2017-18 Catalog

Mathematics

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 or the 4+1 Partnership with the University of Pennsylvania’s School of Engineering and Applied Science.

Faculty

Leslie Cheng, Professor of Mathematics (on leave semester II)
Victor Donnay, Professor of Mathematics on the William R. Kenan, Jr. Chair and Director of Environmental Studies
Erica Graham, Assistant Professor of Mathematics
Peter Kasius, Instructor in Mathematics
Paul Melvin, Professor of Mathematics
Djordje Milićević, Assistant Professor of Mathematics
Amy Myers, Senior Lecturer in Mathematics and Math Program Coordinator
Daisy Sudparid, Instructor
Lisa Traynor, Chair and Professor of Mathematics

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 two writing attentive courses to satisfy the major writing requirement. Courses that are designated as writing attentive are MATH B206, MATH B301, and MATH B303.

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 could enroll in MATH 102 or MATH 201 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.

4+1 Partnership with Penn’s School of Engineering and Applied Science

See the description of the 4+1 Partnership with Penn’s School of Engineering, offered in cooperation with the University of Pennsylvania’s School of Engineering and Applied Science, for beginning work on a Master’s degree in Engineering while still enrolled as an undergraduate at Bryn Mawr.

COURSES

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): Myers,A., Smiley, D.
(Fall 2017)

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. Math 102 assumes familiarity of the content covered in Math 101 or its equivalent.
Approach: Quantitative Methods (QM)
Units: 1.0
Instructor(s): Kasius,P., Myers, A.
(Fall 2017, Spring 2018)

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): Sudparid,D.
(Fall 2017, Spring 2018)

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., Donnay,V.
(Fall 2017)

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): Donnay,V., Kasius, P.
(Spring 2018)

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)
Major Writing Requirement: Writing Attentive
Units: 1.0
Instructor(s): Myers, A.
(Spring 2018)

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): Graham, E.
(Spring 2018)

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.
(Spring 2018)

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
Instructor(s): Cheng,L.
(Fall 2017)

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 2017-2018)

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 2017-2018)

MATH B295 Select Topics in Mathematics
This is a topics course. Course content varies.
Prerequisite: MATH B102 or the equivalent (merit score on the AP Calculus BC Exam or placement).
Approach: Quantitative Methods (QM)
Units: 1.0
Instructor(s): Graham,E., Donnay, V.

Fall 2017: Computational Modeling. Mathematical models are constructed to describe the world within and around us. Computational methods are often employed to visualize and solve these models. Collectively, computational modeling from a mathematical perspective focuses on using computers to simulate dynamics that are described mathematically. This course will provide an introduction to programming in R and mathematical modeling. Topics may include discrete and continuous dynamical systems, data fitting, regression, and simulation techniques.

Spring 2018: Math Modeling and Sustainability. This course will explore how to create mathematical models of problems in sustainability such as CO2 levels, ground water flow, energy use in transportation and heating, and energy generation via wind and solar power. This is a Praxis II course in which students will work in teams and use their mathematical knowledge to carry out a sustainability project of use to a community partner. Not open to first year students.

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): Donnay,V., Milićević,D.
(Fall 2017)

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): Milićević,D.
(Spring 2018)

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): Melvin,P., Cheng,L.
(Fall 2017)

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): Melvin,P.
(Spring 2018)

MATH B308 Applied Mathematics I
This course will provide a general introduction to methods and modeling in applied mathematics. A variety of mathematical tools will be used to develop and study a wide range of models, including deterministic, discrete, and stochastic methods. Additional emphasis will be placed on techniques for analyzing mathematical models, including phase plane methods, stability analysis, dimensional analysis, bifurcation theory, and computer simulations. Applications to biology, physics, chemistry, engineering, and the social sciences may be discussed. Prerequisite: Math 203 and Math 210, or permission from instructor
Units: 1.0
(Not Offered 2017-2018)

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
(Not Offered 2017-2018)

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 2017-2018)

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 2017-2018)

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): Traynor,L.
(Spring 2018)

MATH B390 Number Theory
Study of integers with an emphasis on their multiplicative structure and topics related to analysis, and a first course in analytic number theory. Core topics: divisibility and primes, arithmetic functions, average and extremal orders, techniques of analytic number theory, Riemann zeta function, prime number theorem, Dirichlet characters, L-functions. Possible additional topics may include approximations by rational numbers, geometry of numbers, algebraic numbers and class numbers, sums of squares, and the idea of modular forms.Prerequisite: Math 201, and some familiarity with writing proofs (such as Math 206, Math 301/303 as a co-requisite, or permission of instructor).
Approach: Quantitative Methods (QM)
Units: 1.0
Instructor(s): Milićević,D.
(Fall 2017)

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., Donnay,V., Traynor,L., Graham,E., Milićević,D.
(Fall 2017)

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., Donnay,V., Traynor,L., Graham,E., Milićević,D.
(Spring 2018)

MATH B398 Senior Conference
A seminar for seniors majoring in mathematics. Topics vary from year to year.
Prerequisite: BM undergraduate Math major.
Units: 1.0
Instructor(s): Graham,E.
(Fall 2017)

MATH B399 Senior Conference
A seminar for seniors majoring in mathematics. Topics vary from year to year.
Prerequisite: BM undergraduate Math major.
Units: 1.0
Instructor(s): Graham,E., Milićević,D.
(Spring 2018)

MATH B403 Supervised Work
Units: 1.0
(Fall 2017)

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 Lp spaces.
Units: 1.0
(Not Offered 2017-2018)

MATH B502 Graduate Real Analysis II
This course is a continuation of Math 501.
Units: 1.0
(Not Offered 2017-2018)

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): Melvin,P.
(Fall 2017)

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): Melvin,P.
(Spring 2018)

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 2017-2018)

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
Instructor(s): Traynor,L.
(Fall 2017)

CHEM B221 Physical Chemistry I
Introduction to quantum theory and spectroscopy. Atomic and molecular structure; molecular modeling; rotational, vibrational, electronic and magnetic resonance spectroscopy. Lecture three hours. Prerequisites: CHEM B104 and MATH B201.
Approach: Quantitative Methods (QM)
Counts towards: Biochemistry and Molecular Biology
Units: 1.0
Instructor(s): Francl,M.
(Fall 2017)

CMSC 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 or H107.
Approach: Quantitative Methods (QM)
Units: 1.0
Instructor(s): Eisenberg,R.
(Fall 2017)

CMSC B310 Computational Geometry
A study of algorithms and mathematical theories that focus on solving geometric problems in computing, which arise naturally from a variety of disciplines such as Computer Graphics, Computer Aided Geometric Design, Computer Vision, Robotics and Visualization. The materials covered sit at the intersection of pure Mathematics and application-driven Computer Science and efforts will be made to accommodate Math majors and Computer Science majors of varying math/computational backgrounds. Topics include: graph theory, triangulation, convex hulls, geometric structures such as Voronoi diagrams and Delaunay triangulations, as well as curves and polyhedra surface topology. Prerequisite: CMSC B231/ MATH B231.
Approach: Quantitative Readiness Required (QR)
Units: 1.0
(Not Offered 2017-2018)

CMSC B340 Analysis of Algorithms
This course will cover qualitative and quantitative analysis of algorithms and their corresponding data structures from a precise mathematical point of view. Topics include: performance bounds, asymptotic and probabilistic analysis, worst case and average case behavior and correctness and complexity. Particular classes of algorithms will be studied in detail.
Prerequisite: CMSC B206 or H106 or H107, and B231 or H231 or permission from instructor.
Approach: Quantitative Readiness Required (QR)
Major Writing Requirement: Writing Intensive
Units: 1.0
Instructor(s): Xu,D.
(Fall 2017)

ECON B304 Econometrics
The econometric theory presented in ECON 253 is further developed and its most important empirical applications are considered. Each student does an empirical research project using multiple regression and other statistical techniques. Prerequisites: ECON 203 or 204 or 253; ECON 200 or both 202 and MATH 201.
Units: 1.0
Instructor(s): Sfekas,A.
(Spring 2018)

MATH B425 Praxis III
Counts towards: Praxis Program
Units: 1.0
(Not Offered 2017-2018)

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
(Not Offered 2017-2018)

MATH B701 Supervised Work
Prerequisite: GSAS students only.
Units: 1.0
Instructor(s): Melvin,P., Cheng,L., Donnay,V., Traynor,L., Graham,E., Milićević,D.
(Fall 2017, Spring 2018)

MATH B701 Supervised Work
Prerequisite: GSAS students only.
Units: 1.0
(Not Offered 2017-2018)

MATH B702 Research Seminar
Units: 1.0
Instructor(s): Melvin,P., Cheng,L., Donnay,V., Traynor,L., Graham,E., Milićević,D.
(Fall 2017, Spring 2018)

MATH B702 Research Seminar
Units: 1.0
(Not Offered 2017-2018)

PHYS B306 Mathematical Methods in the Physical Sciences
This course presents topics in applied mathematics useful to students, including physicists, engineers, physical chemists, geologists, and computer scientists studying the natural sciences. Topics are taken from Fourier series, integral transforms, advanced ordinary and partial differential equations, special functions, boundary-value problems, functions of complex variables, and numerical methods. Lecture three hours and additional recitation sessions as needed. Prerequisite: MATH 201 and 203.
Units: 1.0
Instructor(s): Schaffner,D.
(Fall 2017)