## Mathematics Department

**AP: **Students receiving one unit of AP credit based on either the AB or BC Mathematics AP Examination or the calculus credit examination administered by the Department of Mathematics may not be granted credit for MATH 121. Students receiving one unit of AP credit based on the Statistics AP Examination may not be granted credit for MATH 141.

**Advanced Course Placement: **The department recommends that students who have earned a 4 or 5 on the BC examination enroll in MATH 220. Students with a 5 on the AB examination or a 3 on the BC examination generally are advised to elect MATH 220 also, after conferring with the department. Students with a 4 on the AB examination ordinarily are advised to enroll in MATH 127 but should consult with the department.

### Programs

**Major**

**Correlate Sequence in Mathematics**

### Courses

## Mathematics: I. Introductory

### 101a or b. Introduction to Calculus (1)

A course intended for students not majoring in mathematics or the physical sciences who need a working knowledge of calculus. The course emphasizes techniques and applications with relatively little attention to the rigorous foundations. The department.

Not open to students with AP credit in mathematics or students who have completed MATH 121 or its equivalent.

Prerequisite: at least three years of high school mathematics.

Does not generally serve as a prerequisite for MATH 122, MATH 126, MATH 127, or 200-level mathematics courses, consult with the department for more information.

### 102a or b. Topics in Calculus (1)

A continuation of MATH 101. Topics may include: matrix methods, use of differentiation and integration, differential equations, and partial differentiation. Emphasis is on techniques and applications. The department.

Prerequisite: MATH 101 or equivalent.

Not open to those who have had MATH 126, MATH 127. Does not serve as a prerequisite for 200-level mathematics courses.

### 121a. Single Variable Calculus (1)

The calculus of one variable and its applications are discussed. Topics include: limits, continuity, derivatives, applications of derivatives, transcendental functions, the definite integral, applications of definite integrals, approximation methods, differential equations, sequences, and series. The department.

Prerequisite: a minimum of three years of high school mathematics, preferably including trigonometry.

Mathematics 121 is not open to students with AP credit in mathematics or students who have completed MATH 101 or its equivalent.

Yearlong course sequence 121, MATH 126/MATH 127.

### 122a. (DELETE) Single Variable Calculus (1)

The calculus of one variable and its applications are discussed. Topics include: limits, continuity, derivatives, applications of derivatives, transcendental functions, the definite integral, applications of definite integrals, approximation methods, differential equations, sequences, and series.

Yearlong course MATH 121/122.

Three 50-minute periods.

### 125a. (DELETE) Topics in Single Variable Calculus (1)

Material from MATH 121/MATH 122 presented in one semester for students with previous experience with calculus. Topics in second semester calculus are fully developed and topics in first semester calculus are reviewed.

Three 50-minute periods.

### 126a and b. Calculus IIA: Integration Theory (0.5)

In this course, we expand and build upon basic knowledge of differential and integral calculus. Various techniques and applications of integration will be studied. The calculus of transcendental functions---such as the exponential, logarithmic, and inverse trigonometric functions---will also be developed. A main theme in this course is the many ways functions can be defined, and arise naturally in problems in the mathematical sciences.

Prerequisite: MATH 121 or its equivalent.

First or second six-week course.

### 127a and b. Calculus IIB: Sequences and Series (0.5)

Real numbers may be represented as infinite decimals. In this course we generalize this representation by studying the convergence of sequences and of series of real numbers. These notions further generalize to the convergence of sequences and series of functions. We study these ideas and their relation to the Calculus.

Prerequisite: MATH 121 or its equivalent.

First or second six-week course.

### 131a. Numbers, Shape, Chance, and Change (1)

What is the stuff of mathematics? What do mathematicians do? Fundamental concepts from arithmetic, geometry, probability, and the calculus are explored, emphasizing the relations among these diverse areas, their internal logic, their beauty, and how they come together to form a unified discipline. As a counterpoint, we also discuss the "unreasonable effectiveness" of mathematics in describing a stunning range of phenomena from the natural and social worlds. The department.

Prerequisites: at least three years of high school mathematics.

Open only to freshmen; satisfies the college requirement for a Freshman Writing Seminar.

Two 50-minute periods and one 50-minute discussion period.

### 132a. Mathematics and Narrative (1)

To most, mathematics and narrative live in opposition-narrative is ubiquitous while mathematics is perceived as inscrutably esoteric and obscure. In fact, narrative is a fundamental part of mathematics. Mathematical proofs, problems and solutions, textbooks, and journal articles tell some sort of story. Conversely, many literary works (*Arcadia, Proof*, and *Uncle Petros and the Goldbach Conjecture*) use mathematics as an integral part of their narrative. Movie and television narratives such as *Good Will Hunting* and *Numb3rs* are also mathematically based. Nonfiction works about mathematics and mathematical biographies like *Chaos*, Fermat's *Enigma*, and *A Beautiful Mind* provide further examples of the connection between mathematics and narrative. We use this course to explore this connection by reading and writing a variety of mathematical narratives.

Open only to freshmen; satisfies college requirement for a Freshman Writing Seminar.

Not offered in 2015/16.

### 141a and b. Introduction to Statistics (1)

(Same as BIOL 141) The purpose of this course is to develop an appreciation and understanding of the exploration and interpretation of data. Topics include display and summary of data, introductory probability, fundamental issues of study design, and inferential methods including confidence interval estimation and hypothesis testing. Applications and examples are drawn from a wide variety of disciplines. When cross-listed with biology, examples will be drawn primarily from biology.

Prerequisite: three years of high school mathematics.

Not open to students with AP credit in statistics or students who have completed ECON 209 or PSYC 200.

### 142a. Statistical Sleuthing: Personal and Public Policy Decision-Making in a World of Numbers (1)

The world inundates us with numbers and pictures intended to persuade us towards certain beliefs about our health, public policy, or even which brand of product to buy. How can we make informed decisions in this context? The goal of this course is for us to become statistical sleuths who critically read and summarize a piece of statistical evidence. We read articles from a variety of sources, while using basic statistical principles to guide us. Course format: mixture of discussion and lecture, with regular reading and writing assignments. The department.

Open only to freshmen; satisfies the college requirement for a Freshman Writing Seminar.

Three 50-minute periods.

## Mathematics: II. Intermediate

Prerequisites for all intermediate courses: MATH 126 and MATH 127, or permission of the department, unless otherwise indicated.

### 220a and b. Multivariable Calculus (1)

This course extends differential and integral calculus to functions of several variables. Topics include: partial derivatives, gradients, extreme value problems, Lagrange multipliers, multiple integrals, line and surface integrals, the theorems of Green and Gauss.

Prerequisite: MATH 126 and MATH 127 or equivalent.

### 221a and b. Linear Algebra (1)

The theory of higher dimensional space. Topics include: geometric properties of n-space, matrices and linear equations, vector spaces, linear mappings, determinants. The department.

Prerequisite: MATH 126 and MATH 127 or equivilent, or permission of the department.

### 228b. Methods of Applied Mathematics (1)

Survey of techniques used in the physical sciences. Topics include: ordinary and partial differential equations, series representation of functions, integral transforms, Fourier series and integrals. The department.

Prerequisite: MATH 126 and MATH 127, or permission of the department.

### 231b. Topics in Geometry (1)

Topics to be chosen from: conic sections, transformational geometry, Euclidean geometry, affine geometry, projective geometry, inversive geometry, relativistic geometry, non-Euclidean geometry, spherical geometry, convexity, fractal geometry, solid geometry, foundations of geometry. The department. With departmental permission, course may be repeated for credit when the topic changes.

Prerequisite: MATH 126 and MATH 127, or permission of the department.

### 241a. Probability Models (1)

This course in introductory probability theory covers topics including combinatorics, discrete and continuous random variables, distribution functions, joint distributions, independence, properties of expectations, and basic limit theorems. The department.

Prerequisite: MATH 126 and MATH 127, or permission of the department.

### 242a. Applied Statistical Modeling (1)

Applied Statistical Modeling is offered as a second course in statistics in which we present a set of case studies and introduce appropriate statistical modeling techniques for each. Topics may include: multiple linear regression, logistic regression, log-linear regression, survival analysis, an introduction to Bayesian modeling, and modeling via simulation. Other topics may be substituted for these or added as time allows. Students will be expected to conduct data analyses in R. The department.

Prerequisites: MATH 126 and MATH 127; MATH 141.

### 261a. Introduction to Number Theory (1)

Topics include: divisibility, congruence, modular arithmetic, diophantine equations, number-theoretic functions, distribution of the prime numbers. The department.

Prerequisite: MATH 126 and MATH 127, or permission of the department.

### 263b. Discrete Mathematics (1)

Mathematical induction, elements of set theory and logic, permutations and combinations, relations, topics in graph theory, generating functions, recurrence relations, Boolean algebras. The department.

Prerequisite: MATH 126 and MATH 127, or permission of the department.

### 268b. Protecting Information: Applications of Abstract Algebra (1)

In today's information age, it is vital to secure messages against eavesdropping or corruption by noise. Our study begins by surveying some historical techniques and proceeds to examining some of the most important codes currently being used to protect information. These include various public key cryptographic schemes (RSA and its variants) that are used to safeguard sensitive internet communications, as well as linear codes, mathematically elegant and computationally practical means of correcting transmissions errors. The department.

Prerequisite: MATH 126 and MATH 127, or permission of the department.

Not offered in 2015/16.

### 290a or b. Field Work (0.5to1)

### 297a. Topics in Mathematics (0.5)

Reading Course

Prerequisite: MATH 221 or equivalent, and permission of the instructor.

### 298a or b. Independent Work (0.5to1)

Election should be made in consultation with a department adviser.

## Mathematics: III. Advanced

Prerequisites for all advanced courses: MATH 220 and MATH 221, or permission of the department, unless otherwise indicated.

### 301b. Senior Seminar (0.5to1)

Areas of study and units of credit vary from year to year. The department.

Open only to seniors who have a declared major in mathematics. It is strongly recommended that MATH 361 be completed before enrolling in Mathematics 301.

### 321a. Real Analysis (1)

A rigorous treatment of topics in the classical theory of functions of a real variable from the point of view of metric space topology including limits, continuity, sequences and series of functions, and the Riemann-Stieltjes integral. The department.

Prerequisite for all advanced courses: MATH 220 and MATH 221, unless otherwise indicated.

### 324b. Complex Analysis (1)

Integration and differentiation in the complex plane. Topics include: holomorphic (differentiable) functions, power series as holomorphic functions, Taylor and Laurent series, singularities and residues, complex integration and, in particular, Cauchy's Theorem and its consequences. The department.

Prerequisite for all advanced courses: MATH 220 and MATH 221, unless otherwise indicated.

### 327b. Advanced Topics in Real Analysis (1)

Continuation of MATH 321. Measure theory, the Lebesgue integral, Banach spaces of measurable functions. The department.

Prerequisite: MATH 321.

Not offered in 2015/16.

### 328b. Theory of Differential Equations and Dynamical Systems (1)

Existence and uniqueness theorems for ordinary differential equations; general theory and eigenvalue methods for first order linear systems. The department.

Prerequisite: MATH 321 or permission of the instructor.

### 335a. Differential Geometry (1)

The geometry of curves and surfaces in 3-dimensional space and an introduction to manifolds. The department.

Prerequisite: MATH 321.

### 339a. Topology (1)

Introductory point-set and algebraic topology; topological spaces, metric spaces, continuous mappings, connectedness, compactness and separation properties; the fundamental group; simplicial homology. The department.

Prerequisite: MATH 321 or MATH 361.

Not offered in 2015/16.

### 341b. Mathematical Statistics (1)

An introduction to statistical theory through the mathematical development of topics including resampling methods, sampling distributions, likelihood, interval and point estimation, and introduction to statistical inferential methods. The department.

Prerequisites: MATH 220 and MATH 241.

### 342a. Applied Statistical Modeling (1)

For students who have completed MATH 341. Students in this course attend the same lectures as those in MATH 242, but will be required to complete extra reading and problems. Ms. An.

Prerequisites: MATH 126 and MATH 127, MATH 341.

Three 50-minute periods.

### 351a. Mathematical Logic (1)

An introduction to mathematical logic. Topics are drawn from computability theory, model theory, and set theory. Mathematical and philosophical implications also are discussed. The department.

Prerequisite: MATH 321 or MATH 361.

Not offered in 2015/16.

### 361b. Modern Algebra (1)

The theory of groups and an introduction to ring theory. Topics in group theory include: isomorphism theorems, generators and relations, group actions, Sylow theorems, fundamental theorem of finite abelian groups. The department.

Prerequisite for all advanced courses: MATH 220 and MATH 221, unless otherwise indicated.

### 364a. Advanced Linear Algebra (1)

Further study in the theory of vector spaces and linear maps. Topics may include: scalar products and dual space; symmetric, hermitian and unitary operators; eigenvectors and eigenvalues; spectral theorems; canonical forms. The department.

Prerequisite for all advanced courses: MATH 220 and MATH 221, unless therwise indicated.

### 367a. Advanced Topics in Modern Algebra (1)

Continuation of MATH 361. Rings and fields, with a particular emphasis on Galois theory. The department.

Prerequisite: MATH 361.

Not offered in 2015/16.

### 380a. Topics in Advanced Mathematics (1)

Advanced study in an area of mathematics. The department.

Not offered in 2015/16.

### 399a or b. Senior Independent Work (0.5to1)

Election requires the approval of a departmental adviser and of the instructor who supervises the work.