### Mathematics

Professor Wong; Associate Professors Greer (chair), Jayawant, and Ross; Assistant Professors Ott and Salerno; Visiting Assistant Professor Montgomery; Visiting Instructor Balcomb; 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 a 100-level course, MATH 205, MATH 206, or a more advanced course. For students unsure whether to start in MATH 105, 106, or beyond, the department recommends taking the online placement examination.

The mathematics department offers a major and a minor in mathematics. More information on the mathematics department is available on the website.

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) All of the following:

MATH 205. Linear Algebra.

MATH 206. Multivariable Calculus.

MATH s21. Introduction to Abstraction, which the department encourages taking during Short Term of the first year

2) Both of the following, which the department encourages taking before beginning a senior thesis or the senior seminar:

MATH 301. Real Analysis.

MATH 309. Abstract Algebra I.

3) Three elective mathematics courses numbered 200 or higher during a semester, or 30 or higher during Short Term, not including MATH 360, 457, 458, 495, or s50. One of the following courses may count as one of these electives:

ECON 255. Econometrics.

PHYS 301. Mathematical Methods of Physics.

4) A fourth elective mathematics course numbered at the 300 or higher, not including MATH 360, 457, 458. 495, or s50.

5) Completion of either a one-semester thesis (MATH 457 or 458), a two-semester thesis (MATH 457–458), or the senior seminar (MATH 495). The thesis option requires departmental approval. Students satisfy their [W3] requirement by successful completion of either a thesis or MATH 495, the senior seminar.

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 MATH 301 (Real Analysis) and MATH 309 (Abstract Algebra I) must be completed at Bates.

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

3) At least two of the four elective mathematics courses must be completed at Bates, and listed or cross-listed in the Bates mathematics department.

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.

The minor in mathematics consist of:

1) MATH 105. Calculus I.

MATH 106. Calculus II.

Successful completion of only MATH 106, or of MATH 206, satisfies this requirement, even if no course credit has been granted by Bates.

2) MATH 205. Linear Algebra.

MATH 206. Multivariable Calculus.

3) Three elective mathematics courses numbered 200 or higher during a semester, or 20 or higher during Short Term, not including 360, 457, 458, or S50.

Minor Courses Taken Elsewhere. No courses outside the mathematics department can be counted toward the mathematics minor. At least one of the three elective courses must be taken at Bates.

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

#### 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 BIO 244, ECON 250 or 255, ENVR 181, MATH 215 or 315, PSYC 218, or SOC 305. Enrollment limited to 40. [Q] Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 102. Mathematics across the Sciences.

This course makes connections between the mathematics learned in math class and the mathematics used in several science courses. For example, how does the formula for the equation of a line relate to a set of data collected in physics, chemistry, biology, geology, or environmental studies? Students who wish to apply their mathematical knowledge to better understand scientific problems and students who want to strengthen their core mathematical skills are ideally suited to take this course. Not open to students who have received credit for MATH 106 or MATH 205 or higher. Enrollment limited to 40. [Q] M. Greer.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 computational 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 questions page before registering. Prior calculus work generally excludes students from this course. Not open to students who have received equivalent course credit through AP, IB, or A-Level examination scores. Successful completion of MATH 106 or 206 fulfills all Bates requirements for MATH 105. Enrollment limited to 25 per section. [Q] Normally offered every semester. Staff.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 questions page before registering. Not open to students who have received equivalent course credit throught AP, IB, or A-Level examination scores. Successful completion of MATH 106 fullfills all Bates requirements for MATH 105. Successful completion of MATH 206 fulfills all Bates requirements for MATH 106. Enrollment limited to 25 per section. [Q] Normally offered every semester. 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 40. [Q] 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): MATH 105, 106, or BI/MA 255A. Open to first-year students. Enrollment limited to 25 per section. Normally offered every semester. S. Ross, P. Wong, Staff.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): MATH 106. Open to first-year students. Enrollment limited to 40. 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, MATH 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): MATH 206. Enrollment limited to 40. [Q] Normally offered every year. P. Jayawant.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 215. Statistics.

The sequel to MATH 214. This course covers estimation theory and hypothesis testing. Prerequisite(s): MATH 214. Enrollment limited to 30. [Q] G. Coulombe.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, graphical approaches, numerical methods, and applications such as population modeling and mechanical vibrations. Prerequisite(s): MATH 205 and 206. Enrollment limited to 30. [Q] Normally offered every year. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 225. Number Theory.

The theory of numbers is concerned with the properties of the integers, one of the most basic mathematical sets. Seemingly naive questions of number theory stimulated much of the development of modern mathematics and still provide rich opportunities for investigation. Topics include classical ones such as primality, congruences, quadratic reciprocity, and Diophantine equations, as well as more recent applications to cryptography. Additional topics such as computational methods, elliptic curves, or an introduction to analytic methods may be included. Prerequisite(s): MATH 205. Course renumbered from MATH s45N beginning Winter 2015. Not open to students who have received credit for MATH s45N. A. Salerno.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. [Q] 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 discrete and continuous 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 analyzing data. Prerequisite(s): MATH 205. Open to first-year students. Enrollment limited to 30. [Q] Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 255B. Mathematical Modeling.

This course uses mathematical techniques to study the behavior of real-world processes such as weather or population growth. Often this involves learning enough about that process to set up a corresponding system of equations of some kind the solutions of which can inform us about the behavior of process over time. A model can be used to predict future behavior and help us understand the extent to which changes in various elements of the process may affect that future. Finding solutions of the equations by analytic techniques, graphs, numerical methods, computers or some combination, and interpreting the results and their worthiness, are all part of the art of modeling. Prerequisite(s): MATH 205 and 206. Enrollment limited to 30. S. Ross.Concentrations

This course is referenced by the following General Education Concentrations

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

With varying subject matter, this writing-attentive 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): MATH s21. [W2] Normally offered every year.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 295A. Topics in Elementary Analysis.

This seminar examines subtle topics that are normally defined as additional reading material in the first-year calculus sequence. Proper rigor in mathematical proofs, logical presentation of mathematical concepts, and standard mathematical terminology are introduced. Students are expected to articulate mathematics in logical and precise ways through oral presentations and written assignments. The standard word processing tool LaTeX is used to enhance students' learning of mathematical terminology and mathematical writing. Prerequisite(s): MATH 105, 106, and s21. Enrollment limited to 30. [W2] P. Wong.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): MATH 205, 206, and s21. Enrollment limited to 30. Normally offered every year. P. Wong.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): MATH 205 and 206. Enrollment limited to 25. A. Salerno.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): MATH 205 and s21. Enrollment limited to 30. Normally offered every year. Staff.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): MATH 205 and 206. Enrollment limited to 25. 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 may include metric spaces, topological spaces, continuity, compactness, connectedness, homotopy theory, fixed-point theorems, and applications of these topics in areas such as geographic information systems, robotic, and game theory. Prerequisite(s): MATH 205, 206, and s21. Enrollment limited to 25. P. Jayawant.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): MATH 205, 206, and one economics course. Enrollment limited to 20. 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 students deepen their understanding of mathematics while gaining appreciation for computers' capabilities as well as deficiencies. Normally one or two MATH 355 seminars are offered annually.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): MATH 106 and 205. Enrollment limited to 25. [Q] S. Ross.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 355B. Graph Algorithms.

How can we create a newtwrk of cables between houses with minimum cost? Under what circumstances can a mail delivery van traverse all the streets in the neighborhood it serves without repeating any street? Graph theory is the branch of mathematics that provides the framework to answer such questions. 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 then write computer programs to provide solutions to questions such as the ones mentioned before. Prerequisite(s): MATH s21. Enrollment limited to 25. P. Jayawant.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 a field that 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): MATH s21. Enrollment limited to 25. S. Ross.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 355E. Algorithms in Arithmetic Geometry.

Arithmetic geometry is the study of sets of solutions of equations in several variables. Recently powerful results in this area have been conjectured and proved using the aid of computer algebra systems. These results in turn have provided essential tools for applications in areas like cryptography and coding theory. The main goal of this course is to acquaint students with the uses of a computer algebra system through specific examples in the areas of number theory and algebraic geometry. Topics may include number factoriation, primality testing, polynomial factoriation, elliptic curves, counting points on varieties, and some applications. Prerequisite(s): MATH 205 and s21. Enrollment limited to 25. A. Salerno.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. 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 379. Abstract Algebra II.

This course is a continuation of MATH 309, Abstract Algebra I. Advanced topics in group theory, ring theory, and field theory are covered. Applications include geometric constructions, crystallography, and algebraic coding theory. Prerequisite(s): MATH 309. New course beginning Fall 2014. Normally offered every other year. P. Wong.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 457. Senior Thesis.

Prior to entrance into MATH 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 MATH 457 in the fall 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 MATH 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 MATH 458 in the winter semester. [W3] Normally offered every year. Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 495. Senior Seminar.

While the subject matter varies, the writing-attentive 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. Instructor permission is required. [W3] Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 495D. 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): MATH 301. Instructor permission is required. [W3] S. Ross.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 495H. Elliptic Curve Cryptography.

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, and prominently in cryptography. After an introduction to the basics, each student studies various methods of encryption using elliptic curves, drawing from previous knowledge in abstract algebra, analysis, geometry, and number theory. Computer algebra systems also are used to illustrate the applications. Prerequisite(s): MATH 301 and 309. Instructor permission is required. [W3] A. Salerno.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 495I. Fair Division with Combinatorics and Topology.

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? Such fair division questions can be answered with a variety of mathematical techniques from combinatiries and topology. 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, their discrete versions, and other combinatorics methods. Prerequisite(s): MATH 301. Instructor permission is required. [W3] P. Jayawant.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 495J. Advanced Topics in Biomathematics.

Biology is one of the most fertile sources of new mathematics. Research may be based on computation and data, or it may rely entirely on theorems and proofs. It may require calculus, linear algebra, graph theory, differential equations, or numerical analysis. Students in this seminar read biology-inspired mathematical research and present their findings to each other. The students in the seminar and their mathematical interests influence the selection of research papers to be investigated. No previous course in biology or mathematical modeling is required. Prerequisite(s): MATH 301. Instructor permission is required. [W3] M. Greer.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 495K. Generalized Symmetric Spaces.

Symmetric spaces are unique structures studied in mathematics and mathematical physics. In fact, most mathematicians, scientists, and students have unknowingly worked with symmetric spaces many times. The concept is a basic extension of group theory studied in abstract algebra. This course introduces symmetric spaces using concepts from linear algebra, abstract algebra, and some combinatorics. Through current articles and texts, students decipher and discover the patterns and determine the classification of symmetric spaces in various finite groups. As a relatively new field of inquiry, the study of symmetric spaces challenges students to engage in original research during the course. Prerequisite(s): MATH 309. Instructor permission is required. [W3] Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 495L. The Fundamental Theorem of Algebra.

Over the centuries, there have been numerous proofs of the Fundamental Theorem of Algebra (FTA), which asserts that every polynomial of degree n must have at most n distinct roots over the complex numbers. The great German mathematician Carl F. Gauss (1777-1855) published no fewer than four different proofs of the result. While the name of the theorem foregrounds algebra, none of the known proofs is purely algebraic. Over the centuries, techniques from complex analysis, topology, and field extensions have been employed to give new proofs of the FTA. In this seminar, students explore some of these proofs where the methods are drawn from various subfields in mathematics. Recommended background: MATH 208. Prerequisite(s): MATH 309. Enrollment limited to 15. Instructor permission is required. Normally offered every other year. P. Wong.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.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s45. Seminar in Mathematics.

The content varies. Normally offered every year. Staff.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): MATH 105. Enrollment limited to 25. Instructor permission is required. [Q] 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. Prerequisite(s): MATH 205. Recommended background: MATH s21. P. Jayawant.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s45N. Number Theory.

The theory of numbers is concerned with the properties of the integers, one of the most basic 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 computational methods, elliptical curves, or an introduction to analytic methods may be included. Prerequisite(s): MATH s21. Not open to students who have received credit for MATH 225. Enrollment limited to 30. A. Salerno.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s45Q. p-adic Numbers.

This course is an introduction to p-adic numbers, which are fascinating but lesser known than rational, real, and complex numbers. The study of p-adic numbers blends together many parts of mathematics, and has numerous applications, including to physics. These numbers are an analogue of the real numbers, and as such many of the concepts from calculus extend naturally. However, differences in the geometry and the calculus of these numbers lead to interesting discoveries and insights; in this geometry, for example, all triangles are isosceles, and every point inside a circle is the center of the circle. Topics may include the study of absolute values on a field, basic properties and geometry of p-adic numbers, sequences and series, functions and their derivatives, and power series. Recommended background: MATH 309 and s21. Prerequisite(s): MATH 205. New course beginning Short Term 2014. Enrollment limited to 30. A. Salerno.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH s45R. Introduction to Geometric Group Theory.

In the 1980s, M. Gromov introduced a new approach to the study of infinite groups, namely the study of groups as geometric objects. Since then, the field of geometric group theory has flourished and is one of the most active areas of current mathematical research. This course presents the rudiments of geometric group theory. Recommended background: MATH 309. Prerequisite(s): MATH s21. New course beginning Short Term 2015. Enrollment limited to 30. Normally offered every other year. P. Wong.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