# Natural Sciences: Mathematics and Statistics, 2012-2013

### Description

The Department of Mathematics offers courses that promote rigorous thinking in a systematic, deductive, intellectual discipline; present to the student the direction and scope of mathematical development; foster technical competence in mathematics as an aid to the better comprehension of the physical, biological and social sciences guide and direct mathematics majors toward an interest in mathematical research.

The department offers several intermediate-level courses in multivariable mathematics designed for both majors and non-majors. These include MATH 121 and 215–216, which provide an important foundation for more advanced work in mathematics and other sciences. MATH 114 or 115 (or equivalent advanced placement) is sufficient background for any of these courses. A program that includes courses such as MATH 113, 114, 203, 210, 215, and STAT 286 is especially appropriate for majors in the social sciences. Students planning graduate study in economics should consider taking MATH 317.

The Department of Mathematics and Statistics houses the major in mathematics, the minor in mathematics and the minor in statistics, and collaborates with other departments in several concentrations (see below). Mathematics majors take a 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. We urge mathematics majors to gain facility in the use of computers, either through the introductory courses CMSC 105, 106, through applied math electives (like MATH 210, 218, 222 or 397) or through independent work.

Mathematics minors take the same core sequence as do the majors, though not necessarily to the same depth, followed by a selection of electives tailored to the student's interest. Statistics minors take a separate core sequence in probability and statistics, with later flexibility in pursuing either a more applied or a more theoretical track.

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 a 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 listedunder their own headings in this catalog.

Mathematics students (either majors or minors) preparing for a teaching career in mathematics should take one elective in probability and statistics (STAT 203 or MATH 218) and one in geometry or topology (MATH 205 or 335). Students preparing for employment in industry immediately after college should take electives in statistics (STAT 203, 286 or 328) and mathematical modeling (MATH 210 or 222). Students 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 and MATH 335 and 337 for geometry and topology).

### Faculty

*Professor Emeritus* **William C. Davidon**

*Professor Emeritus* **Yung-sheng Tai**

*Professor* **Lynne Butler**

*J. McLain King 1928 Professor of Mathematics* **Curtis Greene**

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

*Associate Professor* **Weiwen Miao**

*Associate Professor* **Joshua Sabloff,** Chair

*Senior Lecturer* **Jeffrey Tecosky-Feldman**

*Visiting Assistant Professor* **Tim DeVries**

*Visiting Assistant Professor* **Ellen Gasparovic**

*Visiting Assistant Professor* **David Lippel**

### Major Requirement

- MATH 215, and either MATH 121 or 216.
- MATH 317 and 333, and either MATH 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. Students may not count MATH 299, 399, 460 nor 480 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.

Students may substitute equivalent courses in mathematics at Bryn Mawr for any requirement, subject to advisor approval.

### Minor Requirements

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

### Requirements for Honors

The department grants Honors 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

#### 221 Number Systems and Computer Arithmetic NA/QU (Cross-listed in Computer Science)

*S.Lindell*

*Prerequisite*: MATH 231 (Discrete Math) or a 200-level mathematics course that includes proofs.

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

*S.Lindell*

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

#### 340 Analysis of Algorithms NA (Cross-listed in Computer Science)

*S.Lindell*

*Prerequisite*: CMSC 106. 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.

### MATHEMATICS CORE COURSES

#### 113 Calculus I NA/QU

*Staff*

This is 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. Does not count toward the major. *Prerequisite*: A solid background in precalculus. Typically offered every Fall.

#### 114 Calculus II NA/QU

*J.Sabloff*

This continuation of MATH 113 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. It may include other topics, e.g., differential equations, parametric curves, polar coordinates or complex numbers. Does not count toward the major. *Prerequisite*: MATH 113 or advanced placement. Typically offered every semester.

#### 115 Enriched Calculus II NA/QU

*C.Greene*

This is 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 mathematics that is distinct from calculus e.g., probability or discrete math. Does not count toward the major. *Prerequisite*: Advanced placement (equivalent to mastery of AB Calculus). Typically offered every Fall.

#### 121 Calculus III NA/QU

*T.DeVries*

This is an introduction to functions of several variables, vector geometry, partial derivatives, maxima and 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

*C.Greene*

This is an introduction to linear algebra: vector spaces, linear transformations and matrices, determinants, quadratic forms and eigenvalue problems. We discuss applications to differential equations and linear models. *Prerequisite*: Either MATH 114, 115 or 121 or equivalent placement. Typically offered every semester.

#### 216 Advanced Calculus NA

*J. Tecosky-Feldman*

This course works with calculus of several variables: continuous and differentiable functions on Euclidean spaces, extreme value problems, inverse and implicit function theorems, and multiple integration, Green's and Stokes' Theorems. *Prerequisite*: MATH 215. Typically offered every Spring.

#### 317 Analysis I NA

*J.Sabloff*

This is 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 the instructor’s consent. Corequisite: MATH 299, for students who have not had MATH 216 or taken a 300-level mathematics course. Typically offered every Fall.

#### 318 Analysis II NA

*J. Sabloff/D. Lippel*

This is 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). We include other advanced topics (such as measure theory, the Lebesgue integral, calculus of variations, Fourier transforms, approximation theorems or fixed point theorems) according to instructor and student interest. *Prerequisite*: MATH 317. Typically offered every spring..

#### 333 Algebra I NA

*E. Beazley*

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 the instructor's consent. *Co-requisite*: Math 299, for students who have not had MATH 216 or taken a 300-level mathematics course. Typically offered every fall.

#### 334 Algebra II NA

*C. Greene/D. Lippel*

This is a continuation of MATH333a. Topics include: Sylow's theorems for groups, finite abelian groups, finite fields, Galois theory, modules and advanced linear algebra. *Prerequisite*: MATH 333 or the instructor’s consent. Typically offered every spring.

#### 399 Senior Seminar NA

*Staff*

This is a seminar for students writing senior papers, dealing with the oral and written exposition of advanced material. Open to senior mathematics majors.

### MATHEMATICS ELECTIVE COURSES

#### 104 Calculus: Concepts and History NA/QU (Cross-listed in Independent College Programs)

*J.Tecosky-Feldman*

This is 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 follows the historical progression of the concepts of function, derivative and integral, including developments, such as fractals. In addition to regular problem sets, students are 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 students cannot substitute this course for MATH 113 in order to take any course (such as MATH 114) requiring MATH 113 as a prerequisite. Not ordinarily open to students who have studied calculus previously. Offered occasionally.

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

*J.Tecosky-Feldman*

This is 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, and differential equations. We explore applications to biology, economics and physics. Does not count toward the major.

#### 202 Introduction to Number Theory NA

*Staff*

This is 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 and Pell's equation. Time permitting, we discuss arithmetic functions related to the distribution of prime numbers. The course emphasizes learning to generalize from examples to precise conjectures. *Prerequisite*: MATH 115 or the instructor’s consent. Offered occasionally.

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

*L.Butler*

*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*

This course examines ordinary differential equations: the general theory of first-order equations, linear equations of higher order, qualitative analysis of nonlinear systems, and computational methods. It may also include other topics, such as series solutions or an introduction to partial differential equations and Fourier series. 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 or 115, or advanced placement. Offered occasionally.

#### 205 Topics in Geometry NA/QU

*J.Tecosky-Feldman*

This is an introduction to several areas in classical and modern geometry: analytic geometry, conic sections, Platonic solids and polyhedra, tessellations of the plane and projective, hyperbolic and differential geometry. Students see how symmetry groups serve as a unifying theme in geometry. This course introduces students to the skill of writing formal mathematical proofs. *Prerequisite*: MATH 114 or 115, advanced placement, or the instructor’s consent. Typically offered in alternate years.

#### 210 Linear Optimization and Game Theory NA/QU (Cross-listed in Computer Science and Economics)

*L.Butler*

This course covers in depth the mathematics of optimization problems with a finite number of variables subject to constraints. We review applications of linear programming to the theory of matrix games and network flows, 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 115. Corequisite: MATH 215. Typically offered in alternate years.

#### 218 Probability NA/QU

*L.Butler*

This course offers 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 216 or 121 or the instructor’s consent. Typically offered in alternate years.

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

*R.Manning*

This is 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.

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

*W.Miao*

*Prerequisite*: MATH215 and one of the following: ECON 204, MATH 203, PSYC 200, or SOCL 215. Typically offered in alternate years.

#### 299 Bridge to Advanced Mathematics NA

*E.Beazley*

This is an introduction to deductive reasoning, mathematical proofs and fundamental ideas of higher mathematics. This course emphasizes 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. Does not count toward the major. *Prerequisite*: MATH 121 and MATH 215. Corequisite: MATH 317 or MATH 333.

#### 335 Topology I NA

*J.Sabloff*

This course generalizes topological concepts from Euclidean spaces to arbitrary topological spaces and introduces elements of algebraic topology. Topics include continuity, connectedness and compactness. The course culminates in an exploration of the fundamental group and covering spaces. *Prerequisite*: MATH 317 and 333, or the instructor’s consent. Typically offered yearly in alternation with Bryn Mawr

#### 337 Differential Geometry NA

*J.Sabloff*

This is a study of the differential geometry of curves and surfaces. Topics include both the local theory (including metrics, curvature and geodesics) and the global theory, culminating in the Gauss-Bonnet theorem. *Prerequisite*: MATH 317 or 216 and consent. Typically offered in alternate years.

#### 390 Advanced Topics in Algebra NA

*L.Butler*

Topic for 2011–12: Cryptography *Prerequisite*: MATH 333 or the instructor’s consent. Offered occasionally.

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

*Staff*

*Prerequisite*: MATH 317. Offered occasionally.

#### 392 Advanced Topics in Analysis NA

*T.DeVries*

T*Prerequisite*: MATH 317 or the instructor’s consent. Typically offered yearly in alternation with Bryn Mawr.

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

*D.Lippel*

*Prerequisite*: MATH 317 or 333, or the instructor’s consent.

#### 395 Advanced Topics in Combinatorics NA

*C.Greene*

Topics are chosen from enumerative combinatorics, permutations, Young tableaux and representation theory. Student projects permit a focus on one or more of these areas. *Prerequisite*: MATH 333 or the instructor’s consent. Offered occasionally.

#### 397 Advanced Topics in Applied Mathematics NA

*R.Manning*

*Prerequisite*: MATH 317 or the instructor’s consent. Offered occasionally.

#### 460 Teaching Assistantship in Mathematics NA

*Staff*

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

*Staff*

*Prerequisite*: The instructor’s consent.

### STATISTICS COURSES

#### 103 Introduction to Probability and Statistics NA/QU (Cross-listed in Statistics)

*W.Miao*

This course covers 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.

#### 328 Mathematical Statistics NA/QU (Cross-listed in Statistics)

*W.Miao*

This is an introduction to mathematical theory of statistics. Topics include Estimation, Hypothesis Testing, one-sample inference, two-sample inference and regression. Additional topics may include goodness-of-fit tests and analysis of variance. *Prerequisite*: MATH 218. Typically offered in alternate years.

#### 396 Mathematical Statistics NA (Cross-listed in Statistics)

*W.Miao*

*Prerequisite*: MATH 218 or the instructor’s consent. Typically offered in alternate years.

### 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**