## Mathematics Department

**Chair: **John McCleary; **Professors: **John A Feroe, Benjamin Lotto, John McCleary, Peter Pappas, Charles Steinhorn^{ab}; **Associate Professor: **Natalie Priebe Frank^{a}; **Assistant Professors: **Ming-Wen An^{b}, Kariane Calta, Jan Cameron; **Visiting Assistant Professor: **Matthew S Miller.

^{a}On leave 2010/11, first semester^{b}On leave 2010/11, second semester^{ab}On leave 2010/11

**Requirements for Concentration:** 9 1/2 units above the 100-level after completion of Mathematics 121/122 or its equivalent (125 or Advanced Placement). The 9 1/2 units must include Mathematics 220/221, 301, 321, 361, and two other units at the 300-level. Math 361 must be completed by the end of the junior year. It is recommended that a student complete a course in which methods of proof are introduced and developed (one of Mathematics 231, 261, 263, 324, or 364) before enrolling in Mathematics 321 or 361. Reading courses and other independent work may be counted among the required units only with prior approval of the chair. Work used to satisfy major requirements may not be taken NRO after declaration of the major, and only one course taken NRO may count toward the major. No work at the 300-level for the major may be taken NRO.

**Senior Year Requirements:** Mathematics 301.

**Recommendations:** Majors are strongly urged to elect at least 2 units in fields such as the Natural Sciences, Computer Science, or Economics, where applications of mathematics play a key role, and to consider taking Math 228 and/or 241/341. A reading knowledge of French, German, or Russian is advised for those contemplating graduate study.

**Sequence of Courses for Concentration:** Incoming students will normally elect Mathematics 121/122, 125, or 220/221, but freshman eligible for advanced course placement should confer with the department. Election of advanced courses should be made in consultation with a departmental adviser.

Prospective majors in mathematics should complete Mathematics 121/122 or 125 by the end of the freshman year and Mathematics 220 and 221 by the end of the sophomore year. In any case, the first sequence must be completed by the end of the sophomore year in order to declare the major and Mathematics 220/221 must be completed by the end of the junior year.

**Advisers:** The department.

**Correlate Sequence in Mathematics:** Students majoring in other programs may complement their study by electing a correlate sequence in mathematics. Course selection should be made in consultation with the department and the major adviser to ensure exposure to the mathematics most useful to the field of concentration.

**Requirements for the Correlate Sequence:** Mathematics 121/122 (or 125 or permission of the department to enroll in 220), 4 graded units above the 100-level (that is, these units may not be taken NRO). The 4 units must include Mathematics 220/221 and one unit at the 300-level.

**AP Credit:** Students receiving 1 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 Mathematics 101 or 121. Students receiving one unit of AP credit based on the Statistics AP Examination may not be granted credit for Mathematics 141.

**Advanced Course Placement:** The department recommends that students who have earned a 4 or 5 on the BC examination enroll in Mathematics 220. Students with a 5 on the AB examination or a 3 on the BC examination generally are advised to elect Mathematics 220 also, after conferring with the department. Students with a 4 on the AB examination ordinarily are advised to enroll in Mathematics 125, but should consult with the department.

## I. Introductory

### 100b. Pre-Calculus (1/2)

This sequence, 100-101, is designed for students who wish to take Mathematics 101, Introduction to Calculus, but whose mathematical background is deficient. Students with three years of high school mathematics should begin with Mathematics 101. Topics of Mathematics 100 include the algebra of polynomials, operations with fractions, solving equations and inequalities, exponents and radicals, elements of coordinate geometry, functions and their graphs, logarithms and elements of trigonometry.

Year-long course, 100-101. On the satisfactory completion of Mathematics 101, the student receives 1/2 unit of credit for Mathematics 100.

Not open to students with AP credit in mathematics or students who have completed Mathematics 101 or 121.

Prerequisite: high school mathematics. Advice of the department should be sought before registering for this course.

### 101a. 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 Mathematics 121 or its equivalent.

Does not generally serve as a prerequisite for Mathematics 122, 125, or 200-level mathematics courses, consult with the department for more information.

Prerequisite: at least three years of high school mathematics.

Three 50-minute periods.

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

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

Not open to those who have had Mathematics 122.

Does not serve as a prerequisite for 200-level mathematics courses.

Prerequisite: Mathematics 101 or equivalent.

### 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.

Year-long course, 121/122.

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

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

Three 50-minute periods.

### 122b. 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.

Year-long course, 121/122.

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

Three 50-minute periods.

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

Material from Mathematics 121/122 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. The department.

Three 50-minute periods.

### 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.

Two 50-minute lectures and one 50-minute discussion per week.

### 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. Mr. Lotto.

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

Two 75-minute periods.

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

(Same as Biology 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.

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

Prerequisite: three years of high school mathematics.

### 142a or b. 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.

## II. Intermediate

### 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 122 or 125 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 for all intermediate courses: Mathematics 122, 125, or permission of the department, unless otherwise indicated.

### 228a or b. 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 for all intermediate courses: Mathematics 122, 125, or permission of the department, unless otherwise indicated.

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

Topics to be chosen from: conic sections, transformational geometry, Euclidean geometry, affine geometry, projective geometry, inversive 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 for all intermediate courses: Mathematics 122, 125, or permission of the department, unless otherwise indicated.

### 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.

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

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

### 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.

### 290a or b. Field Work (1/2 or 1)

### 297a. Topics in Mathematics (1/2)

Reading course.

Prerequisite: Mathematics 221 or equivalent, and permission of instructor.

### 298a or b. Independent Work (1/2 or 1)

Election should be made in consultation with a department adviser.

## III. Advanced

### 301b. Senior Seminar (1/2 or 1)

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 Mathematics 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: Mathematics 220 and 221, unless otherwise indicated.

### 324a or b. 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: Mathematics 220 and 221, unless otherwise indicated.

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

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

Prerequisite: Mathematics 321.

### 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: Mathematics 321 or permission of instructor.

### 335a or b. Differential Geometry (1)

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

Prerequisite: Mathematics 321.

### 336a or b. Algebraic Geometry (1)

An introduction to the study of algebraic geometry. Topics may include: projective space, homogeneous coordinates, plane curves, Bezout's theorem, elliptic curves, affine and projective varieties, the Zariski topology, coordinate rings, functions on varieties. The department.

Prerequisite: Mathematics 361.

### 339a or b. 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: Mathematics 321 or 361.

### 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.

Prerequisite: Mathematics 220 and 241.

### 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: Mathematics 321 or 361.

### 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: Mathematics 220 and 221, unless otherwise indicated.

### 364a or b. 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: Mathematics 220 and 221, unless otherwise indicated.

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

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

Prerequisite: Mathematics 361.

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

Advanced study in an area of mathematics. The department.

Alternate years.

### 399a or b. Senior Independent Work (1/2 or 1)

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