Students may complete a major or minor in Mathematics. Within 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**

Leslie Cheng, Associate Professor

Victor Donnay, Professor

Helen Grundman, Professor (on leave semesters I and II)

Peter Kasius, Instructor

**Paul Melvin, Professor and Chair**

Djordje Millicevic, Assistant Professor

Amy Myers, Lecturer and Math Program Coordinator

Gregory Schneider, Lecturer

Lisa Traynor, Professor

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

The analysis and algebra sequences, MATH 301/302 and MATH 303/304, both have a strong proof writing focus. Consequently, students often find it useful to take a course such as MATH 206 (Transition to Higher Mathematics) before they enroll in these sequences, and in any case should consult with the instructor if they are unsure about their level of preparation.

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. A student may also, in consultation with a major adviser, 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 units of work at the 300 level or above (which may include one or two units of MATH 395/396 or MATH 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 possibly five) years.

**3-2 Program in Engineering and Applied Science**

Visit www.brynmawr.edu/catalog/2012-13/program/opportunities/32engineering.htmlfor 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.

**COURSES**

**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.

Units: 1.0

(Not Offered 2012-13)

Review of arithmetic and introduction to the basics of elementary and intermediate algebra for students whose mathematical backgrounds require such support. This course prepares students to take either MATH 001 or MATH 104 immediately thereafter. Placement in this course is by advice of the department. This is a half-credit course.

Units: 0.5

(Not Offered 2012-13)

A first course in one-variable calculus: Functions, limits, continuity, the derivative, differentiation formulas, applications of the derivative, the integral, integration by substitution, fundamental theorem of calculus. May include a computer component. Prerequisite: adequate score on calculus placement exam, or permission of the instructor.

Requirement(s): Division II and Quantitive

Approach: Quantitative Methods (QM); Quantitative Readiness Required (QR)

Units: 1.0

Instructor(s): Myers,A., Schneider,G.

(Fall 2012)

A continuation of Calculus I: Transcendental functions, techniques of integration, applications of integration, infinite sequences and series, convergence tests, power series. May include a computer component. Prerequisite: merit grade in MATH 101, adequate score on calculus placement exam, or permission of the instructor.

Requirement(s): Division II and Quantitive

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Myers,A., Schneider,G., Melvin,P.

(Spring 2013)

**MATH B104 Basic 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.

Requirement(s): Quantitative

Approach: Quantitative Methods (QM); Quantitative Readiness Required (QR)

Units: 1.0

Instructor(s): Kasius,P.

(Spring 2013)

**MATH B151 Introduction to Math and Sustainability
**

The world faces many sustainability challenges: climate change, energy, over-population, natural resource depletion. Using techniques of mathematical modeling including dynamical systems and bifurcation theory (tipping points), we will study quantitative aspects of these problems. No advanced mathematics beyond high school mathematics (pre-calculus) is required.

Requirement(s): Division II and Quantitive

Approach: Quantitative Methods (QM)

Units: 1.0

(Not Offered 2012-13)

**MATH B201 Multivariable Calculus
**

Vectors and geometry in two and three dimensions, partial derivatives, extremal problems, double and triple integrals, vector analysis (gradients, curl and divergence), line and surface integrals, the theorems of Gauss, Green and Stokes. May include a computer component. Prerequisite: MATH 102 or permission of instructor.

Requirement(s): Division II and Quantitive

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Kasius,P., Myers,A.

(Fall 2012)

Systems of linear equations, matrix algebra, determinants, vector spaces and subspaces, linear independence, bases and dimension, linear transformations and their representation by matrices, eigenvectors and eigenvalues, orthogonality, and applications of linear algebra. Pre or corequisite: MATH 102, or permission of the instructor

Requirement(s): Division II and Quantitive

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Myers,A., Schneider,G.

(Spring 2013)

**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.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Kasius,P.

(Spring 2013)

**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.

Requirement(s): Division II and Quantitive

Approach: Quantitative Methods (QM)

Counts toward: Environmental Studies

Units: 1.0

Instructor(s): Donnay,V., Schneider,G.

(Fall 2012)

**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.

Requirement(s): Division II and Quantitive

Approach: Quantitative Methods (QM)

Units: 1.0

(Not Offered 2012-13)

**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.

Requirement(s): Division II and Quantitive

Approach: Quantitative Methods (QM)

Crosslisting(s): CMSC-B231

Units: 1.0

Instructor(s): Xu,D.

(Fall 2012)

**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.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Units: 1.0

(Not Offered 2012-13)

**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.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Milicevic,D.

(Spring 2013)

**MATH B295 Select Topics in Mathematics
**

This year’s topic is mathematical modeling of real world problems. We will examine a variety of different types of models with a focus on discrete time systems. Prerequisites: MATH 102 and MATH 203 or permission of the instructor.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Counts toward: Environmental Studies

Units: 1.0

Instructor(s): Donnay,V.

(Spring 2013)

A first course in real analysis, providing a rigorous development of single variable calculus, with a strong focus on proof writing. Topics covered: the real number system, elements of set theory and topology, limits, continuous functions, the intermediate and extreme value theorems, differentiable functions and the mean value theorem, uniform continuity, the Riemann integral, the fundamental theorem of calculus. Possible additional topics include analysis on metric spaces or dynamical systems. Prerequisite: MATH 201. Some students also find it helpful to have taken a transitional course such as MATH 206 before enrolling in this course.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Melvin,P., Traynor,L.

(Fall 2012)

A continuation of Real Analysis I: Infinite series, power series, sequences and series of functions, pointwise and uniform convergence, and additional topics selected from: Fourier series, calculus of variations, the Lebesgue integral, dynamical systems, and calculus in higher dimensions. Prerequisite: MATH 301.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Traynor,L.

(Spring 2013)

A first course in abstract algebra, including an introduction to groups, rings and fields, and their homomorphisms. Topics covered: cyclic and dihedral groups, the symmetric and alternating groups, direct products and finitely generated abelian groups, cosets, Lagrange’s Theorem, normal subgroups and quotient groups, isomorphism theorems, integral domains, polynomial rings, ideals, quotient rings, prime and maximal ideals. Possible additional topics include group actions and the Sylow Theorems, free abelian groups, free groups, PIDs and UFDs. Prerequisite: MATH 203. Some students also find it helpful to have taken a transitional course such as MATH 206 before enrolling in this course.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Cheng,L.

(Fall 2012)

A continuation of Abstract Algebra I. Vector spaces and linear algebra, field extensions, algebraic and transcendental extensions, finite fields, fields of fractions, field automorphisms, the isomorphism extension theorem, splitting fields, separable and inseparable extensions, algebraic closures, and Galois theory. Also, if not covered in Abstract Algebra I: group actions and Sylow theorems, free abelian groups, free groups, PIDs and UFDs. Possible additional topic: finitely generated modules over a PID and canonical forms of matrices. Prerequisite: MATH 303.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Cheng,L.

(Spring 2013)

**MATH B308 Applied Mathematics I
**

Course content varies.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Units: 1.0

(Not Offered 2012-13)

**MATH B310 Introduction to the Mathematics of Financial Derivatives
**

An introduction to the mathematics utilized in the pricing models of derivative instruments. Topics to be covered may include Arbitrage Theorem, pricing derivatives, Wiener and Poisson processes, martingales and martingale representations, Ito’s Lemma, Black-Scholes partial differentiation equation, Girsanov Theorem and Feynman-Kac Formula. Prerequisite: MATH 201 or permission of instructor.

Requirement(s): Division II: Natural Science

Units: 1.0

Instructor(s): Cheng,L.

(Spring 2013)

An introduction to geometry with an emphasis that varies from year to year. Prerequisites: MATH 201 and 203 (or equivalent) or permission of instructor.

Requirement(s): Division II: Natural Science

Units: 1.0

(Not Offered 2012-13)

**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.

Requirement(s): Division II: Natural Science

Approach: Quantitative Methods (QM)

Units: 1.0

(Not Offered 2012-13)

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.

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Melvin,P., Cheng,L., Donnay,V., Traynor,L.

(Fall 2012)

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.

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Melvin,P., Cheng,L., Donnay,V., Traynor,L.

(Spring 2013)

A seminar for seniors majoring in mathematics. Topics vary from year to year.

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Donnay,V., Milicevic,D.

(Fall 2012)

A seminar for seniors majoring in mathematics. Topics vary from year to year.

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Donnay,V., Traynor,L.

(Spring 2013)

Units: 1.0

(Fall 2012, Spring 2013)

**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.

Units: 1.0

Instructor(s): Milicevic,D.

(Fall 2012)

**MATH B502 Graduate Real Analysis II
**

This course is a continuation of Math 501.

Units: 1.0

Instructor(s): Milicevic,D.

(Spring 2013)

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 will include categories, groups, rings, modules, and linear algebra.

Units: 1.0

(Not Offered 2012-13)

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.

Units: 1.0

(Not Offered 2012-13)

This is the first course of a 2 semester sequence, covering the basic notions of algebraic topology. The focus will be on homology theory, which will be introduced axiomatically (via the Eilenberg-Steenrod axioms) and then studied from a variety of points of view (simplicial, singular and cellular homology). The course will also treat cohomology theory and duality (on manifolds), and the elements of homotopy theory.

Units: 1.0

Instructor(s): Melvin,P.

(Spring 2013)

**MATH B506 Graduate Topology II**

Math 505 and Math 506 offer an introduction to topology at the graduate level. These courses can be taken in either order. Math 506 focuses on differential topology. Topics covered include smooth manifolds, smooth maps, and differential forms.

Units: 1.0

(Not Offered 2012-13)

Units: 1.0

(Fall 2012, Spring 2013)

Units: 1.0

Instructor(s): Melvin,P.

(Spring 2013)

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