality and transcendence of e and 𝝅.

Units: 1

Max Enrollment: 20

Prerequisites: MATH 302 or MATH 305; open to students with advanced background in physics, computer science, or related fields with permission of the instructor.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Not Offered

Notes:

Linear algebra at this more advanced level is a basic tool in many areas of mathematics and other fields. The course begins by revisiting some linear algebra concepts from MATH 206 in a more sophisticated way, making use of the mathematical maturity picked up in MATH 305. Such topics include vector spaces, linear independence, bases, and dimensions, linear transformations, and inner product spaces. Then we will turn to new notions, including dual spaces, reflexivity, annihilators, direct sums and quotients, tensor products, multilinear forms, and modules. One of the main goals of the course is the derivation of canonical forms, including triangular form and Jordan canonical forms. These are methods of analyzing matrices that are more general and powerful than diagonalization (studied in MATH 206). We will also discuss the spectral theorem, the best example of successful diagonalization, and its applications.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 305.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Not Offered

Notes:

Graph Theory has origins both in recreational mathematics problems (i.e., puzzles and games) and as a tool to solve practical problems in many areas of society. Topics covered will include trees and distance, connectivity and paths, network flow, graph coloring, directed graphs, and tournaments. In addition, students will gain a sense of what it means to do research in graph theory.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 225 and either MATH 305 or MATH 302; or permission of the instructor.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Spring

Notes: Majors can fulfill the major presentation requirement in this course in 2023-24.

This course covers questions of enumerations, existence, and construction in combinatorics, building on the fundamental ideas introduced in MATH 225. Topics include: famous number families, combinatorial and bijective proofs, counting under equivalence, combinatorics on graphs, combinatorial designs, error-correcting codes, and partially ordered sets.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 225. Co-requisite - MATH 305.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Every other year

Semesters Offered this Academic Year: Spring

Notes: Majors can fulfill the major presentation requirement with this course.

In this course, students will leverage their prior mathematical knowledge to communicate complex mathematical ideas to audiences ranging from the general public to other mathematicians. Each week, students will research a new topic and produce a piece of writing explaining this topic in a specific context. Assignments may include research abstracts, book reviews, interviews with mathematicians, newspaper articles, and technical documentation. Class time will be devoted to discussing the mathematical content behind each assignment as well as workshopping students' writing. This course will give students the opportunity to ground (and expand on) the mathematics they have learned and make connections across the discipline. Moreover, this course's unique format will help students develop their research and independent learning skills.

Units: 1

Max Enrollment: 12

Prerequisites: MATH 302 and MATH 305, or permission of the instructor.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Other Categories: CSPW - Calderwood Seminar in Public Writing

Typical Periods Offered: Spring

Semesters Offered this Academic Year: Spring

Notes:

Topic for Spring 2025: Introduction to Cryptography

The goal of this course is to use modern public-key cryptography as a vehicle for learning various important concepts in advanced mathematics. Topics will include Diffie-Hellman key exchange, RSA cryptosystem, NTRU cryptosystem, elliptic curve cryptography, discrete logs, DES and AES, digital signatures, hash functions, error correcting codes and quantum cryptography. To understand these ideas, we will need to study ring theory, probability, number theory over a finite field, elliptic curves, linear algebra of lattices, and  NP-Completeness. There will also be a computational component to the course - encryption, decryption, and attacks on cryptosystems using a computer.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 302 or permission of the instructor.

Distribution Requirements: MM - Mathematical Modeling and Problem Solving

Typical Periods Offered: Fall

Semesters Offered this Academic Year: Fall; Spring

Notes: This is a topics course and can be taken more than once for credit as long as the topic is different each time.

Units: 1

Max Enrollment: 25

Prerequisites: Permission of the instructor. Open to juniors and seniors.

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

The first course in a two-semester investigation of a significant research project, culminating in the preparation of a thesis and defense of that thesis before a committee of faculty from the Mathematics department. If sufficient progress is made, students may continue with thesis research (365) in the second semester. This route does not lead to departmental honors.

Units: 1

Max Enrollment: 15

Prerequisites: Permission of the Department. GPA in Mathematics Courses above 100-level must exceed 3.0.

Typical Periods Offered: Fall and Spring

Semesters Offered this Academic Year: Fall; Spring

Notes:

Units: 1

Max Enrollment: 25

Prerequisites: Permission of the department.

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes: Students enroll in Senior Thesis Research (360) in the first semester and carry out independent work under the supervision of a faculty member. If sufficient progress is made, students may continue with Senior Thesis (370) in the second semester.

The second course a two-semester investigation of a significant research project, culminating in the preparation of a thesis and defense of that thesis before a committee of faculty from the Mathematics department. This route does not lead to departmental honors.

Units: 1

Max Enrollment: 15

Prerequisites: MATH 355 and permission of the Department.

Typical Periods Offered: Fall and Spring

Semesters Offered this Academic Year: Fall; Spring

Notes:

Units: 1

Max Enrollment: 25

Prerequisites: MATH 360 and permission of the department.

Typical Periods Offered: Spring; Fall

Semesters Offered this Academic Year: Fall; Spring

Notes: Students enroll in Senior Thesis Research (360) in the first semester and carry out independent work under the supervision of a faculty member. If sufficient progress is made, students may continue with Senior Thesis (370) in the second semester.