2011-2012 Undergraduate Catalog

Mathematics

Students may complete a major or minor in Mathematics. With the major, students may complete the requirements for secondary school certification. Majors may complete an M.A. in Mathematics, if accepted into the combined A.B./M.A. program, or may enter the 3-2 Program in Engineering and Applied Science at the California Institute of Technology.

Faculty

Emi Arima, Instructor
Leslie Cheng, Associate Professor and Chair
Victor Donnay, Professor
Helen Grundman, Professor
Peter Kasius, Instructor
Paul Melvin, Professor
Amy Myers, Lecturer
Gregory Schneider, Instructor
Lisa Traynor, Professor (on leave semester II)

The Mathematics curriculum is designed to expose students to a wide spectrum of ideas in modern mathematics, train students in the art of logical reasoning and clear expression, and provide students with an appreciation of the beauty of the subject and of its vast applicability.

Major Requirements

A minimum of 10 semester courses is required for the major, including the six core courses listed below and four electives at or above the 200 level.

Core Requirements:

MATH B201 Multivariable Calculus (H121 or H216)
MATH B203 Linear Algebra (H215)
MATH B301 Real Analysis I (H317)
MATH B303 Abstract Algebra I (H333)
MATH B302 Real Analysis II (H318) or MATH B304 Abstract Algebra II (H334)
MATH B398 or B399 Senior Conference

With the exception of Senior Conference, equivalent courses at Haverford or elsewhere may be substituted for Bryn Mawr courses with approval of the major adviser. In consultation with a major adviser, a student may also petition the department to accept courses in fields outside of mathematics as electives if these courses have serious mathematical content appropriate to the student's program.
Mathematics majors are encouraged to complete their core requirements other than Senior Conference by the end of their junior year. Senior Conference must be taken during the senior year. Students considering the possibility of graduate study in mathematics or related fields are urged to go well beyond the minimum requirements of the major. In such cases, a suitable program of study should be designed with the advice of a major adviser.

Honors

A degree with honors in mathematics will be awarded by the department to students who complete the major in mathematics and also meet the following further requirements: at least two additional semesters of work at the 300 level or above (this includes Supervised Work 403), completion of a meritorious project consisting of a written thesis and an oral presentation of the thesis, and a major grade point average of at least 3.6, calculated at the end of the senior year.

Minor Requirements

The minor requires five courses in mathematics at the 200 level or higher, of which at least two must be at the 300 level or higher.

Advanced Placement

Students entering with a 4 or 5 on the Calculus AB advanced placement test will be given credit for MATH 101 and should enroll in MATH 102 as their first mathematics course. Students entering with a 4 or 5 on the Calculus BC advanced placement test will be given credit for MATH 101 and 102, and should enroll in MATH 201 as their first mathematics course. All other students are strongly encouraged to take the Mathematics Placement Exam so they can be best advised.

A.B./M.A. Program

For students entering with advanced placement credits it is possible to earn both the A.B. and M.A. degrees in an integrated program in four or five years.

3-2 Program in Engineering and Applied Science

See the description of the 3-2 Program in Engineering and Applied Science, offered in cooperation with the California Institute of Technology, for earning both an A.B. at Bryn Mawr and a B.S. at Cal Tech.

MATH B001 Fundamentals of Mathematics

Basic techniques of algebra, analytic geometry, graphing, and trigonometry for students who need to improve these skills before entering other courses that use them, both inside and outside mathematics. Placement in this course is by advice of the department and permission of the instructor.
1.0 units
Staff

MATH B101, B102 Calculus with Analytic Geometry I and II

Differentiation and integration of algebraic and elementary transcendental functions, with the necessary elements of analytic geometry and trigonometry; the fundamental theorem, its role in theory and applications, methods of integration, applications of the definite integral, infinite series. May include a computer lab component. Prerequisite: math readiness or permission of the instructor. Students in the calculus sequence need a grade of 2.0 or better to continue with the next course.
Division II and Quantitative Skills
Quantitative Methods (QM)
1.0 units
Arima,E., Schneider,G., Traynor,L.

MATH B104 Elements of Probability and Statistics

This course introduces students to key concepts in both descriptive and inferential statistics. Students learn how to collect, describe, display, and interpret both raw and summarized data in meaningful ways. Topics include summary statistics, graphical displays, correlation, regression, probability, the law of averages, expected value, standard error, the central limit theorem, hypothesis testing, sampling procedures, and bias. Students learn to use statistical software to summarize, present, and interpret data. This course may not be taken after any other statistics course. Prerequisite: math readiness or permission of instructor.
Quantitative Skills
Quantitative Methods (QM)
1.0 units
Grundman,H.

MATH B201 Multivariable Calculus

Vectors and geometry in two and three dimensions, partial derivatives, extremal problems, double and triple integrals, line and surface integrals, Green's and Stokes' Theorems. May include a computer lab component. Prerequisite: MATH 102 or permission of instructor.
Division II and Quantitative Skills
Quantitative Methods (QM)
1.0 units
Kasius,P.

MATH B203 Linear Algebra

Matrices and systems of linear equations, vector spaces and linear transformations, determinants, eigenvalues and eigenvectors, inner product spaces and quadratic forms. May include a computer lab component. Prerequisite: MATH 102 or permission of instructor.
Division II and Quantitative Skills
Quantitative Methods (QM)
1.0 units
Staff, Kasius,P.

MATH B206 Transition to Higher Mathematics

An introduction to higher mathematics with a focus on proof writing. Topics include active reading of mathematics, constructing appropriate examples, problem solving, logical reasoning, and communication of mathematics through proofs. Students will develop skills while exploring key concepts from algebra, analysis, topology, and other advanced fields. Corequisite: MATH 203; not open to students who have had a 300-level math course.
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Staff

MATH B210 Differential Equations with Applications

Ordinary differential equations, including general first-order equations, linear equations of higher order and systems of equations, via numerical, geometrical, and analytic methods. Applications to physics, biology, and economics. Corequisite: MATH 201 or 203.
Division II and Quantitative Skills
Quantitative Methods (QM)
1.0 units
Schneider,G.

MATH B221 Introduction to Topology and Geometry

An introduction to the ideas of topology and geometry through the study of knots and surfaces in three-dimensional space. The course content may vary from year to year, but will generally include some historical perspectives and some discussion of connections with the natural and life sciences. Corequisite: MATH 201 or 203.
Division II and Quantitative Skills
Quantitative Methods (QM)
1.0 units
Arima,E.

MATH B231 Discrete Mathematics

An introduction to discrete mathematics with applications to computer science. Topics include set theory, functions and relations, propositional logic, proof techniques, recursion, counting techniques, difference equations, graphs, and trees.
Division II and Quantitative Skills
Quantitative Methods (QM)
CROSS-LISTED AS CMSC-B231
1.0 units
Cheng,L.

MATH B261 Introduction to Harmonic Analysis and Wavelets

A first introduction to harmonic analysis and wavelets. Topics to be covered: Fourier series, Fourier transform, wavelets, and their applications, including signal processing and medical imaging. Prerequisite: MATH 203 or permission of instructor.
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Cheng,L.
Not offered in 2011-12.

MATH B290 Elementary Number Theory

Properties of the integers, divisibility, primality and factorization, congruences, Chinese remainder theorem, multiplicative functions, quadratic residues and quadratic reciprocity, continued fractions, and applications to computer science and cryptography. Prerequisite: MATH 102.
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Staff
Not offered in 2011-12.

MATH B295 Select Topics in Mathematics

What is enumerative combinatorics? It is a collection of techniques for enumerating a set of objects (saying how many) without counting them (listing all the possibilities). Combinatorial techniques are often applied to questions of probability in situations when all outcomes are equally likely. Although combinatorial problems can often be stated in the language of puzzles and games, the results have applications throughout mathematics, both pure and applied. Prerequisite: MATH 102
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Myers,A.

MATH B301, B302 Introduction to Real Analysis I and II

The real number system, elements of set theory and topology, continuous functions, uniform convergence, the Riemann integral, power series, Fourier series and other limit processes. Prerequisite: MATH 201.
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Donnay,V., Cheng,L.

MATH B303, B304 Abstract Algebra I and II

Groups, rings, and fields and their homomorphisms. Quotient groups, quotient rings, and the isomorphism theorems. Standard examples including symmetric groups, free groups, and finitely generated abelian groups; integral domains, PID's and UFD's, and polynomial rings; finite and infinite fields. Sylow theory and field extensions. Additional topics may include: Galois theory, modules and canonical forms of matrices, algebraic closures, and localization. Prerequisite: MATH 203.
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Melvin,P.

MATH B311 Partial Differential Equations

Heat and wave equations on bounded and unbounded domains, Laplace's equation, Fourier series and the Fourier transform, qualitative behavior of solutions, computational methods. Applications to the physical and life sciences. Prerequisite: MATH 301 or permission of instructor.
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Staff
Not offered in 2011-12.

MATH B312 Topology

General topology (topological spaces, continuity, compactness, connectedness, quotient spaces), the fundamental group and covering spaces, introduction to geometric topology (classification of surfaces, manifolds). Typically offered yearly in alternation with Haverford. Corequisite: MATH 301, MATH 303, or permission of instructor.
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Staff
Not offered in 2011-12.

MATH B315 Geometry

Differential geometry is a deep area of mathematics which focuses on using the tools of integral and differential calculus to study geometric and topological properties of objects and spaces. By restricting ourselves to "low-dimensional" objects––curves and surfaces in 3-space––we can easily adapt techniques familiar from multivariable calculus to explore the rich structure underlying them, while taking a glimpse into the goals and approaches of the field as a whole. Prerequisite: MATH 201 or permission of instructor.
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Schneider,G.

MATH B322 Functions of Complex Variables

Analytic functions, Cauchy's theorem, Laurent series, calculus of residues, conformal mappings, Moebius transformations. Prerequisite: MATH 301 or permission of instructor.
Division II: Natural Science
Quantitative Methods (QM)
1.0 units
Melvin,P.

MATH B395 Research Seminar

A research seminar for students involved in individual or small group research under the supervision of the instructor. With permission, the course may be repeated for credit. Prerequisite: Permission of instructor.
Quantitative Methods (QM)
1.0 units
Melvin,P., Cheng,L., Donnay,V., Grundman,H., Traynor,L.

MATH B396 Research Seminar

A research seminar for students involved in individual or small group research under the supervision of the instructor. With permission, the course may be repeated for credit. Prerequisite: Permission of instructor.
Quantitative Methods (QM)
1.0 units
Melvin,P., Cheng,L., Donnay,V., Grundman,H.

MATH B398 Senior Conference

A seminar for seniors majoring in mathematics. Topics vary from year to year.
Quantitative Methods (QM)
1.0 units
Cheng,L.

MATH B399 Senior Conference

A seminar for seniors majoring in mathematics. Topics vary from year to year.
Quantitative Methods (QM)
1.0 units
Cheng,L.

MATH B403 Supervised Work

1.0 units
Staff

MATH B501 Graduate Real Analysis I

In this course we will study the theory of measure and integration. Topics will include Lebesgue measure, measurable functions, the Lebesgue integral, the Riemann-Stieltjes integral, complex measures, differentiation of measures, product measures, and L^p spaces. Cheng,L., Donnay,V.
Not offered 2011-12

MATH B502 Graduate Real Analysis II

This course is a continuation of MATH 501.
Cheng,L.
Not offered 2011-12

MATH B503 Graduate Algebra I

This is the first course in a two course sequence providing a standard introduction to algebra at the graduate level. Topics in the first semester include categories, groups, rings, modules, and linear algebra. Grundman,H.

MATH B504 Graduate Algebra II

This course is a continuation of MATH 503, the two courses providing a standard introduction to algebra at the graduate level. Topics in the second semester will include linear algebra, fields, Galois theory, and advanced group theory.
Grundman,H.

MATH B505 Graduate Topology I (Algebraic Topology)

MATH 505 and MATH 506 offer an introduction to topology at the graduate level. These courses can be taken in either order. Topics covered include homology and cohomology theory and applications; duality on manifolds.
Melvin,P.
Not offered 2011-12

MATH B506 Graduate Topology II (Differential Topology)

MATH 505 and MATH 506 offer an introduction to topology at the graduate level. These courses can be taken in either order. Topics covered include smooth manifolds and maps; transversality; differential forms and integration on manifolds.
Traynor,L.