For the most recent information about courses in the Department of Mathematics, or to print out a complete list of all mathematics course descriptions, please refer to the Mathematics section of the Wellesley College Course Catalog. For descriptions of the courses offered in the Department of Mathematics, click on the links below.
An introduction to the fundamental ideas and methods of statistics for analyzing data. Topics include descriptive statistics, basic probability, inference and hypothesis testing. Emphasis on understanding the use and misuse of statistics in a variety of fields, including medicine and both the physical and social sciences. This course is intended to be accessible to those students who have not yet had calculus.
In this course, students use probability and statistics to examine the risks that we encounter every day. The focus is on personal medical decision-making and the impact of our environment on our health. Students will address questions such as: How concerned should we be about pesticide use? How can we make informed decisions about women's health issues, including contraception and sexually transmitted diseases? How much of an impact does diet have on health? Why did different studies of hormone replacement therapy come to contradictory conclusions, and how can we read reports on such studies intelligently and skeptically? Topics include descriptive statistics, basic probability, inference and hypothesis testing.
Not offered in 2011-2012 This course explores several areas of mathematics which have application in the physical and social sciences, yet which require only high-school mathematics as a prerequisite. The areas covered will be chosen from systems of linear equations, linear programming, probability, game theory, and stochastic processes. Students will solve problems on topics ranging from medical testing to economics, with the results demonstrating the value of mathematical reasoning. May not be counted toward the major.
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.
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'Hopital's rule, improper integrals, geometric and other applications of integration, theoretical basis of limits and continuity, 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.
This course is a variant of 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'Hopital's rule, and applications of integration.
Mathematics has the distinction of dealing with truths that aren't changed by political revolution, toppled by new observations, or eroded by neglect. In this seminar we will investigate the key component to the permanence of these discoveries: mathematical proof. Through a careful examination of fundamental mathematical objects such as sets, relations and functions, students will learn a handful of proof techniques which are robust enough to prove nearly everything in the mathematical spectrum. Along the way we will uncover plenty of unexpected results --- from the fact that there is more than one size of infinity, to the fact that there are mathematical statements whose truth value is provably indeterminate. Students will leave the course with the ability to make and understand reasoned, logical arguments.
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.
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 MATH 115 and MATH 116 to functions of more than one variable. Topics include vectors, matrices, determinants, polar, cylindrical, and spherical coordinates, curves, functions of several variables, partial and directional derivatives, gradients, Lagrange multipliers, multiple integrals, line integrals, and Green's Theorem.
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, eigenvalues.
Introduction to theory and solution of ordinary differential equations, with applications to such areas as physics, ecology, and economics. Includes linear and nonlinear differential equations and equation systems, existence and uniqueness theorems, and such solution methods as power series, Laplace transform, and graphical and numerical methods.
Not offered in 2011-2012. An introduction to the differential geometry of curves and surfaces. Topics include curvature of curves and surfaces, first and second fundamental forms, equations of Gauss and Codazzi, the fundamental theorem of surfaces, geodesics, and surfaces of constant curvature. Normally offered in alternate years.
Not offered in 2011-2012. A rigorous treatment of the fundamentals of two-dimensional geometry: Euclidean, spherical, elliptic and hyperbolic. The course will present the basic classical results of plane geometry: congruence theorems, concurrence theorems, classification of isometries, etc. and their analogues in the non-Euclidean settings. The course will provide a link between classical geometry and modern geometry, preparing for study in group theory, differential geometry, topology, and mathematical physics. The approach will be analytical, providing practice in proof techniques. This course is strongly recommended for prospective teachers of mathematics.
This course is tailored to the needs and preparations of students considering majors in the sciences. It presents techniques of applied mathematics relevant to a broad range of scientific studies, from the life sciences to physics and astronomy. The topics of study include complex numbers, ordinary differential equations, an introduction to partial differential equations, linear algebra (matrices, systems of linear equations, vector spaces, eigenvalue problems), and Fourier series. The course emphasizes mathematical techniques and presents applications from all the sciences. Some familiarity with vectors (e.g. dot products) is assumed.
This course is about the mathematics of uncertainty, where we use the ideas of probability to describe patterns in chance phenomena. Probability is the basis of statistics and game theory, and is immensely useful in many fields including business, social and physical sciences, and medicine. The first part of the course focuses on probability theory (random variables, conditional probability, probability distributions), using integration and infinite series. The second part discusses topics from statistics (sampling, estimation, confidence interval, hypothesis testing). Applications are taken from areas such as medical diagnosis, quality control, gambling, political polls, and others.
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? Are there infinitely many primes? 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 305. Majors can fulfill the major presentation requirement in this course in 2008-09.
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.
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. We also study metric spaces, which generalize the familiar notion of distance to a broader setting. Topics include metric spaces; compact, complete, and connected spaces; continuous functions; differentiation and integration; limits and sequences; and interchange of limit operations as time permits.
Not offered in 2010-11. A continuation of MATH 302. Topics chosen from the theory of Riemann integration, measure theory, Lebesgue integration, Fourier series, and calculus on manifolds. Offered in alternate years.
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.
Topic for 2011-12: 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. Majors can fulfill the major presentation requirement in
this course in 2011-12.
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).
Not offered in 2011-12. 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, Zorn's Lemma, ordinal numbers and cardinal numbers. We also study Godel's incompleteness theorem, which asserts that any consistent system containing arithmetic has questions that cannot be answered within the system. Offered in alternate years.
This course offers a rigorous treatment of complex analysis of one variable. Topics include complex numbers and functions, analyticity, Cauchy's integral formula and its consequences, Taylor and Laurent series, the residue theorem, the principle of the argument and Rouche's theorem. Other subjects may include conformal mappings, asymptotic series and infinite products. The course will be conducted at the level of both theory and computation.
Differential geometry has two aspects. Classical differential geometry, which shares origins with the beginnings of calculus, is the study of local properties of curves and surfaces. Local properties are those properties which depend only on the behavior of the curve or the surface in a neighborhood of point. The other aspect is global differential geometry: here we see how these local properties influence the behavior of the entire curve or surface. The main idea is that of curvature. What is curvature? It can be intrinsic or extrinsic. What's the difference? What does it mean to have greater or smaller (or positive or negative) curvature? We will answer these questions for surfaces in 3-space, as well as for abstract manifolds.Topics include curvature of curves and surfaces, first and second fundamental forms, equations of Gauss and Codazzi, the fundamental theorem of surfaces, geodesics, and surfaces of constant curvature.
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 dimension, linear transformations, and inner product spaces. Then we will turn to new notions, including dual spaces, reflexivity, annihilators, direct sums and quotients, tensor products and multilinear forms. One of the main goals of the course is the derivation of canonical forms, including triangular form and Jordan canonical form. These are methods of analyzing matrices that are more general and powerful than diagonalization (studied in Math 206). These will be introduced in the context of modules, a powerful generalization of vector spaces. We will also discuss the spectral theorem, the best example of successful diagonalization and its applications.
Not offered in 2011-2012. This is an advanced course in number theory from the algebraic point of view. The course begins with the notion that every integer can be factored uniquely into primes. We will then explore these notions of primeness and unique factorization in other, more general, number systems. Topics covered will include number fields, algebraic integers, Diophantine equations, cyclotomic extensions and class number.
Not offered in 2010-11. 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 include: trees, connectivity, Hamiltonian cycles, directed graphs and tournaments, vertex and edge coloring, matchings, extremal graph theory. Students will be expected to experiment and formulate conjectures. Majors can fulfill the major presentation requirement in this course in 2009-10.
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, abstract linear algebra surrounding lattices, NP-Completeness, and various other subjects. There will also be a computational component to the course -- encryption, decryption, and attacks on cryptosystems using a computer.
This is a course on linear and nonlinear optimization. In optimization, we seek to maximize or minimize a function of several variables, where the variables may be required to satisfy some constraints. When the function and constraints are linear we solve this with linear programming, which is based on linear algebra and convexity and is one of the most widely used methods of applied mathematics. Nonlinear problems use methods based on multivariable calculus and are often solved by approximation. We will focus on the theory underlying these various optimization techniques, on the convergence properties of the algorithms, and on applications. Applications will be selected from a range of areas, such as production, inventory, scheduling, investment, transportation, and distribution.
Recent independent study courses have included Functional Analysis, Representation Theory, Advanced Differential Equations, Advanced Group Theory, Advanced Complex Analysis, Knot Theory, Differential Geometry and Stochastic Processes.