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**

Isabel Averill, Lecturer in Math

Leslie Cheng, Chair and Professor of Mathematics

Victor Donnay, William R. Kenan, Jr. Chair, Professor of Mathematics and Director of Environmental Studies

Erica Graham, Assistant Professor of Mathematics

Helen Grundman, Professor of Mathematics

Peter G. Kasius, Instructor in Mathematics

Paul Melvin, Professor of Mathematics

Djordje Milicevic, Assistant Professor of Mathematics (on leave semesters I and II)

Amy Myers, Senior Lecturer in Mathematics and Math Program Coordinator

Lisa Traynor, Professor of Mathematics (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

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.

**Major Writing Requirement**

Students will take MATH B301 and MATH B303, two writing attentive courses, to satisfy the major writing requirement.

**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. A draft of the written thesis should be submitted to the Math Department Office one week before the last day of classes.

**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**

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.

**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 2015-2016)

**MATH B101 Calculus I**

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. Students should have a reasonable command of high school algebra, geometry and trigonometry.

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

Units: 1.0

Instructor(s): Averill,I., Myers,A.

(Fall 2015)

**MATH B102 Calculus II**

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: MATH B101 with merit grade (2.0 or higher), adequate score on Calculus placement exam, or permission of the instructor.

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Averill,I., Graham,E.

(Fall 2015, Spring 2016)

**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: Quantitative Readiness Required.

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

Units: 1.0

Instructor(s): Averill,I., Myers,A.

(Fall 2015, Spring 2016)

**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.

Approach: Quantitative Methods (QM)

Units: 1.0

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

(Fall 2015)

**MATH B203 Linear Algebra**

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. Prerequisite or corequisite: MATH 102, or permission of the instructor.

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Kasius,P., Donnay,V.

(Spring 2016)

**MATH B205 Theory of Probability with Applications**

The course analyzes repeatable “experiments,” such as coin tosses or die rolls, in which the short-term outcomes are uncertain, but the long-run behavior is predictable. Such random processes are used as models for real-world phenomena to solve problems such as determining the effectiveness of a new drug, or deciding whether a series of record-high temperatures is due to the natural variation in weather or rather to climate change. Topics include: random variables, discrete distributions (binomial, geometric, negative binomial, Poisson, hypergeometric, Benford), continuous densities (exponential, gamma, normal, Maxwell, Rayleigh, chi-squared), conditional probability, expected value, variance, the Law of Large Numbers, and the Central Limit Theorem. Prerequisite: MATH B102 or the equivalent (merit score on the AP Calculus BC Exam or placement).

Approach: Quantitative Methods (QM)

Units: 1.0

(Not Offered 2015-2016)

**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.

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Myers,A.

(Spring 2016)

**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. Co-requisite: MATH 201 or 203.

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Averill,I.

(Fall 2015)

**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. Co-requisite: MATH 201 or 203.

Approach: Quantitative Methods (QM)

Units: 1.0

Instructor(s): Traynor,L.

(Fall 2015)

**MATH B225 Introduction to Financial Mathematics**

Topics to be covered include market conventions and instruments, Black-Scholes option-pricing model, and practical aspects of trading and hedging. All necessary definitions from probability theory (random variables, normal and lognormal distribution, etc.) will be explained. Prerequisite: MATH 102. ECON 105 is recommended.

Approach: Quantitative Methods (QM)

Units: 1.0

(Not Offered 2015-2016)

**MATH B231 Discrete Mathematics**

An introduction to discrete mathematics with strong applications to computer science. Topics include propositional logic, proof techniques, recursion, set theory, counting, probability theory and graph theory. Co-requisites: CMSC B110 or H105.

Approach: Quantitative Methods (QM)

Crosslisting(s): CMSC-B231

Units: 1.0

Instructor(s): Xu,D.

(Spring 2016)

**MATH B251 Chaotic Dynamical Systems**

Topics to be covered may include iteration, orbits, graphical and computer analysis, bifurcations, symbolic dynamics, fractals, complex dynamics and applications. Prerequisite: MATH B102

Units: 1.0

(Not Offered 2015-2016)

**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.

Approach: Quantitative Methods (QM)

Units: 1.0

(Not Offered 2015-2016)

**MATH B295 Select Topics in Mathematics**

This is a topics course. Course content varies. Prerequisite: MATH B102.

Approach: Quantitative Methods (QM)

Units: 1.0

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

Spring 2016:Advanced Linear Algebra. This course will cover vector spaces over general fields, linear transformations and matrices, multi-linear forms and determinants, inner product spaces and orthogonality, dual vector spaces, rational and Jordan canonical forms, and other possible topics as time and interest allow. Prerequisite: Math 203 (Linear Algebra).Combinatorics. Enumerative combinatorics is a collection of techniques for enumerating a set of objects (saying how many) without listing all the possibilities, and graph theory considers the structure of the relationships within a set of objects. Although combinatorial problems can often be stated in the language of puzzles and games, the results have applications throughout mathematics, both pure and applied. Topics include: permutations, combinations, binomial identities, generating functions, recurrence relations, inclusion-exclusion, planar graphs, Hamilton circuits, Euler cycles, graph coloring, and trees.

Computational Modeling. TBA.

**MATH B301 Real Analysis I**

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.

Major Writing Requirement: Writing Attentive

Units: 1.0

Instructor(s): Melvin,P.

(Fall 2015)

**MATH B302 Real Analysis II**

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.

Units: 1.0

Instructor(s): Melvin,P.

(Spring 2016)

**MATH B303 Abstract Algebra I**

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.

Major Writing Requirement: Writing Attentive

Units: 1.0

Instructor(s): Cheng,L., Kasius,P.

(Fall 2015)

**MATH B304 Abstract Algebra II**

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.

Units: 1.0

Instructor(s): Cheng,L.

(Spring 2016)

**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.

Units: 1.0

Instructor(s): Cheng,L.

(Fall 2015)

**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.

Units: 1.0

(Not Offered 2015-2016)

**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. Co-requisite: MATH 301, MATH 303, or permission of instructor.

Units: 1.0

(Not Offered 2015-2016)

**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.

Units: 1.0

Instructor(s): Melvin,P.

(Spring 2016)

**MATH B361 Harmonic Analysis and Wavelets**

A first introduction to harmonic analysis and wavelets. Topics to be covered include Fourier series on the circle, Fourier transforms on the line and space, Discrete Wavelet Transform, Fast Wavelet Transform and filter-bank representation of wavelets. Prerequisite: MATH B203 or permission of instructor.

Units: 1.0

(Not Offered 2015-2016)

**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. This is a topics course. Prerequisite: Permission of instructor.

Units: 1.0

Instructor(s): Melvin,P., Cheng,L., Grundman,H., Traynor,L.

(Fall 2015)

**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.

Units: 1.0

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

(Spring 2016)

**MATH B398 Senior Conference**

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

Units: 1.0

Instructor(s): Traynor,L.

(Fall 2015)

**MATH B399 Senior Conference**

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

Units: 1.0

Instructor(s): Donnay,V., Grundman,H.

(Spring 2016)

**MATH B403 Supervised Work**

Units: 1.0

(Fall 2015, Spring 2016)

**MATH B425 Praxis III**

Counts towards: Praxis Program

Units: 1.0

(Not Offered 2015-2016)

**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

(Not Offered 2015-2016)

**MATH B502 Graduate Real Analysis II**

This course is a continuation of Math 501.

Units: 1.0

(Not Offered 2015-2016)

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

Units: 1.0

Instructor(s): Grundman,H.

(Fall 2015)

**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. Prerequisite: MATH B503.

Units: 1.0

Instructor(s): Grundman,H.

(Spring 2016)

**MATH B505 Graduate Topology I**

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

(Not Offered 2015-2016)

**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 2015-2016)

**MATH B670 Graduate Perspectives in Mathematics Pedagogy**

This course will cover a spectrum of topics in mathematics pedagogy of importance for graduate students serving as mathematics teaching assistants as well as those preparing to teach high school, community college, or university-level mathematics. It will meet every other week for three hours following a seminar format combining some lectures and guest speakers with extended discussion.

Units: 1.0

Instructor(s): Grundman,H.

(Fall 2015, Spring 2016)

**MATH B701 Supervised Work**

Units: 1.0

Instructor(s): Melvin,P., Cheng,L., Grundman,H., Traynor,L., Milicevic,D.

(Fall 2015, Spring 2016)

**MATH B701 Supervised Work**

Units: 1.0

Instructor(s): Traynor,L.

(Fall 2015)

**MATH B702 Research Seminar**

Units: 1.0

Instructor(s): Melvin,P., Cheng,L., Grundman,H., Traynor,L., Milicevic,D., Donnay,V.

(Fall 2015, Spring 2016)

**MATH B702 Research Seminar**

Units: 1.0

(Fall 2015)

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