## Mathematics

**Professors: **John Feroe, John McCleary (Chair), Peter C. Pappas, Charles I. Steinhorn; **Associate Professor: **Benjamin A. Lotto;**Assistant Professors: **Jason Cantarella, Heather Johnston; **Visiting Assistant Professor: **Natalie Priebe; **Adjunct Instructor: **Doris Haas*

**Requirements for Concentration: **Mathematics 121/122 (or 125 or permission of the department to enroll in 221), 9^{1}/_{2} units above the 100–level including Mathematics 221/222, 301, 321, 361, and two other units at the 300–level. Reading courses are not counted among the required units. Work used to satisfy major requirements may not be taken NRO after declaration of the major. At most one unit at the 300–level taken NRO prior to declaration of the major may be used to satisfy major requirements.

**Senior Year Requirements: **Mathematics 301.

**Recommendations: **Majors are strongly urged to elect at least 2 units in applications of mathematics to other fields. 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, 221/222, but freshman eligible for Advanced Placement should confer with the department. Election of advanced courses should be made in consultation with a departmental adviser.

Prospective majors in mathematics are strongly advised to complete Mathematics 121/122 or Mathematics 125 by the end of the freshman year and Mathematics 221/222 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 221/222 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 221), 4 graded units above the 100level including 221/222 and 1 unit at the 300level.

**Advanced Placement: **Students receiving 1 unit of Advanced Placement credit based on either the AB or BC Mathematics Advanced Placement Examination or the calculus credit examination administered by the Department of Mathematics may not be granted credit for Mathematics 101 or 121.

The department recommends that students who have earned a 4 or 5 on the BC examination enroll in Mathematics 221. Students with a 5 on the AB examination or a 3 on the BC examination generally are advised to elect Mathematics 221 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**

**100a–101b. PreCalculus and Introduction to Calculus ( ^{1}/_{2}, 1)**

This sequence 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. Ms. Haas.

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.

**101b/102a.
101b. 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 those who have had Mathematics 121 or its equivalent.

Does not serve as a prerequisite for Mathematics 122, 125, or 200–level mathematics courses.

Prerequisite: at least three years of high school mathematics.

Three 50–minute periods.

**102a. 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 200level mathematics courses.

Prerequisite: Mathematics 101 or equivalent.

**121a/122b. Single Variable Calculus (1)**

The calculus of one variable and 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.

Not open to those who have had Mathematics 101.

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

Thr ee 50minute periods; one 50minute problem session.

**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 secondsemester calculus are fully developed and topics in firstsemester calculus are reviewed. The department.

Three 50minute periods; one 50minute problem session.

**180a. 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.

**II. Intermediate**

Prerequisite for all intermediate courses: Mathematics 122, 125 or equivalent, unless otherwise indicated.

**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.

**222a and b. Multivariable Calculus (1)**

Continuation of Mathematics 221. Differential calculus of vector functions, implicit function theorem, extreme values, multiple integrals, vector field theory. The department.

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

**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.

**[231a or b. Topics in Geometry] (1)**

Topics to be chosen from: conic sections, transformational geometry, Euclidean geometry, affine geometry, projective geometry, inversive geometry, nonEuclidean geometry, spherical geometry, convexity, fractal geometry, solid geometry, foundations of geometry. The department.

Not offered in 2001/02.

**241a or b. Probability Models (1)**

A presentation of commonly applied discrete and continuous probability distributions, including the use of expectation, independence, conditional probability, and related statistical concepts. The department.

**261a or b. Introduction to Number Theory (1)**

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

**263a or b. 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.

**280b. Protecting Information: Applications of Abstract Algebra (1)**

Living in the early decades of the information age, we find ourselves depending more and more on codes that protect messages against either noise of eavesdropping. We begin this course by studying the history of this subject, including, for example, the story of the Enigma code from WWII. We then examine some of the most important codes currently being used to protect information, including linear codes, which in addition to being mathematically elegant are the most practical codes for error correction, and the RSA public key cryptographic scheme, popular nowadays for internet applications. Looking ahead by a decade or more, we chow how a "quantum computer" could crack any RSA code in short order, and how quantum cryptographic devices will achieve security through the Heisenberg uncertainty principle. Mr. Lotto

**290. Field Work ( ^{1}/_{2} or 1)**

**Reading Courses**

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

297.01. Topics in Mathematics (^{1}/_{2})

**298. Independent Work ( ^{1}/_{2} or 1)**

Election should be made in consultation with a department adviser.

**III. Advanced**

Prerequisite for all advanced courses: Mathematics 222, unless otherwise indicated.

**301b. Senior Seminar ( ^{1}/_{2})**

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

Open only to seniors whose major is mathematics.

**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.

**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.

**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.

Alternate years: offered in 2001/02.

**[328b. Theory of Differential Equations] (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.

Alternate years: not offered in 2001/02.

**[335a or b. Topics in Differential Geometry and Topology] (1)**

Aspects of the elementary geometry and topology of differentiable manifolds. Topics vary from year to year. The department.

Prerequisite: Mathematics 321.

Alternate years: not offered in 2001/02.

**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.

Alternate years: offered in 2001/02.

**[339a or b. Topology] (1)**

Introductory pointset 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.

Not offered in 2001/02.

**341b. Mathematical Statistics (1)**

The rigorous development of topics in mathematical statistics: probability and distributions; multivariate distributions; special distributions; distributions of functions of several variables; limiting distributions; introduction to statistical inference. Additional topics drawn from sufficient statistics, estimation theory, statistical testing, and inferences about normal models. The department

Prerequisite: Mathematics 222 and 241.

Alternate years: offered in 2001/02.

**351a. Foundations of Mathematics (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.

Alternate years: offered in 2001/02.

**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.

**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.

**[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.

Alternate years: not offered in 2001/02.

**[380a or b. Topics in Advanced Mathematics] (1)**

Advanced study in an area of mathematics. The department.

Not offered in 2001/02.

**399. Senior Independent Work ( ^{1}/_{2} or 1)**

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