### Mathematics

Professors Haines and Wong; Associate Professors Ross and Shulman (chair); Assistant Professors Greer and Jayawant; Visiting Assistant Professor Shor; Lecturers Harder and Coulombe

A dynamic subject, with connections to many disciplines, mathematics is an integral part of a liberal arts education, and is increasingly vital in understanding science, technology, and society. Entry-level courses introduce students to basic concepts and hint at some of the power and beauty behind these fundamental results. Upper-level courses and the capstone experience provide majors with the opportunity to explore mathematical topics in greater depth and sophistication, and to delight in the fascination of this important discipline.

During new-student orientation the department assists students planning to study mathematics in choosing an appropriate starting course. Based on a student's academic background and skills, the department recommends Mathematics 101, 105, 106, 110, 205, 206, or a more advanced course.

The mathematics department offers a major and a minor in mathematics. More information on the mathematics department is available on the Web site (www.bates.edu/MATH.xml).

Major Requirements. The mathematics major requirements accommodate a wide variety of interests and career goals. The courses provide broad training in undergraduate mathematics, preparing majors for graduate study, and for positions in government, industry, and the teaching profession.

The major in mathematics consists of:

1) Mathematics 205 and 206;

2) Mathematics s21, which should be taken during Short Term of the first year;

3) Mathematics 301 and 309, which should be taken before beginning a senior thesis or the senior seminar;

4) four elective mathematics courses numbered 200 or higher, not including 360, 395, 457, 458, or s50;

5) completion of either a one-semester thesis (Mathematics 457 or 458), a two-semester thesis (Mathematics 457-458), or the senior seminar (Mathematics 395). The thesis option requires departmental approval.

Any mathematics Short Term unit numbered 30 or above may be used as one of the electives in 4). One elective may also be replaced by a departmentally approved course from another department.

While students must consult with their major advisors in designing appropriate courses of study, the following suggestions may be helpful: For majors considering a career in secondary education the department suggests Mathematics 312, 314, 315, and 341. Students interested in operations research, business, or actuarial science should consider Mathematics 218, 314, 315, and 341. Students interested in applied mathematics in the physical and engineering sciences should consider Mathematics 218, 219, 308, 314, 315, and 341. Majors planning on graduate study in pure mathematics should particularly consider Mathematics 308, 313, and 457-458. Mathematics majors may pursue individual research either through Independent Study (360 or s50), or Senior Thesis (457 and/or 458).

Pass/Fail Grading Option. Pass/fail grading may not be elected for courses applied toward the major.

Minor in Mathematics. Designed either to complement another major, or to be pursued for its own sake, the minor in mathematics provides a structure for obtaining a significant depth in mathematical study. It consists of seven mathematics courses, four of which must be Mathematics 105, 106, 205, and 206. (Successful completion of Mathematics 206 is sufficient to fulfill the requirements for Mathematics 105 and 106, even if no course credit for these has been granted by Bates.)

The other three courses must be mathematics courses at the 150-level or above (or units at the s20 level or above). At least one of these three must be taken at Bates.

Elective courses should relate to each other with a common theme. Possible combinations include:

1) Analysis:

MATH 218. Numerical Analysis.

MATH 301. Real Analysis.

MATH 308. Complex Analysis.

2) Geometry:

MATH 312. Geometry.

MATH 313. Topology.

3) Mathematical Biology:

BI/MA 155. Mathematical Models in Biology.

MATH 219. Differential Equations.

MATH 341. Mathematical Modeling.

4) Actuarial Science:

MATH 218. Numerical Analysis.

MATH 314. Probability.

MATH 315. Statistics.

5) Statistics:

MATH 301. Real Analysis.

MATH 314. Probability.

MATH 315. Statistics.

6) Applied/Engineering Mathematics:

MATH 218. Numerical Analysis.

MATH 219. Differential Equations.

MATH 308. Complex Analysis.

MATH 341. Mathematical Modeling.

The following do not count toward the mathematics minor: Mathematics 360, 457, 458, and s50.

Pass/Fail Grading Option. Pass/fail grading may not be elected for courses applied toward the minor in mathematics.

General Education Information for the Classes of 2008, 2009, and 2010. The quantitative requirement is satisfied by any of the mathematics courses or units and FYS 355. Advanced Placement, International Baccalaureate, or A-Level credit awarded by the department for mathematics, computer science, or statistics may also satisfy the quantitative requirement.

#### MATH 101. Working with Data.

Techniques for analyzing data are described in ordinary English without emphasis on mathematical formulas. The course focuses on graphical and descriptive techniques for summarizing data, design of experiments, sampling, analyzing relationships, statistical models, and statistical inference. Applications are drawn from everyday life: drug testing, legal discrimination cases, and public opinion polling. Not open to students who have received credit for Biology 244, Economics 250 or 255, Environmental Studies 181, Mathematics 315, Psychology 218, or Sociology 305. Enrollment limited to 30 per section. [Q] Normally offered every year. M. Harder, B. Shulman.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 105. Calculus I.

While the word calculus originally meant any method of calculating, it has come to refer more specifically to the fundamental ideas of differentiation and integration that were first developed in the seventeenth century. The subject's early development was intimately connected with understanding rates of change within the context of the physical sciences. Nonetheless, it has proven to be widely applicable throughout the natural sciences and some social sciences, as well as crucial to the development of most modern technology. This course develops the key notions of derivatives and integrals and their interrelationship, as well as applications. An emphasis is placed on conceptual understanding and interpretation, as well as on calculational skills. Graphing calculators are used in the course. Students are required to attend approximately six additional 50-minute laboratory sessions at times to be arranged. Students must read the mathematics department calculus FAQs before registering (http://abacus.bates.edu/acad/depts/math/faq.html). Enrollment limited to 25 per section. [Q] Normally offered every semester. M. Greer, C. Shor, B. Shulman.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 106. Calculus II.

A continuation of Calculus I. Further techniques of integration, both symbolic and numerical, are studied. The course then treats applications of integration to problems drawn from fields such as physics, biology, chemistry, economics, and probability. Differential equations and their applications are also introduced, as well as approximation techniques and Taylor series. Graphing calculators are used in the course. Students are required to attend approximately six additional 50-minute laboratory sessions at times to be arranged. Recommended background: Math 105 or equivalent. Students must read the mathematics department calculus FAQs before registering (http://abacus.bates.edu/acad/depts/math/faq.html). Enrollment limited to 25 per section. [Q] Normally offered every semester. P. Jayawant, D. Haines, P. Wong.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 110. Great Ideas in Mathematics.

Is mathematics composed of impenetrable formulas to be memorized, a series of insurmountable cliffs to be scaled? Are there individuals who can think logically and creatively, but never "do math"? In this course, students are asked to use their imagination to grapple with challenging mathematical concepts. The process enables them to master techniques of effective thinking, experience the joy of discovering new ideas, and feel the power of figuring out things on their own. Together they contemplate some of the greatest and most intriguing creations of human thought, from Pythagoras to the fourth dimension, from chaos to symmetry. Enrollment limited to 30. [Q] Normally offered every year. G. Coulombe.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 111. Mathematics across Time and Cultures.

How could we count if we had no numbers? The Incas used quipus as a sophisticated counting device to keep tally in the absence of a written number system. Mathematical ideas arise in many different cultural and historical contexts. Many of these ideas can be recast using contemporary Western mathematical theory. For example, Sona drawings of the Chokwe people of South Central Africa can be understood using graph theory. In this course students investigate selected cultural activities and practices (e.g., kin relations, symmetries in art, counting devices) as a way to explore contemporary mathematical topics (e.g., group theory, geometry, number systems). Enrollment limited to 10. Offered in the Hughes Summer Scholars Program. G. Coulombe.Concentrations

This course is referenced by the following General Education Concentrations

#### BI/MA 155. Mathematical Models in Biology.

Mathematical models are increasingly important throughout the life sciences. This course provides an introduction to deterministic and statistical models in biology. Examples are chosen from a variety of biological and medical fields such as ecology, molecular evolution, and infectious disease. Computers are used extensively for modeling and for analyzing data. Recommended background: a course in biology. Enrollment limited to 30. Normally offered every other year. B. Shulman.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 205. Linear Algebra.

Vectors and matrices are introduced as devices for the solution of systems of linear equations with many variables. Although these objects can be viewed simply as algebraic tools, they are better understood by applying geometric insight from two and three dimensions. This leads to an understanding of higher dimensional spaces and to the abstract concept of a vector space. Other topics include orthogonality, linear transformations, determinants, and eigenvectors. This course should be particularly useful to students majoring in any of the natural sciences or economics. Prerequisite(s): Mathematics 105, 106, or 155. Open to first-year students. Enrollment limited to 25 per section. Normally offered every semester. M. Greer, S. Ross.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 206. Multivariable Calculus.

This course extends the ideas of Calculus I and II to deal with functions of more than one variable. Because of the multidimensional setting, essential use is made of the language of linear algebra. While calculations make straightforward use of the techniques of single-variable calculus, more effort must be spent in developing a conceptual framework for understanding curves and surfaces in higher-dimensional spaces. Topics include partial derivatives, derivatives of vector-valued functions, vector fields, integration over regions in the plane and three-space, and integration on curves and surfaces. This course should be particularly useful to students majoring in any of the natural sciences or economics. Prerequisite(s): Mathematics 106 and 205. Open to first-year students. Normally offered every semester. D. Haines, P. Wong.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 218. Numerical Analysis.

This course studies the best ways to perform calculations that have already been developed in other mathematics courses. For instance, if a computer is to be used to approximate the value of an integral, one must understand both how quickly an algorithm can produce a result and how trustworthy that result is. While students implement algorithms on computers, the focus of the course is the mathematics behind the algorithms. Topics may include interpolation techniques, approximation of functions, solving equations, differentiation and integration, solution of differential equations, iterative solutions of linear systems, and eigenvalues and eigenvectors. Prerequisite(s): Mathematics 106 and 205. [Q] Normally offered every other year. M. Greer.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 219. Differential Equations.

A differential equation is a relationship between a function and its derivatives. Many real-world situations can be modeled using these relationships. This course is a blend of the mathematical theory behind differential equations and their applications. The emphasis is on first- and second-order linear equations. Topics include existence and uniqueness of solutions, power series solutions, numerical methods, and applications such as population modeling and mechanical vibrations. Prerequisite(s): Mathematics 206. [Q] Normally offered every year. M. Greer.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 301. Real Analysis.

An introduction to the foundations of mathematical analysis, this course presents a rigorous treatment of fundamental concepts such as limits, continuity, differentiation, and integration. Elements of the topology of the real numbers are also covered. Prerequisite(s): Mathematics 206 and s21. [W2] Normally offered every year. M. Greer.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 308. Complex Analysis.

This course extends the concepts of calculus to deal with functions whose variables and values are complex numbers. Instead of producing new complications, this leads to a theory that is not only more aesthetically pleasing, but is also more powerful. The course should be valuable to those interested in pure mathematics, as well as those who need additional computational tools for physics or engineering. Topics include the geometry of complex numbers, differentiation and integration, representation of functions by integrals and power series, and the calculus of residues. Prerequisite(s): Mathematics 206. Normally offered every other year. P. Wong.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 309. Abstract Algebra I.

An introduction to basic algebraic structures common throughout mathematics. These include the integers and their arithmetic, modular arithmetic, rings, polynomial rings, ideals, quotient rings, fields, and groups. Prerequisite(s): Mathematics 205 and s21. Normally offered every year. D. Haines.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 312. Geometry.

This course studies geometric concepts in Euclidean and non-Euclidean geometries. Topics include isometries, arc lengths, curvature of curves and surfaces, and tessellations, especially frieze and wallpaper patterns. Prerequisite(s): Mathematics 206. Normally offered every other year. P. Wong.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 313. Topology.

The notion of "closeness" underlies many important mathematical concepts, such as limits and continuity. Topology is the careful study of what this notion means in abstract spaces, leading to a thorough understanding of continuous mappings and the properties of spaces that they preserve. Topics include metric spaces, topological spaces, continuity, compactness, and connectedness. Additional topics, such as fundamental groups or Tychonoff's theorem, may also be covered. Prerequisite(s): Mathematics 206 and s21. Normally offered every other year. S. Ross.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 314. Probability.

Probability theory is the foundation on which statistical data analysis depends. This course together with its sequel, Mathematics 315, covers topics in mathematical statistics. Both courses are recommended for math majors with an interest in applied mathematics and for students in other disciplines, such as psychology and economics, who wish to learn about some of the mathematical theory underlying the methodology used in their fields. Prerequisite(s): Mathematics 106. [Q] Normally offered every other year. M. Harder.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 315. Statistics.

The sequel to Mathematics 314. This course covers estimation theory and hypothesis testing. Prerequisite(s): Mathematics 314. [Q] Normally offered every other year. M. Harder.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 341. Mathematical Modeling.

Often, analyzing complex situations (like the weather, a traffic flow pattern, or an ecological system) is necessary to predict the effect of some action. The purpose of this course is to provide experience in the process of using mathematics to model real-life situations. The first half examines and critiques specific examples of the modeling process from various fields. During the second half each student creates, evaluates, refines, and presents a mathematical model from a field of his or her own choosing. Prerequisite(s): Mathematics 206. Normally offered every other year. D. Haines.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 360. Independent Study.

Students, in consultation with a faculty advisor, individually design and plan a course of study or research not offered in the curriculum. Course work includes a reflective component, evaluation, and completion of an agreed-upon product. Sponsorship by a faculty member in the program/department, a course prospectus, and permission of the chair are required. Students may register for no more than one independent study per semester. This course may not be used to fulfill requirements for the mathematics major or concentration in mathematics. Normally offered every semester. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 365. Special Topics.

Content varies from semester to semester. Possible topics include chaotic dynamical systems, number theory, mathematical logic, measure theory, algebraic topology, combinatorics, and graph theory.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 365B. Number Theory.

The theory of numbers is concerned with the properties of the integers, one of the most basic of mathematical sets. Seemingly naive questions of number theory stimulated much of the development of modern mathematics and still provide rich opportunities for investigation. Topics studied include classical ones such as primality, congruences, quadratic reciprocity, and Diophantine equations, as well as more recent applications to cryptography. Additional topics such as continued fractions, elliptical curves, or an introduction to analytic methods may be included. Prerequisite(s): Mathematics s21. Offered with varying frequency. C. Shor.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 365D. Graph Algorithms.

Finding a path with certain characteristics (such as the shortest path between two locations) is important in many applications such as communications networks, design of integrated circuits, and airline scheduling. Graph theory is the branch of mathematics that provides the framework to find these paths. Topics may include definitions and properties of graphs and trees, Euler and Hamiltonian circuits, shortest paths, minimal spanning trees, network flows, and graph coloring. Some of the class meetings are devoted to learning to program in Maple. Students implement one or more of the path algorithms in a computer program at the end of the semester. Prerequisite(s): Mathematics s21. Enrollment limited to 30. Offered with varying frequency. P. Jayawant.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 365E. Computers and Abstract Mathematics.

The computational and representational power of computers has had a great impact on mathematics. In this course students use the functional programming language Scheme to represent mathematical ideas and construct computational solutions to abstract mathematical problems. Topics may include: logic; orders of growth; representation of and arithmetic on number systems; testing integers for primality; representation of abstract structures such as sets, groups, and graphs; symbolic operations from calculus; algebra of polynomials, rational functions, and matrices; representation of geometric figures; random number generators; noncomputable problems; and representation of the infinite. Prerequisite(s): Any two mathematics courses or units. Offered with varying frequency. D. Haines.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 395. Senior Seminar.

While the subject matter varies, the seminar addresses an advanced topic in mathematics. The development of the topic draws on students' previous course work and helps consolidate their earlier learning. Students are active participants, presenting material to one another in both oral and written form, and conducting individual research on related questions. Required of all majors not writing a thesis. Not open to students who have received credit for a mathematics thesis.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 395C. History of the Proof.

In this senior seminar students examine notions of rigor and proof in mathematics. Through readings of original sources, students trace particularly the evolution of the "epsilon-delta" proofs in calculus. They read excerpts (in translation) from Cauchy, Weierstrass, Dedekind, and others. Students also choose their own readings to present to the class. Prerequisite(s): Math 301. Corequisite: Math 309. Not open to students who have received credit for Mathematics 457 or 458. Enrollment limited to 10. Offered with varying frequency. B. Shulman.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 395E. Wavelets and Their Applications.

Several questions in the areas of science and technology center on interpreting or transferring large volumes of data in which minute details play a key role. These include downloading online video quickly without losing clear picture or sound; teaching machines to recognize faces or reproduce speech; removing unwanted noise from a long-distance telephone connection; and determining which brain activity visible via EEG is triggered by a particular stimulus. Wavelet theory provides insight into the above examples and many more. Students explore basic ideas of wavelets, then read papers and compare projects on detailed uses of wavelet theory. Prerequisite(s): Mathematics 301. Corequisite(s): Mathematics 309. Corequisite(s): Mathematics 309. Not open to students who have received credit for Mathematics 457 or 458. Offered with varying frequency. M. Greer.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 395G. Game Theory: The Mathematics of Conflict and Cooperation.

Game theory provides a mathematical framework for analyzing situations where individuals (or companies, political parties, or nations) are faced with the prospect of maximizing their own well-being, dependent on the decisions of others. How can we decide on the best strategy? How can we mathematically model such abstract notions as fairness and rationality? After an introduction to the basics of game theoretic methods, students independently explore its applications to anthropology, warfare, economics, politics, philosophy, biology, and the NFL draft. Prerequisite(s): Math 206 and one of the following: Math s21, 301, or 309. Not open to students who have received credit for Mathematics 457, 458 or s45J. Enrollment limited to 10. Offered with varying frequency. B. Shulman.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 395H. Elliptic Curves.

An elliptic curve is defined as the set of points of a cubic polynomial in two variables. Interestingly, one can add any two points on the curve to get another point on the curve. In this way, the points form a group. These groups are used in various areas of mathematics, including cryptography, primality testing, and the proof of Fermat's last theorem. After an introduction to the basics, each student studies various aspects of elliptic curves, drawing from previous knowledge in abstract algebra, analysis, geometry, and number theory. Prerequisite(s): Mathematics 301. Corequisite(s): Mathematics 309. Not open to students who have received credit for Mathematics 457 or 458. Enrollment limited to 10. Offered with varying frequency. C. Shor.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 457. Senior Thesis.

Prior to entrance into Mathematics 457, students must submit a proposal for the work they intend to undertake toward completion of a thesis. Open to all majors upon approval of the proposal. Required of candidates for honors. Students register for Mathematics 457 in the fall semester and Mathematics 458 in the winter semester. [W3] Normally offered every year. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 457, 458. Senior Thesis.

Prior to entrance into Mathematics 457, students must submit a proposal for the work they intend to undertake toward completion of a thesis. Open to all majors upon approval of the proposal. Required of candidates for honors. Students register for Mathematics 457 in the fall semester and Mathematics 458 in the winter semester. [W3] Normally offered every year. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 458. Senior Thesis.

Prior to entrance into Mathematics 457, students must submit a proposal for the work they intend to undertake toward completion of a thesis. Open to all majors upon approval of the proposal. Required of candidates for honors. Students register for Mathematics 457 in the fall semester and Mathematics 458 in the winter semester. [W3] Normally offered every year. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s21. Introduction to Abstraction.

An intensive development of the important concepts and methods of abstract mathematics. Students work individually, in groups, and with the instructors to prove theorems and solve problems. Students meet for up to five hours daily to explore such topics as proof techniques, logic, set theory, equivalence relations, functions, and algebraic structures. The unit provides exposure to what it means to be a mathematician. Prerequisite(s): one semester of college mathematics. Required of all majors. Enrollment limited to 30. Normally offered every year. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s31. Applied Abstract Algebra.

This unit examines the applications of basic concepts of abstract and linear algebra to problems of the real world, such as cryptography, error-correcting codes, and symmetry groups. The major goal of the unit is to show that algebraic abstractions can be used to refine our understanding of many scientific phenomena. This unit is oriented to both math majors and science majors. The former learn about applied examples of pure mathematical objects, and the latter consider what mathematical principles govern certain scientific ideas. Previous knowledge of abstract algebra is helpful but not assumed. Prerequisite(s): Mathematics 205 and Mathematics s21. Recommended background: Mathematics 309. Offered with varying frequency. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s45. Seminar in Mathematics.

The content varies. Recent topics have included number theory and an introduction to error-correcting codes. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s45D. Introduction to Knot Theory.

Over a century ago, Lord Kelvin's theory of the atom suggested that understanding the knotting phenomenon that occurs between atoms would provide insights into chemistry. Since then, sophisticated mathematical tools have been developed in order to classify knots. Recent works of V. Jones (1985) and of E. Witten (1989) have made important contributions to chemistry, molecular biology, and theoretical physics. This unit introduces the mathematics behind the classical theory of knots. Combinatorial, geometric, and algebraic techniques are presented. Prerequisite(s): Mathematics 205, 206, and s21. Enrollment limited to 30. Offered with varying frequency. P. Wong.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s45J. Game Theory: The Mathematics of Conflict and Cooperation.

Game theory provides a mathematical framework for analyzing situations where individuals (or companies, political parties, or nations) are faced with the prospect of maximizing their own well-being, dependent on the decisions of others. How can one decide on the best strategy? How can we mathematically model such abstract notions as fairness and rationality? After an introduction to the basics of game theoretic methods, students consider its applications to anthropology, warfare, economics, politics, philosophy, biology, and the NFL draft. Prerequisite(s): Mathematics 105. Not open to students who have received credit for Mathematics 395G. Enrollment limited to 30. Offered with varying frequency. B. Shulman.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s45K. Roller Coasters: Theory, Design, and Properties.

Amusement park roller coasters excite us, scare us, and capture our imagination. What records will designers break next? How do they create rides that are exhilarating, yet physically safe? A scientific contemplation of these questions requires math and physics concepts such as vectors, parametric equations, curvature, energy, gravity, and friction. Students consider these ideas, gaining background in basic and more advanced math and physics. During the second half of the unit, students conceive and design projects to study specific aspects of roller coasters. Prerequisite(s): Mathematics 105. Enrollment limited to 25. Offered with varying frequency. M. Greer.Concentrations

This course is referenced by the following General Education Concentrations