# Catalog

### Mathematics

Professor Wong; Associate Professors Greer, Jayawant, Ott, Ross, and Salerno (chair); Assistant Professor Boateng; Visiting Assistant Professors Chari and Montgomery; Lecturers Balcomb 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.

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 (bates.edu/mathematics).

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 a wide variety of careers, such as 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 300 or higher, not including MATH 360, 457, 458, 495, or any Short Term course.

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 selection of thesis or senior seminar requires departmental approval. The thesis or senior seminar requirement must be completed during the student's last year at Bates. Students satisfy their [W3] requirement by successful completion of either a thesis or a 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 consists 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 Bates courses lacking the mathematics subject code (MATH or MA) can be counted toward the mathematics minor. At least two of the three elective mathematics 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, PLTC 218, PSYC 218, or SOC 305. Enrollment limited to 39. [Q] Staff.#### 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 higher. Enrollment limited to 39. [Q] M. Greer.#### 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. Since then, 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 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. Not open to students who have received credit for MATH 206. Enrollment limited to 25 per section. Normally offered every semester. [Q] [QF] S. Chari, A. Salerno.#### 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. 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 through AP, IB, or A-Level examination scores. Successful completion of MATH 106 fulfills all Bates requirements for MATH 105. Successful completion of MATH 206 fulfills all Bates requirements for MATH 106. Not open to students who have received credit for MATH 206. Enrollment limited to 25 per section. Normally offered every semester. [Q] [QF] P. Jayawant, P. Wong, S. Balcomb, S. Ross.#### 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. Not open to students who have received credit for ECON 250 or 255 or any math course numbered 200 or above. Enrollment limited to 39. [Q] S. Balcomb, A. Salerno.#### 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 or 106. Open to first-year students. Enrollment limited to 25 per section. Normally offered every semester. [QF] P. Jayawant, 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. 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. Prerequisite(s): MATH 106. Open to first-year students. Enrollment limited to 29. Normally offered every semester. [QF] S. Chari.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 29. Normally offered every year. [Q] [QF] G. Coulombe.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 215. Statistics.

The sequel to MATH 214. This course explores inferential methods in statistics. Topics may include sampling distributions, point and interval estimation, hypothesis testing, analysis of variance, and linear regression. While applications are discussed, considerable emphasis is placed on the mathematical theory of statistics. Prerequisite(s): MATH 214. Enrollment limited to 29. [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, graphical approaches, numerical methods, and applications such as population modeling and mechanical vibrations. Prerequisite(s): MATH 205 and 206. Enrollment limited to 29. Normally offered every year. [Q] [QF] M. Montgomery.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. Not open to students who have received credit for MATH s45N. Enrollment limited to 29. A. Salerno.#### 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. [Q]#### BI/MA 255A. Mathematical Models in Biology.

Mathematical models are increasingly important throughout the life sciences. This course provides an introduction to a variety of models in biology, with concrete examples chosen from biological and medical fields. Students work both theoretically and with computer software to analyze models, compute numerical results, and visualize outcomes. Prerequisite(s): MATH 205. Open to first-year students. Enrollment limited to 29. [Q] M. Greer.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 255D. Mathematical Modeling with Monte Carlo Methods.

This course introduces mathematical modeling using Monte Carlo methods. Applications are drawn from the natural and social sciences, engineering, and mathematics. The course covers several Monte Carlo methods and students learn to write sample simulation problems using Matlab. The course also covers important ideas from probability theory needed to understand Monte Carlo methods. Prerequisite(s): MATH 205 and 206. Enrollment limited to 29. [Q] H. Boateng.#### MA/PH 255E. Nonlinear Models and Chaos.

A model is a simplified description of a system in mathematical and/or conceptual terms. Models help us understand how systems work and behave. The goals of this course are threefold: building models of natural systems, exploring their underlying mathematical structures and similarities, and simulating them with computers. Concepts acquired from simple systems in physics are applied to more complex systems in areas of biology, environment, climate, and social dynamics. Prerequisite(s): MATH 105 or 106 and PHYS 107. Open to first-year students. Enrollment limited to 29. Normally offered every year. [Q] [S] R. Saha.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 255F. Agent-Based Modeling with NetLogo.

This course uses mathematical techniques to study the behavior of real-world systems. The focus is on agent-based models (ABMs). ABMs are especially useful for understanding dynamics in systems that are affected by individual agents within the system, and the variations, interactions, decisions, and adaptations of those agents. To model using ABMs, students use the free software NetLogo. Students learn the guiding principles of ABM modeling and how to build models in NetLogo, then have the opportunity to create and analyze their own models. Prerequisite(s): MATH 205 and 206. Enrollment limited to 29. [Q] [QF] M. Greer.#### 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. Enrollment limited to 19. [W2] A. Salerno.#### 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 19. Normally offered every year. [QF] M. Greer.#### MATH 302. Topics in Analysis.

This course is a continuation of MATH 301. The content varies semester to semester. Possible topics include sequences and series of functions, Fourier series, metric spaces, measure theory, theories of integration, and functions of several variables. Prerequisite(s): MATH 301. Enrollment limited to 29. K. Ott.#### MATH 305. Applied Partial Differential Equations with Boundary Value Problems.

Partial differential equations (PDEs) arise in several fields of study in the natural and social sciences. This course provides an introduction to the theory and applications of partial differential equations. Students learn solution techniques and qualitative analysis for linear partial differential equations including initial and boundary value problems for second-order linear partial differential equations. Prerequisite(s): MATH 219. Enrollment limited to 25. [Q] Staff.#### MATH 306. Applied and Computational Linear Algebra.

This course explores techniques for solving huge linear systems, covering both the theory behind the techniques and the computation. The course reviews and further develops concepts from MATH 205 and uses them to efficiently solve problems across natural and social sciences. Problems are drawn from numerical analysis, mathematical biology, data analysis and machine learning, imaging and signal processing, chemistry, physics, economics, computer science, engineering, and other disciplines. Prerequisite(s): MATH 205 and 206. Enrollment limited to 29. Staff.#### 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. Staff.#### MATH 309. Abstract Algebra I.

An introduction to basic algebraic structures common throughout mathematics. These may 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 29. Normally offered every year. M. Montgomery.#### 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.#### 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. Wong.#### DC/MA 316. PIC Math: Topics in Industrial Mathematics.

This PIC Math (Preparation for Industrial Careers in Mathematical Sciences) course is intended for students with a strong interest in industrial applications of mathematics and computation. Students work in teams on a research problem identified by a community partner from business, industry, or government. Students develop their mathematical and programming skills as well as skills and traits valued by employers of STEM professionals, such as teamwork, effective communication, independent thinking, problem solving, and final products. Prerequisite(s): MATH 205 and 206. New course beginning winter 2020. Enrollment limited to 15. Instructor permission is required. (Community-Engaged Learning.) One-time offering. A. Salerno.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 19. 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 offered every year.#### 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] Staff.Concentrations

This course is referenced by the following General Education Concentrations

#### MATH 355B. Graph Algorithms.

How can we create a network 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, the field of 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): MATH s21. Enrollment limited to 25. 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. Staff.#### 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.#### MATH 379. Abstract Algebra II.

This course is a continuation of MATH 309. 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. [QF] P. Wong.#### 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.#### 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.#### MATH 495. Senior Seminar.

Prior to entrance into MATH 495, students must submit a proposal for the section of senior seminar they wish to undertake. 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. Instructor permission is required. [W3] Staff.#### MATH 495C. Numerical Linear Algebra.

This course is an introduction to numerical linear algebra, which is fundamental to scientific computing. Through guided and self-directed explorations, students examine direct and iterative solutions for linear systems problems including computation of eigenvalues and eigenvectors. The course focuses on both theoretical study of convergence of the numerical methods and practical implementations of these methods. Prerequisite(s): MATH 301. Enrollment limited to 15. Instructor permission is required. [W3] Staff.#### 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.#### MATH 495E. Networks.

A variety of networks such as social networks of people in a community, neural networks of organisms, and the network of web pages in the World Wide Web can be represented using graphs from the field of graph theory. A graph is a collection of nodes in which certain pairs of nodes are joined by edges between them. Graph theory provides a framework for the study of properties of networks. After a basic introduction to graph theory, students choose a network or particular properties shared by different types of networks to study further through readings from books and research papers. Prerequisite(s): MATH 301. Enrollment limited to 15. Instructor permission is required. [W3] P. Jayawant.#### MATH 495H. Cryptography.

Public key cryptography is at the center of most secure transactions these days, from using a credit card online to sending and signing secure messages. The security of a cryptosystem relies on finding solutions to difficult math problems like factorization of large numbers and the discrete logarithm problem. After an introduction to the basics, each student studies various methods of encryption like the RSA cryptosystem, the Diffie-Hellman key exchange, Elliptic Curve Cryptography, and various methods of breaking these encryptions. These topics draw 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.#### 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. Students' 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.#### 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. Prerequisite(s): MATH 309. Enrollment limited to 15. Instructor permission is required. [W3] P. Wong.#### 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 29. Normally offered every year. P. Wong, P. Jayawant.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. Enrollment limited to 29. P. Jayawant.Concentrations

This course is referenced by the following General Education Concentrations

#### DC/MA s45T. Mathematical Image Processing.

Digital image processing is a field essential to many disciplines, including medicine, astronomy, astrophysics, photography, and graphics. It is also an active area of mathematical research with ideas stemming from numerical linear algebra, Fourier analysis, partial differential equations and statistics. This course introduces mathematical methods in digital image processing, including basic image processing tools and techniques with an emphasis on their mathematical foundations. Students implement the theory using MATLAB. Topics may include image compression, image enhancement, edge detection, and image filtering. Students conceive and complete projects—either theoretical or practical—on an aspect of digital image processing. Prerequisite(s): MATH 205. Enrollment limited to 29. K. Ott.Concentrations

This course is referenced by the following General Education Concentrations