# Mathematics: 2007-2008

### Description

The aims of courses in mathematics are: (1) to promote rigorous thinking in a systematic, deductive, intellectual discipline; (2) to present to the student the direction and scope of mathematical development; (3) to foster technical competence in mathematics as an aid to the better comprehension of the physical, biological, and social sciences; and (4) to guide and direct the mathematics majors toward an interest in mathematical research.

Mathematics majors take a three-year core sequence of courses in calculus, linear algebra, abstract algebra, and analysis, designed to provide a foundation for further study in the major areas of modern mathematics. Students with substantial advanced placement may complete this sequence by the end of their sophomore year. Students who have completed the core sequence may take advanced courses in algebra, analysis, topology, or other special topics.

The department offers several intermediate-level courses designed for both majors and non-majors. These include Mathematics 121, 204 and 215-216, which provide an important foundation for more advanced work in mathematics and other sciences. Mathematics 113, 114 and 115 (or equivalent advanced placement) is sufficient background for any of these courses. A program including Mathematics 113, 114, 116, 203, 210, and 215 is especially appropriate for majors in the social sciences. Students planning graduate study in economics should consider taking Mathematics 317.

Mathematics majors are urged to gain facility in the use of computers, either through the introductory courses Computer Science 105, 206, through applied math electives (like Math 204, 218, or 222), or through independent work. Students interested in pursuing computer science in depth as part of a mathematics major should consider the possibility of a concentration in computer science. Mathematics majors interested in applying their skills to economic problems have the option of pursuing an area of concentration in mathematical economics. Students interested in teaching mathematics can concentrate in educational studies. The requirements for concentration in computer science, mathematical economics, and educational studies are described under their own headings in this catalog.

Mathematics minors preparing for a mathematics teaching career should take one elective in probability and statistics (Math 203 or 218) and one in geometry or topology (Math 205 or 335). Minors preparing for employment in industry immediately after college should take electives in probability and statistics (Math 203 or 218) and mathematical modelling (Math 204, 210, or 222). Minors preparing for graduate work in physical chemistry or theoretical physics should take Complex Analysis (Math 220) and Analysis II (Math 318). Minors desiring a deep understanding of an area of pure math should take 300-level courses in that area (Math 318 and 392 for analysis, Math 334 and 390 for algebra, Math 335 and 336 for topology).

### Faculty

Professor Emeritus William C. Davidon

Professor Lynne Butler

J. McLain King Professor of Mathematics Curtis Greene (on leave semester II)

Associate Professor Yung-sheng Tai

William H. and Johanna A. Harris Associate Professor of Computational Science Robert Manning

Associate Professor Weiwen Miao

Assistant Professor Joshua Sabloff

Senior Lecturer Jeffrey Tecosky-Feldman (on leave semester I)

Visiting Assistant Professor Stephen Wang

### Major Requirements

- Mathematics 215, and either Mathematics 121 or Mathematics 216.
- Mathematics 317, 333 and one of Mathematics 318 or 334.
- Four additional electives in mathematics or approved related courses at the 200 level or higher. At least one of these must be at the 300 level. None of Math 299, Math 399, Math 460 and Math 480 used for senior paper preparation may be counted toward these requirements.
- The senior seminar, Fall and Spring.
- A senior paper and oral presentation.

Students planning graduate study in mathematics or related fields are strongly advised to take additional courses at the 300 level. Equivalent courses in mathematics at Bryn Mawr College may be substituted for any requirement, subject to advisor approval.

### Minor Requirements

- Mathematics 215 and either Mathematics 121 or Mathematics 216.
- Mathematics 317 and 333.
- Two additional electives in mathematics at the 200 level or higher.

### Requirements for Honors

Honors are granted to those senior mathematics majors who, by means of their course work, senior paper, and oral presentation, have given evidence of their ability, initiative, and interest in the study of mathematics. High Honors are awarded to the exceptionally able students.

### Courses

- 113
**Calculus I**NA/QU

*R.Manning, J.Sabloff*

An introduction to calculus of a single variable. Topics include limits, differentiation and integration, and the fundamental theorem of calculus with applications to the natural and social sciences. The intensive section offered each spring, Math 113b, is designed for students who need and desire extra help with precalculus concepts; it meets for three 2-hour sessions each week. Prerequisite: A solid background in precalculus. - 114
**Introductory Integral Calculus**NA/QU

*R.Manning, S.Wang*

An introduction to the theory and applications of the definite and indefinite integral. Includes numerical and analytical techniques for computing integrals and applications to differential equations. NOTE: This is a quarter-long course, typically offered in the first half of both the fall and spring semesters. Students typically take either Math 115 or 116 or 117 in the second half of the semester in which they take Math 114. Prerequisite: Math 113 or advanced placement. - 115
**Calculus Applications: Series and Complex Numbers**NA/QU

*L.Butler, C.Greene, R.Manning, S.Wang*

Infinite sequences and series, Taylor approximations, polar coordinates and complex numbers. The significance of these topics in mathematics and their applications in the natural sciences are discussed. NOTE: This is a quarter-long course. It is typically offered either in the 2nd half of a semester as a followup to Math 114, or in the 1st half of a semester, typically followed by either Math 116 or 117. Prerequisite: Math 114 or advanced placement. - 116
**Calculus Applications: Probability Distributions**NA/QU

*L.Butler, C.Greene*

Probability distributions and their applications in the natural and social sciences: the concept of probability and conditional probability; discrete and continuous random variables; expected value and variance; applications of the binomial, Poisson, exponential and normal distributions; and the Central Limit Theorem. NOTE: This is a quarter-long course. It is typically offered in the second half of a semester as a followup to Math 114 or 115. Prerequisite: Math 114 or advanced placement. - 117
**Calculus Applications: Multivariable Optimization**NA/QU

*Staff*

Introduction to multivariable differential calculus: partial derivatives and gradients; unconstrained and constrained optimization; and applications of the Lagrange Multiplier Theorem to economic models. Students who earn a half credit for Math 117 may not also earn a full credit for Math 121, since the first half of Math 121 duplicates most of what is taught in Math 117. NOTE: This is a quarter-long course. It is typically offered in the second half of a semester, as a followup to either Math 114 or 115. Prerequisite: Math 114 or advanced placement. - 121
**Calculus III**NA/QU

*Y.Tai, J.Tecosky-Feldman*

An introduction to functions of several variables, vector geometry, partial derivatives, maxima & minima, Taylor's Theorem, infinite series, multiple integrals, line integrals, and Green's and Stokes' Theorem. Students who earn a half credit for Math 117 may not also earn a full credit for Math 121, since the first half of Math 121 duplicates most of what is taught in Math 117. Prerequisite: Math 114 and 115 or 116 or 117, or advanced placement. - 215
**Linear Algebra**NA/QU

*C.Greene, S.Wang*

An introduction to linear algebra: vector spaces, linear transformations and matrices, determinants, quadratic forms and eigenvalue problems. Applications to differential equations and linear models are discussed. Prerequisite: Math 114 and 115 or 116 or 117, or 121, or advanced placement. - 216
**Advanced Calculus**NA

*Y.Tai*

Calculus of several variables: continuous and differentiable functions on Euclidean spaces, extreme value problems, inverse and implicit function theorems, multiple integration, Green's and Stokes' Theorems. Prerequisite: Math 215. - 317
**Analysis I**NA

*S.Wang*

A rigorous development of topics in calculus, including detailed treatment of the axioms of the real number line, cardinality, topology of normed spaces, compactness, and various notions of convergence. This course also serves as a thorough introduction to clear, correct writing of mathematical proofs. Prerequisite: Math 215 and either Math 121 or 216, or instructor consent. Corequisite of Math 299 for students who have not had Math 216 or math at the 300 level. - 318
**Analysis II**NA

*S.Wang*

A continuation of Math 317, focusing particularly on sequences and series of functions with applications (e.g., Fourier series, existence and uniqueness of solutions to differential equations). Other advanced topics (such as the Lebesgue integral, calculus of variations, Fourier transforms, special functions, approximation theorems or fixed point theorems) are included according to instructor and student interest. Prerequisite: Math 317. - 333
**Algebra I**NA

*Y.Tai*

A rigorous treatment of fundamental algebraic structures. Topics include: axioms for integers, modular arithmetic, polynomials, rings, fields, and introduction to groups. Prerequisite: Math 215 and either Math 121 or 216, or instructor consent. Corequisite of Math 299 for students who have not had Math 216 or math at the 300 level. - 334
**Algebra II**NA

*Y.Tai*

A continuation of Math 333a. Topics include: Sylow's theorems for groups, finite abelian groups, finite fields, Galois theory, modules, and advanced linear algebra. Prerequisite: Math 333 or consent of the instructor. - 399
**Senior Seminar**NA

*L.Butler, R.Manning, W.Miao, J.Sabloff, Y.Tai, S.Wang*

Seminar for students writing senior papers, dealing with the oral and written exposition of advanced material. Prerequisite: Open to senior mathematics majors.

**ELECTIVES**

- 103
**Introduction to Probability and Statistics**NA/QU

*W.Miao*

Basic concepts and methods of elementary probability and quantitative reasoning, with practical applications. Topics include: sample average and standard deviation, normal curves, regression, expected value and standard error, confidence intervals and hypothesis tests. - 104
**Calculus: Concepts and History**NA/QU (Cross-listed in Independent College Programs)

*J.Tecosky-Feldman*

An introduction to the history and development of the ideas of calculus, one of the most beautiful and useful creations of the human intellect. Beginning with a study of achievements of Archimedes and his predecessors, the course will follow the historical progression of the concepts of function, derivative and integral, including developments, such as fractals. In addition to regular problem sets, students will be required to write essays explaining the important concepts of the course. This course is suitable for students interested in a nontechnical survey of the ideas of calculus. In particular, it does not cover the same amount of material as Math 113, and cannot substitute for Math 113 in any course requiring Math 113 as a prerequisite (such as Math 114). Prerequisite: Not ordinarily open to students who have studied calculus previously. - 123
**Community Math Teaching Project**NA/QU (Cross-listed in Education)

*J.Sabloff*

A service-learning course in which students teach "math labs" to high school geometry students. Students will develop effective teaching methods through pedagogical theory and practice, and will explore the context in which mathematics is taught in high school. - 202
**Introduction to Number Theory**NA

*L.Butler*

An introduction to the classical theory of numbers. Topics include: primes and divisibility, congruences, the Chinese Remainder Theorem, quadratic reciprocity, sums of squares, Diophantine equations, continued fractions, approximation by rationals, Pell's equation. Time permitting, we will discuss arithmetic functions related to the distribution of prime numbers. Emphasis will be placed on learning to generalize from examples to precise conjectures. Prerequisite: MATH 115 or consent. Typically offered in alternate years. - 203
**Statistical Methods and Their Applications**NA/QU

*W.Miao*

An introduction to statistical methods used to analyze data in the natural and social sciences. It covers probability distributions, the binomial and Poisson distributions, the exponential and normal distributions, expected value and variance, confidence intervals and hypothesis testing, comparison of two samples, regression, and analysis of variance. Prerequisite: Math 114 or advanced placement. Typically offered in alternate years. - 204
**Differential Equations**NA/QU

*J.Tecosky-Feldman*

Ordinary differential equations: the general theory of first-order equations, linear equations of higher order, qualitative analysis of nonlinear systems, and computational methods. Other topics, such as series solutions or an introduction to partial differential equations and Fourier series, may be included. Elements of linear algebra are developed as needed. Emphasis is on applications, especially on differential equations as mathematical models in the physical, biological and social sciences. Prerequisite: Math 114 and 115 or 116 or 117, or advanced placement. - 205
**Topics in Geometry**NA/QU

*J.Tecosky-Feldman*

An introduction to several areas in classical and modern geometry: analytic geometry, conic sections, Platonic solids and polyhedra, tessellations of the plane, projective, hyperbolic, and differential geometry. Students will see how symmetry groups serve as a unifying theme in geometry. This course will introduce students to the skill of writing formal mathematical proofs. Prerequisite: Math 121 or instructor consent. Typically offered in alternate years. - 210
**Linear Optimization and Game Theory**NA/QU (Cross-listed in Computer Science and Economics)

*L.Butler*

Covers in depth the mathematics of optimization problems with a finite number of variables subject to constraints. Applications of linear programming to the theory of matrix games and network flows are covered, as well as an introduction to nonlinear programming. Emphasis is on the structure of optimal solutions, algorithms to find them, and the underlying theory that explains both. This course is designed for students interested in computer science, economics, or mathematics. Prerequisite: Math 121 and instructor consent, or Math 215. Typically offered in alternate years. - 218
**Probability**NA/QU

*C.Greene*

Probabilistic techniques with applications: The concept of probability and conditional probability, random variables, stochastic processes, applications to statistics, Markov chains and processes, and queuing theory. Prerequisite: Math 116 or 121, or instructor consent. Typically offered in alternate years. - 220
**Elementary Complex Analysis**NA/QU

*Staff*

Line integrals; complex derivatives; Cauchy's theorem and residue calculations; elementary conformal mapping; harmonic functions. Prerequisite: Math 121 or 215. Typically offered yearly in alternation with Bryn Mawr. - 222
**Introduction to Scientific Computing**NA

*R.Manning*

A survey of major algorithms in modern scientific computing (including root-finding, optimization, Monte Carlo, discretization of differential equations, and search algorithms) and their application across the natural and social sciences. Prerequisite: Math 121 or 216, and experience with Mathematica or a programming language, or permission of the instructor. Typically offered in alternate years. - 235
**Information and Coding Theory**NA (Cross-listed in Computer Science)

*S.Lindell*

Prerequisite: Math 215 or equivalent (may be taken concurrently). Offered occasionally. - 299
**Bridge to Advanced Mathematics**NA

*J.Sabloff*

An introduction to deductive reasoning, mathematical proof, and fundamental ideas of higher mathematics. Emphasis will be placed on developing strategies for understanding and constructing proofs. Topics include basic logic, set theory, and relations. Prerequisite: Math 121 and Math 215. Concurrent registration in Math 317 or Math 333. - 335
**Topology I**NA

*J.Sabloff*

Generalizes topological concepts from Euclidean spaces to arbitrary topological spaces, and introduces elements of algebraic topology. Concepts covered include continuity, connectedness, and compactness. The course culminates in an exploration of the fundamental group and covering spaces. Prerequisite: Math 317 and 333 or instructor consent. Typically offered yearly in alternation with Bryn Mawr. - 336
**Topology II**NA

*J.Sabloff*

Algebraic topology and its applications to low-dimensional topology. The course investigates surfaces, knots, and 3-manifolds using the fundamental group, basic simplicial homology, and the mapping class group. Prerequisite: Math 335. Typically offered yearly in alternation with Bryn Mawr. - 340
**Analysis of Algorithms**NA (Cross-listed in Computer Science)

*S.Lindell*

Prerequisite: CMSC 206 Typically offered in alternate years. - 345
**Theory of Computation**NA (Cross-listed in Computer Science)

*S.Lindell*

Prerequisite: Computer Science/Math 231. Typically offered in alternate years. - 349
**Mathematical Methods in Economics**NA (Cross-listed in Economics)

Prerequisite: Math 317 or instructor consent. - 390
**Advanced Topics in Algebra**NA

*Staff*

Prerequisite: Math 333 or instructor consent. - 391
**Advanced Topics in Geometry and Topology**NA

*Staff*

Prerequisite: Math 318 or consent. - 392
**Advanced Topics in Analysis**NA

*Staff*

Prerequisite: Math 317 or equivalent. - 394
**Advanced Topics in Theoretical Computer Science & Discrete Mathematics**NA (Cross-listed in Computer Science)

*L.Butler* - 395
**Advanced Topics in Combinatorics**NA

*Staff*

Prerequisite: Math 333 or instructor consent. - 396
**Advanced Topics in Probability and Statistics**NA

*W.Miao*

Prerequisite: Math 218 or permission of instructor. - 397
**Advanced Topics in Applied Mathematics**NA

*R.Manning*

Prerequisite: Math 317 or instructor consent. - 460
**Teaching Assistantship in Mathematics**NA

*J.Tecosky-Feldman*

Students work as assistants to a faculty member in an introductory mathematics course for a semester, offering various kinds of classroom support including problem sessions, review, tutoring, and laboratory assistance. Open to junior and senior majors by invitation. May be taken at most twice. - 480
**Independent Study**NA

*Staff*

Prerequisite: Instructor consent.

**COURSES AT BRYN MAWR COLLEGE **

- 001 Fundamentals of Mathematics
- 101,102 Calculus with Analytic Geometry
- 104 Elements of Probability and Statistics
- 201 Multivariable Calculus
- 203 Linear Algebra
- 206 Transition to Higher Mathematics
- 210 Differential Equations with Applications
- 225 Introduction to Financial Mathematics
- 231 Discrete Mathematics
- 295 Select Topics in Mathematics
- 301, 302 Introduction to Real Analysis
- 303, 304 Abstract Algebra
- 311 Partial Differential Equations
- 312 Topology I
- 390 Number Theory
- 501 Graduate Real Analysis I
- 502 Graduate Real Analysis II
- 505 Graduate Topology I