### Mathematics

Professors Haines and Wong; Associate Professors Ross (chair), Shulman, and Greer; Assistant Professors Jayawant and Salerno; Visiting Instructors Balcomb and Webster; Lecturer 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 course 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 214, 215, 255, and 312. Students interested in operations research, business, or actuarial science should consider Mathematics 214, 215, 255A, 341, and 355A. Students interested in applied mathematics in the physical and engineering sciences should consider Mathematics 214, 215, 219, 255B, 308, and 355A . 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).

Major Courses Taken Elsewhere. Of the nine courses (205, 206, 301, 309, s21 and four electives) required for the mathematics major, up to four may be taken at other institutions, in off-campus study programs, or in other Bates departments. These courses are subject to these limitations:

1) At least one of 301 (Real Analysis) and 309 (Abstract Algebra) must be completed at Bates.

2) s21 (Introduction to Abstraction) must be completed at Bates.

3) At least two of the four elective mathematics courses must be completed at Bates.

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 Short Term courses at the s20 level or above). At least one of these three must be taken at Bates.

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 Class of 2010. The quantitative requirement is satisfied by any of the mathematics courses or Short Term courses and First-Year Seminar 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 215 or 315, Psychology 218, or Sociology 305. Enrollment limited to 30. [Q] Normally offered every year. 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. A. Salerno, D. Haines, S. Ross, M. Greer.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. D. Haines, Staff.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). G. Coulombe.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. S. Ross, P. Jayawant.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. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 214. Probability.

Probability theory is the foundation on which statistical data analysis depends. This course together with its sequel, Mathematics 215, 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. Not open to students who have received credit for Mathematics 314. [Q] Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 215. Statistics.

The sequel to Mathematics 214. This course covers estimation theory and hypothesis testing. Prerequisite(s): Mathematics 214. Not open to students who have received credit for Mathematics 315. [Q] Staff.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 255. Topics in Mathematical Modeling.

Mathematical modeling is a tool used by natural and social scientists, including physicists, biologists, engineers, economists, and political scientists. Mathematical models use the language of mathematics to describe and analyze complex systems. They extract the essential features of real world phenomena and represent the system in one or more mathematical forms. These abstract structures may include differential equations, dynamical systems, statistical models,and game-theoretic models, among others. Normally offered every year.Concentrations

This course is referenced by the following General Education Concentrations

#### BI/MA 255A. 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. Prerequisite(s): Mathematics 105. Not open to students who have received credit for Biology/Mathematics 155. Not open to students who have received credit for BI/MA 155. Open to first-year students. Enrollment limited to 30. B. Shulman.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 255B. 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. Not open to students who have received credit for Mathematics 341. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 255C. Mathematical Models of Social Dilemmas.

The power and utility of mathematical modeling in the physical sciences is well recognized. What is less well-known is that mathematics can also be used to model social dilemmas—situations where the public good is in conflict with individual self-interest. The social dynamics involved in this so-called "tragedy of the commons" are analyzed using a branch of mathematics called game theory. In this course, students create models using classical and evolutionary game theory to gain insight into questions about how cooperation evolves, what is necessary for it to gain a foothold in a group, and under what conditions it will persist. Prerequisite(s): Mathematics 105. Enrollment limited to 30. B. Shulman.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 295. Sophomore-Junior Seminar.

With varying subject matter, this seminar addresses both the oral and written communication of mathematics. The seminar focuses on understanding why rigor is necessary and what constitutes effective communication of mathematical ideas to different audiences. Students practice peer editing and peer reviewing, and learn how to write effective grant and thesis proposals. Prerequisite(s): Mathematics s21. Normally offered every year.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. Normally offered every year. P. Jayawant.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. Staff.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. 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. Continued development of good mathematical writing and/or a presentation is a major component of this course. Prerequisite(s): Mathematics 206 and s21. [W2] S. Ross.Concentrations

This course is referenced by the following General Education Concentrations

#### EC/MA 342. Optimal Control Theory with Economic Applications.

Optimal control theory unifies numerous economic problems related to the creation and use of physical capital. This course introduces optimal control theory as a tool for dynamic optimization and applies that theory to a variety of classic economic problems involving capital. Among the economic problems examined are optimal use of a renewable resource, optimal use of a nonrenewable resource, and optimal economic growth when growth begets pollution. The course includes formally proving Pontryagin's maximum principle, which characterizes dynamic optima, in the special case context of common economic problems. Prerequisite(s): Mathematics 206 and one economics course. M. Murray.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 355. Topics in Computational Mathematics.

Computational mathematics considers how computing can aid the study of mathematics. Computers may be used to perform numerical calculations involved in modeling, provide graphical understanding in analysis, or complete symbolic calculations in algebra. Mathematical study of computer algorithms can lead to more efficient algorithms and actual implementation of algorithms helps student deepen their understanding of mathematics while gaining appreciation for computers' capabilities as well as deficiencies. Normally offered every year.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 355A. 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. Not open to students who have received credit for Mathematics 218. [Q] Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 355B. 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. Not open to students who have received credit for Mathematics 365D. Enrollment limited to 30. P. Jayawant.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 355C. 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 Short Term courses. Not open to students who have received credit for Mathematics 365E. [W2] D. Haines.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 355D. Dynamical Systems and Computer Science.

The study of long-term behaviors of feedback processes, dynamical systems is best understood from both theoretical and computational viewpoints, as each informs the other. Students explore attracting and repelling cycles and witness the complicated dynamics and chaos a simple quadratic function can exhibit. Real and complex functions are considered. Simultaneously, students learn sound computer science fundamentals by writing Visual Basic programs that illustrate the theory of dynamical systems. In particular, students plot both orbit and bifurcation diagrams, Julia sets, and the Mandelbrot set. The course explores both dynamical systems and computer science in depth, thus requiring four meetings per week. Prerequisite(s): Mathematics s21. Not open to students who have received credit for Mathematics 365F. Enrollment limited to 12. S. Ross.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 minor 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. Staff.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. Not open to first-year students, sophomores, or juniors.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): Mathematics 301. Corequisite(s): Mathematics 309. Open to seniors only. Not open to students who have received credit for Mathematics 457 or 458. Enrollment limited to 10. B. Shulman.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 395D. Chaotic Dynamical Systems.

One of the major scientific accomplishments of the last twenty-five years was the discovery of chaos and the recognition that sensitive dependence on initial conditions is exhibited by so many natural and man-made processes. To really understand chaos, one needs to learn the mathematics behind it. This seminar considers mathematical models of real-world processes and studies how these models behave as they demonstrate chaos and its surprising order. Prerequisite(s): Mathematics 301. Corequisite(s): Mathematics 309. Open to seniors only. Not open to students who have received credit for Mathematics 457 or 458. S. Ross.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): Mathematics 206 and one of the following: Mathematics s21, 301, or 309. Open to seniors only. Not open to students who have received credit for Mathematics 457, 458, or s45J. Enrollment limited to 10. 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 and 309. Open to seniors only. Not open to students who have received credit for Mathematics 457 or 458. Enrollment limited to 10. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 395I. Topological Methods in Combinatorics.

How can the rent of a house with differently-sized rooms be divided among a group of people so that each person feels that he or she got the best deal? How can we prove that at this very moment there are two diametrically opposite points on Earth's surface that have the same temperature and the same air pressure? After an introduction to the required basics of topology, geometry, and combinatorics, students independently explore these and related questions using the Borsuk-Ulam theorem, the Brouwer fixed point theorem, and their discrete versions. Prerequisite(s): Mathematics 301 and 309. Open to seniors only. Not open to students who have received credit for Mathematics 457 and 458. Enrollment limited to 10. P. Jayawant.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 457. Senior Thesis.

Prior to entrance into Mathematics 457and/or 458, 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 457and/or 458, 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 and/or 458, 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 course 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. P. Wong, Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s31. Applied Abstract Algebra.

This course 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 course is to show that algebraic abstractions can be used to refine our understanding of many scientific phenomena. This course 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. 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 course 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. 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. 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 course, students conceive and design projects to study specific aspects of roller coasters. Prerequisite(s): Mathematics 105. Enrollment limited to 25. M. Greer.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s45M. Enumerative Combinatorics.

In how many ways can we put twenty letters in addressed envelopes such that no letter goes into the correct envelope? In how many ways can we seat ten people around a table so that two of them who are friends are seated next to each other? Techniques for enumeration have been developed over centuries to answer such questions. This course covers the basic methods of counting via binomial coefficients and bijections. Through projects, students apply the methods to explore algebraic structures such as permutations, magic squares, and symmetric structures that are useful in computer science and coding theory. Recommended background: Mathematics s21. Prerequisite(s): Mathematics 205. P. Jayawant.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s50. 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 during a Short Term. May not be used to fulfill the requirement for the mathematics major or concentration in mathematics. Normally offered every year. Staff.Concentrations

This course is referenced by the following General Education Concentrations