# Natural Sciences: Mathematics, 2010-2011

### 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 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, 203, 210, 215, and 286 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 or in scientific computing (the former being more theoretical, and the latter more applied). 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 concentrations in computer science, scientific computing, 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 statistics (Math 203 or 286) and mathematical modeling (Math 204, 210, or 222). Minors preparing for graduate work in physical chemistry or theoretical physics should take Complex Analysis (Math 392) 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**

*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**

*Visiting Assistant Professor* **David Lippel**

*Visiting Assistant Professor*** Clay Shonkwiler**

### Major Requirement

- Mathematics 215, and either Mathematics 121 or Mathematics 216.
- Mathematics 317 and 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 this requirement.
- 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 student.

### Courses

#### 113 Calculus I NA/QU

*C.Shonkwiler*

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. Discussion 3 hours MWF 3-4 is mandatory. Prerequisite: A solid background in precalculus. Does not count toward the major. Typically offered every Semester.

#### 114 Calculus II NA/QU

*Staff*

A continuation of Math 113 that includes an introduction to the theory and applications of the definite and indefinite integral, as well as an introduction to infinite series and Taylor approximations. May include other topics, e.g., differential equations, parametric curves, polar coordinates, or complex numbers. Prerequisite: Math 113 or advanced placement. Does not count toward the major. Typically offered every Semester.

#### 115 Enriched Calculus II NA/QU

*C.Greene*

A "bridge" course for students who have completed most of a standard college first-year calculus class. Includes a careful treatment of the convergence of sequences and infinite series, the theory of Taylor series, and substantial treatment of an additional topic, often introducing an area of math distinct from calculus, e.g., probability or discrete math. Prerequisite: Advanced placement (equivalent to mastery of AB Calculus). Does not count toward the major.

#### 121 Calculus III NA/QU

*Y.Tai*

An introduction to functions of several variables, vector geometry, partial derivatives, maxima & minima, Taylor's Theorem, multiple integrals, line integrals, and Green's and Stokes' Theorem. Prerequisite: MATH 114 or 115 or equivalent placement. Typically offered every Semester.

#### 215 Linear Algebra NA/QU

*J.Tecosky-Feldman*

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 or 115 or 121 or equivalent placement. Typically offered every Semester.

#### 216 Advanced Calculus NA

*J.Sabloff*

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. Typically offered every Spring.

#### 317 Analysis I NA

*C.Shonkwiler*

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. Typically offered every Fall.

#### 318 Analysis II NA

*C.Shonkwiler*

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 measure theory, the Lebesgue integral, calculus of variations, Fourier transforms, approximation theorems or fixed point theorems) are included according to instructor and student interest. Prerequisite: MATH 317. Typically offered every Spring.

#### 333 Algebra I NA

*L.Butler*

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. Typically offered every Fall.

#### 334 Algebra II NA

*L.Butler*

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 instructor consent. Typically offered every Spring.

#### 399 Senior Seminar NA

*D.Lippel, R.Manning, W.Miao, C.Greene, L.Butler, C.Shonkwiler*

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

### MATHEMATICS ELECTIVE COURSES

#### 103 Introduction to Probability and Statistics NA/QU

*R.Manning*

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. Typically offered in alternate years.

#### 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. Offered occasionally.

#### 105 Applied Modeling with Calculus NA/QU

*J.Tecosky-Feldman*

An introduction to aspects of calculus useful in applied work in the natural and social sciences, with a strong emphasis on developing mathematical modeling skills. Topics include differential calculus of functions of one and several variables, differential equations. Applications to biology, economics, and physics. Does not count toward the major.

#### 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. Offered occasionally.

#### 202 Introduction to Number Theory NA

*Y.Tai*

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 instructor consent. Offered occasionally.

#### 203 Statistical Methods and Their Applications NA/QU

*L.Butler*

An introduction to statistical methods used to analyze data in the natural and social sciences. It covers descriptive statistics, the binomial and normal distributions, expected value and variance, confidence intervals and hypothesis testing, comparison of two samples, regression, and analysis of variance. A required computer lab, using R, is taught alongside this course. Prerequisite: MATH 114 or placement at the level of MATH 115 or higher. Typically offered every Fall.

#### 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. Offered occasionally.

#### 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 115 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 and hidden Markov models. 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 215 or MATH 115 and concurrent registration in MATH 215. Typically offered in alternate years.

#### 218 Probability NA/QU

*W.Miao*

An introduction to probability theory. Topics include: sample spaces, combinatorics, conditional probability, independence, discrete and continuous random variables, functions of random variables, expected value and variance, the moment generating function, and some basic limit theorems. Prerequisite: MATH 116 or 121, or consent. Typically offered in alternate years.

#### 222 Scientific Computing: Continuous Systems NA/QU

*R.Manning*

A survey of major algorithms in modern scientific computing, with a focus on continuous problems. Topics include root-finding, optimization, Monte Carlo methods, and discretization of differential equations, with applications in the natural and social sciences. Prerequisite: MATH 114 or 115 or equivalent placement. Typically offered in alternate years.

#### 235 Information and Coding Theory NA (Cross-listed in Computer Science)

*S.Lindell*

Prerequisite: Math 215 (may be taken concurrently). Offered occasionally.

#### 286 Applied Multivariate Statistical Analysis NA/QU

*W.Miao*

An introduction to multivariate statistical analysis. The course includes methods for choosing, fitting, and evaluating multiple regression models and analysis of variance models. A required computer lab, using R, is taught alongside this course. Prerequisite: One of the following: MATH 203, PSYC 200, or ECON 203 or 204 or consent of instructor. MATH 215 is recommended. Typically offered in alternate years.

#### 299 Bridge to Advanced Mathematics NA

*D.Lippel*

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. This is a quarter-long course; it is taught in the first half of the fall semester. Prerequisite: MATH 121 and MATH 215. Concurrent registration in MATH 317 or MATH 333. Does not count toward the major.

#### 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

*C.Shonkwiler*

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: CMSC/MATH 231. Typically offered in alternate years.

#### 390 Algebraic Number Theory NA

*Y.Tai*

Prerequisite: MATH 333 or consent. Offered occasionally.

#### 391 Advanced Topics in Geometry and Topology NA

*Staff*

Prerequisite: MATH 317. Offered occasionally.

#### 392 Complex Analysis NA

*Staff*

Prerequisite: MATH 317 or consent. Typically offered yearly in alternation with Bryn Mawr.

#### 394 Advanced Topics in Theoretical Computer Science & Discrete Mathematics NA (Cross-listed in Computer Science)

*D.Lippel*

Prerequisite: MATH 317 or 333. Typically offered in alternate years.

#### 395 Advanced Topics in Combinatorics NA

*Staff*

Prerequisite: MATH 333 or consent. Offered occasionally.

#### 396 Advanced Topics in Probability and Statistics NA

*W.Miao*

Prerequisite: MATH 218 or consent. Typically offered in alternate years.

#### 397 Partial Differential Equations NA

*R.Manning*

Prerequisite: MATH 317 or instructor consent. Offered occasionally.

#### 460 Teaching Assistantship in Mathematics NA

*L.Butler*

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. Does not count toward the major.

#### 480 Independent Study NA

*W.Miao*

Prerequisite: Instructor consent.

### MATHEMATICS COURSES AT BRYN MAWR COLLEGE

005 **Math Workshop**

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**

231 **Discrete Mathematics**

290 **Elementary Number Theory**

295 **Select Topics in Mathematics**

301, 302 **Introduction to Real Analysis**

303, 304 **Abstract Algebra**

311 **Partial Differential Equations**

312 **Topology I**