Chair, Professor H. T. (Pete) Hayslett

Professors Hayslett and Thomas Berger; Visiting Professors William Berlinghoff and Ramachandran Bharath; Associate Professors Fernando Gouvêa, Benjamin Mathes, and George Welch; Assistant Professors Leo Livshits, Jan Holly, Weiwen Miao, and Otto Bretscher

The Department of Mathematics offers courses in mathematics and statistics for students who: (1) plan a career in an area of pure or applied mathematics; (2) need mathematics as support for their chosen major; or (3) elect to take mathematics as part of their liberal arts education or to fulfill the area requirement in quantitative reasoning.

The department offers three programs: majors in mathematics and mathematical sciences and a minor in mathematics. Majors in mathematics and mathematical sciences can be taken with honors. In addition, there are interdepartmental joint majors in economics-mathematics and philosophy-mathematics.

Colby mathematics majors in recent years have entered graduate school to do advanced work in mathematics, statistics, computer science, biomathematics, and physics. They also have used the major as a solid foundation for careers in teaching, law, banking, insurance, management, the computer industry, and other areas.

All students who intend to enroll in one of the 100-level calculus courses are required to complete the mathematics placement questionnaire prior to registration.

Requirements for the Major in Mathematics
Completion of one year of calculus, Mathematics 253, 274, 333, 338, plus five additional courses selected from Mathematics 262 and any other three- or four-credit mathematics courses numbered 300 or above, at least one of which must be a 400-level course (excluding Mathematics 484).

Requirements for the Honors Program in Mathematics or Mathematical Sciences
An honors program is available for students majoring in mathematics and mathematical sciences who have a grade point average of at least 3.25 in all mathematics courses numbered 200 or higher and who complete an additional, pre-approved program of independent study in the major (Mathematics 484) culminating in both a written paper and a colloquium presentation. Students who successfully complete the requirements and who receive recommendation of the department will graduate "With Honors in Mathematics" or "With Honors in Mathematical Sciences."

Requirements for the Minor in Mathematics
Six three- or four-credit mathematics courses, including completion of at least one semester of calculus, Mathematics 253, and at least one course at the 300 level or above.

The point scale for retention of the majors and minor applies to all courses in the majors/minors. No requirement for the majors or minor may be taken satisfactory/unsatisfactory.

COURSE OFFERINGS

[101]    Calculus with Precalculus I    Designed for students who enter Colby with insufficient precalculus background for the standard calculus sequence. It is expected that all students who complete Mathematics 101 will enroll in Mathematics 102 in the following January. The combination of 101 and 102 covers the same calculus material as Mathematics 121. Completion of 101 alone does not constitute completion of a College calculus course for any purpose; in particular, it does not qualify a student to take Mathematics 122 nor does it satisfy the quantitative reasoning requirement. Students electing this course must complete the mathematics placement questionnaire prior to registration. Three credit hours.    

[102]    Calculus with Precalculus II    A continuation of Mathematics 101. Successful completion of both Mathematics 101 and 102 is equivalent to completion of Mathematics 121. Prerequisite: Mathematics 101. Three credit hours.  Q.    

111fs    Mathematics As a Liberal Art    The historical and contemporary role of mathematics in culture and intellectual endeavor; the nature of contemporary mathematics; mathematics as a tool for problem solving; logical reasoning; selected topics from modern mathematics. Four credit hours.  Q.    BERLINGHOFF, GOUVEA

112fs    Elementary Statistics    Description of data, confidence intervals, tests of hypotheses, non-parametric statistics, experimental design, analysis of variance, correlation and regression (including multiple regression), use of computer statistical packages. Credit is not given for both Mathematics 112 and 231. Four credit hours.  Q.    BHARATH, HAYSLETT

121fs    Single-Variable Calculus    Differential and integral calculus of one variable: limits and continuity; differentiation and its applications, antiderivatives, the definite integral and its applications; exponential, logarithmic, and trigonometric functions. Students electing this course in the fall term must complete the mathematics placement questionnaire prior to registration. Four credit hours.  Q.    FACULTY

122fs    Series and Multi-Variable Calculus    Further study of differential and integral calculus of one variable; infinite series; vectors and analytic geometry in two and three dimensions; vector calculus; multivariable calculus. Prerequisite: A course in single-variable calculus. Four credit hours.  Q.    FACULTY

131f    Complements to Calculus    Intended for students with some prior exposure to calculus who do not feel prepared to enter Mathematics 122 or 161, the course will reinforce and complement calculus concepts by relating calculus with other areas of mathematics, such as discrete mathematics, linear algebra, and complex variables. For students with no prior exposure to calculus, this course may be taken concurrently with Mathematics 121. Prerequisite: Previous exposure to Calculus. Four credit hours.  Q.    MATHES

161f    Honors Calculus I    Differential calculus of one and several variables: functions, limits, continuity, and differentiation. May not be taken for credit if the student has earned credit for Mathematics 122. First-year students must complete the mathematics placement questionnaire prior to selecting this course. Prerequisite: Students must have had substantial calculus in high school. Four credit hours.  Q.    LIVSHITS

162s    Honors Calculus II    A continuation of Mathematics 161. Integral calculus of one and several variables; infinite series. May not be taken for credit if the student has earned credit for Mathematics 122. Prerequisite: Mathematics 161. Four credit hours.    LIVSHITS

194s    Mathematics Seminar    Informal discussion of topics related to the mathematical sciences. Topics vary but are centered on a single book whose emphasis will generally be on the non-technical, humanistic side of mathematical endeavors. May be repeated for credit. Nongraded. One credit hour.    BERGER, WELCH

231fs    Applied Statistics and Regression Analysis    Elementary probability theory, special discrete and continuous distributions, descriptive statistics, sampling theory, confidence intervals, tests of hypotheses, correlation, linear regression, and multiple linear regression. Examples and applications slanted toward economics. Credit is not given for both Mathematics 112 and 231. Prerequisite: Mathematics 102, 121, or 161. Four credit hours.  Q.    BHARATH

253fs    Linear Algebra    Solutions of linear systems of equations, matrix algebra, determinants. Introduction to abstract vector spaces and linear transformations, eigenvalues, and eigenvectors. Prerequisite: Mathematics 102, 121, 131, or 161. Four credit hours.    BRETSCHER, HOLLY, MATHES

262s    Vector Calculus    An advanced calculus course. Vectors, lines, and planes; limits, continuity, derivatives, and integrals of vector-valued functions; polar, spherical, and cylindrical coordinates; partial and directional derivatives; multiple integrals; line and surface integrals; Green's Theorem; Stokes's Theorem; Fourier series; applications. Typically involves the use of a large computer mathematics package such as Mathematica or Maple. Prerequisite: Mathematics 122 or 162. Four credit hours.    BERGER

274fs    Introduction to Abstract Mathematical Thought    An introduction to fundamental mathematical techniques used in upper-level mathematics courses. The course presents the principles of mathematical logic and uses them to examine standard methods of direct and indirect proof, including mathematical induction. Topics include techniques from finite mathematics, the set theoretic approach to functions and relations, and the theory of infinite sets. Prerequisite: Mathematics 102, 121, 131, or 161. Two semesters of calculus is recommended. Four credit hours.    BERLINGHOFF, MATHES

311s    Introduction to Differential Equations    Theory and solution methods of ordinary differential equations; linear differential equations; first-order linear systems; qualitative behavior of solutions; nonlinear dynamics; existence and uniqueness of solutions; applications. Prerequisite: Mathematics 122 or 162, and 253. Four credit hours.    HOLLY, LANE

[331]    General Topology    Elementary set theory, functions, equivalence relations, topological spaces, basis for a topology, subspaces, concept of neighborhoods, open and closed sets, continuous functions, product topology, connectedness, separation axioms, coverings of spaces, compactness, paracompactness, metric spaces, and identification topology. Prerequisite: Mathematics 122 or 162, and 274. Four credit hours.    

332s    Introductory Numerical Analysis    Solution by numerical methods of linear and nonlinear equations, systems of equations, and differential equations; numerical integration; polynomial approximation; matrix inversion; error analysis. Prerequisite: Mathematics 122 or 162, and 253. Four credit hours.    HOLLY

333f    Abstract Algebra    Introduction to algebraic structures, such as groups, rings, integral domains, and fields. Prerequisite: Mathematics 253 and 274. Four credit hours.    MATHES

[336]    Mathematical Economics    Listed as Economics 336 (q.v.). Prerequisite: Economics 223, 224, and Mathematics 122 or 162. Three or four credit hours.    

338s    Real Analysis    An introduction to real analysis, with special focus on foundational issues. Prerequisite: Mathematics 122 or 162, and 274. Four credit hours.    LIVSHITS

[352]    Complex Variables    The arithmetic and calculus of complex numbers and functions. The properties of analytic functions, including Cauchy's integral theorem and formula, representation by Laurent series, residues and poles, and the elementary functions. Offered in alternate years. Prerequisite: Mathematics 122 or 162, and 274. Four credit hours.    

[357]    Elementary Number Theory    An introduction to the theory of numbers. Factorization and primes: unique factorization, greatest common divisors, the sequence of primes, primality testing and factoring on the computer, connections with cryptography. Congruences: linear congruences, theorems of Fermat, Euler, and Wilson, Chinese remainder theorem, quadratic residues, quadratic reciprocity law. Further topics chosen by the instructor. Offered in alternate years. Prerequisite: Mathematics 102, 121, 131, or 161. Two semesters of calculus or Mathematics 253 is recommended. Four credit hours.    

376s    History of Mathematics    A survey of the history of mathematics from the dawn of civilization to the 20th century. Original sources will be examined. The instructor may choose to focus on one theme or topic and its development throughout the history of mathematics. Prerequisite: Mathematics 274. Four credit hours.  H.    GOUVEA

[378]    Introduction to the Theory of Computation    Listed as Computer Science 378 (q.v.). Prerequisite: Mathematics 274 or Computer Science 231 and either Mathematics 122 or 162. Four credit hours.    

381f    Mathematical Statistics    Random variables, special probability distributions, moment generating functions, maximum likelihood estimators, sampling distributions, regression, tests of hypotheses, confidence intervals, linear models, analysis of variance. Although applications are discussed, the emphasis is on theory. Prerequisite: Mathematics 122 or 162. Four credit hours.    HAYSLETT

382s    Mathematical Statistics    Random variables, special probability distributions, moment generating functions, maximum likelihood estimators, sampling distributions, regression, tests of hypotheses, confidence intervals, linear models, analysis of variance. Although applications are discussed, the emphasis is on theory. Prerequisite: Mathematics 381. Four credit hours.    HAYSLETT

397Af    Geometry    The study of geometry to understand the plane through its symmetries. Euclidean geometry of the plane will be covered, followed by its geometrical symmetries, culminating in the 17 possible types of wallpaper patterns. Prerequisite: Mathematics 253 or permission of the instructor. Four credit hours.    BERGER

397Bf    Game Theory    An introduction to two-person and N-person games as mathematical models of conflict, cooperation, and voting. Both theory and applications will be presented. Examples related to fields such as economics, politics, anthropology, biology, business and philosophy will be considered. Students may be expected to use computing systems and to design practical effective game strategies, to be evaluated in contests. Prerequisite: Mathematics 253. Mathematics 274 is recommended. Four credit hours.    LIVSHITS

[398]    Mathematical Modeling    Application of mathematics to real-life problems in a variety of areas. Interpretation of existing mathematical models, analysis and computer simulation. Formulation and development of mathematical models. Prerequisite: Mathematics 122 or 162, and 253. Four credit hours.    

434s    Topics in Abstract Algebra    A sequel to Mathematics 333. Topics may vary from year to year. May be repeated, with permission of instructor, for credit. Prerequisite: Mathematics 333. Four credit hours.    GOUVEA

439f    Topics in Real Analysis    A sequel to Mathematics 338. Content may vary from year to year, but topics such as topology, measure theory, functional analysis, or related areas may be considered. May be repeated for credit with permission of instructor. Prerequisite: Mathematics 338. Four credit hours.    MATHES

491f, 492s    Independent Study    Independent study in an area of mathematics of particular interest to the student. Prerequisite: Permission of the instructor. One to four credit hours.    FACULTY