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College Catalogue

Mathematics

Faculty
Robert Kantrowitz, Chair
Richard E. Bedient (F)
Debra L. Boutin (FS)
Sally Cockburn
Andrew Dykstra
Courtney Gibbons
Timothy J. Kelly

Larry E. Knop (f)
Chinthaka Kuruwita (S)
Michelle LeMasurier
Robert Redfield (F)

Joshua Wiscons (fs)

Topaz Dent Wiscons (f)

For students matriculating in 2013 or later:
A concentration in mathematics consists of the courses 116, 216, 224, 314, 325, 437, and three electives, of which at least one must be at the 300 level or higher. Concentrators fulfill the Senior Program requirement by taking 437. It must be taken in the fall of the student's senior year, and all lower-numbered required courses, with at most one exception, should be completed prior to that time. Physics 320 or Physics 325, but not both, may be counted toward the concentration as an elective at the 200 level. Students may earn departmental honors by completing courses that satisfy the concentration with an average of 3.6 or higher, by taking a fourth elective that is at the 300 level or higher, and by making a public presentation to the department on a mathematical topic during their junior or senior year.

A minor in mathematics consists of 116, 216, 224 and two mathematics electives.

For students matriculating before 2013:
A concentration in mathematics consists of nine courses including the required courses 113; 114, 115 or 215; 224; 215, 231, 234, 235 or 253; 314; 325; 437; and two electives, of which at least one must be at the 300 level or higher. Concentrators fulfill the Senior Program requirement by taking 437. It should be taken in the fall of the student's senior year, and all lower-numbered required courses, with at most one exception, should be completed prior to that time. Physics 320 or Physics 325, but not both, may be counted as a lower-level elective toward the concentration. Students may earn departmental honors by completing courses that satisfy the concentration with an average of 3.6 or higher, by taking a third elective that is at the 300 level or higher, and by making a public presentation to the department on a mathematical topic during their junior or senior year. A minor in mathematics consists of 113, 224 and three mathematics electives. One of the electives is normally 114 or 215 and at least one of them must have 224 as a prerequisite.

[100] Statistical Reasoning and Data Analysis.
An introductory course intended to develop an understanding of and appreciation for the statistical approach to problems in business and the natural, social and behavioral sciences. (Quantitative and Symbolic Reasoning.) Not open to students who have taken a calculus course, Economics 265 or Psychology 280. May not be counted toward the concentration or the minor. Maximum enrollment, 25.

[101F] Fractal Geometry and Chaos Theory.
A visual introduction to the geometry of fractals and the dynamics of chaos. Study of mathematical patterns repeating on many levels and expressions of these patterns in nature. Extensive use of computers, but no computer expertise assumed. Placement subject to approval of the department. Not open to students who have taken a calculus course or 123. May not be counted toward the concentration or the minor. Maximum enrollment, 20.

[103F] Explorations in Mathematics.
A study of topics selected from scheduling, ways of counting, probability and statistics, geometry, social choice and decision making. Placement subject to approval of the department. Not open to students who have taken a calculus course or 123. May not be counted toward the concentration or the minor.

[108F] Transformation Geometry.
An introduction to transformations of the plane. Topics include line reflections, rotations, glide reflections, groups of isometries and symmetry groups. May not be counted toward the concentration or the minor. Maximum enrollment, 25.

113F,S Calculus I.
Introduction to the differential and integral calculus of a single variable. Topics include limits, continuity, derivatives, max-min problems and integrals. (Quantitative and Symbolic Reasoning.) Four hours of class. The Department.

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116F,S Calculus II.
116 F,S Calculus II – A continuation of the study begun in 113. Methods of integration, improper integrals, applications of integration to volume and arc length, parametric equations, sequences and series, power series, vectors, and an introduction to 3-dimensional coordinate systems with equations of lines and planes. Prerequisite, 113 or placement by the department. Not open to students who have taken 114. The Department. (Quantitative and Symbolic Reasoning.) Completion of 116 with a grade of C- or greater gives Hamilton credit for both 113 and 116 for those students placed into 116. The Department.

201F,S Topics in Mathematics.
Self-designed exploration of mathematical theory or applications that may include concentrated study of a narrowly focused topic, guest lectures, faculty and/or student presentations, independent research in the mathematical literature or a field experience. Course ends with an oral presentation in which the student summarizes the mathematics learned in the process. Prerequisite, consent of instructor. One-quarter course credit based on Satisfactory/Unsatisfactory. May be taken more than once with consent of the department. The Department.

216F,S Multivariable Calculus.
Introduction to functions of more than one variable, partial derivatives, multiple integrals in two and three dimensions, line and surface integrals, Green’s Theorem, curl, divergence, the Divergence Theorem and Stokes’ Theorem. Prerequisite 116 or placement by the department. Not open to students who have taken 114. The Department. (Quantitative and Symbolic Reasoning.) Completion of 216 with a grade of C- or greater carries credit for both 116 and 216 for those students placed into 216. The Department.

224F,S Linear Algebra.
An introduction to linear algebra: matrices and determinants, vector spaces, linear transformations, linear systems and eigenvalues; mathematical and physical applications. (Writing-intensive.) (Quantitative and Symbolic Reasoning.) Prerequisite, 116 or 215 or consent of instructor. Maximum enrollment, 20. The Department.

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231S Linear Optimization.
An introduction to solving optimization problems involving linear functions subject to linear constraints (linear programming). Topics include the simplex method, duality theory, game theory and integer programming. Features applications to economics, computer science and other areas. (Quantitative and Symbolic Reasoning.) Prerequisite, 224. Cockburn.

[234] Combinatorics.
Topics include enumeration, design theory and error correcting codes. Enumeration theory covers methods of counting objects with a given description (used to compute probabilities and to estimate computer program running times). Design theory covers methods for creating collections of sets meeting given criteria (used in experimental design). Error correcting codes covers how small errors can be identified and corrected (used in MP3 players, DVDs, cable TV). Prerequisite, 224.

235F,S Differential Equations.
Theory and applications of differential equations, including first-order equations, second-order linear equations, systems of equations, and qualitative and numerical methods. (Oral Presentations.) Prerequisite, For students matriculating before 2013: 114, 115 or 215, and 224. For students matriculating in 2013 or later: 116 and 224 (216 is recommended but not required). Maximum enrollment, 24. LeMasurier.

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253F,S Statistical Analysis of Data.
An introduction to the principles and methods of applied statistics. Topics include exploratory data analysis, sampling distributions, confidence intervals, hypothesis testing, regression analysis, analysis of variance and categorical data analysis. Extensive reliance on authentic data and statistical computer software. (Quantitative and Symbolic Reasoning.) Prerequisite, 113 or departmental placement. Maximum enrollment, 25. The Department.

265S Fractal Geometry: Concepts and Applications.
Considers the mathematics behind the stunning visual images of fractals. Topics will include self-similarity, dimension, Julia sets, the Mandelbrot set, circle inversions, cellular automata and basins of attraction. (Quantitative and Symbolic Reasoning.) Prerequisite, 224. Bedient.

[313] Knot Theory.
An introduction to knot theory. Topics include classification of different types of knots, the relations between knots and surfaces, and applications of knots to a variety of fields. (Quantitative and Symbolic Reasoning.) Prerequisite, 224.

314F,S Real Analysis.
An introduction to analysis. Topics include completeness of the real numbers, cardinality, sequences, series, real-valued functions of a real variable, limits, and continuity. (Writing-intensive.) Prerequisite, 116 or 216, and 224. Maximum enrollment, 20. Kantrowitz (fall), Cockburn (spring).

[315S] Functional Analysis.
An introduction to functional analysis. Topics include metric and normed linear spaces, including sequence spaces, function spaces, Hilbert and Banach spaces; Fourier series, and bounded linear operators. Prerequisite, 314 or consent of instructor.

318S Complex Analysis.
An introduction to the theory of analytic functions of a complex variable: Cauchy-Riemann equations, contour integration, Cauchy-Goursat theorem, Liouville theorem, Taylor and Laurent expansions, Residue theory. Prerequisite, 314. LeMasurier.

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[322S] Graph Theory.
An introduction to the theory and applications of graph theory. Topics include: trees; connectivity; Eulerian and Hamiltonian graphs; vertex-, edge- and map-colorings; digraphs; tournaments; matching theory; planarity and Ramsey numbers. (Quantitative and Symbolic Reasoning.) Prerequisite, 224.

[324S] Linear Algebra II.
A continuation of 224, with emphasis on the study of linear operators on complex vector spaces, invariant subspaces, generalized eigenvectors and inner product spaces. Prerequisite, 224.

325F,S Modern Algebra.
An introduction to the three fundamental structures of abstract algebra: groups, rings and fields. (Writing-intensive.) Prerequisite, 224. Maximum enrollment, 20. Gibbons (fall), J Wiscons (spring).

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327S Cryptography.
An introduction to cryptography, the study of enciphering messages. Topics covered follow the historical progression of codes, from symmetric key to public key cryptosystems and their implementation. The course also covers the number theory necessary for the study of modern encryption techniques, with special attention paid to modular arithmetic and theorems about prime numbers. Optional topics at the end of the course include quantum cryptography or the study of elliptic curves. (Quantitative and Symbolic Reasoning.) Prerequisite, 314 or 325. Gibbons.

[335] Differential Equations II.
A continuation of 235, with emphasis on techniques for studying nonlinear dynamical systems. Topics include equilibria in nonlinear systems, bifurcations, limit sets, the Poincare-Bendixon theorem, strange attractors, discrete dynamical systems and symbolic dynamics. Prerequisite, 235 and 314.

337S Partial Differential Equations.
Theory and applications of partial differential equations. Topics include separation of variables, Fourier series and transforms, and the Laplace, heat and wave equations. (Quantitative and Symbolic Reasoning.) Prerequisite, Math 224 and Math 235. Dykstra.

351F Probability Theory and Applications.
An introduction to probability theory, including probability spaces, random variables, expected values, multivariate distributions and the central limit theorem, with applications to other disciplines and an emphasis on simulation as an exploratory tool. Prerequisite, 116 or 216, and 224. 224 may be taken concurrently. Kelly.

352S Mathematical Statistics and Applications.
Study of the mathematical theory underlying statistical methodology. Topics include the law of large numbers, estimation, hypothesis testing, linear models, experimental design, analysis of variance and nonparametric statistics, with applications to a variety of disciplines. Prerequisite, 351. Kuruwita.

[361] Number Theory.
Number theory is the study of the properties of the positive integers. Topics include divisibility, congruences, quadratic reciprocity, numerical functions, diophantine equations, continued fractions, distribution of primes. (Quantitative and Symbolic Reasoning.) Prerequisite, 325 or consent of instructor.

[362S] Geometries.
A survey of geometries including Euclidean, hyperbolic, and spherical. Discrete geometry (triangulations of spaces), and possible geometries of the universe will also be discussed. A geometric/pictorial approach will be emphasized. The course will include reading assignments, discussions, and student presentations. (Quantitative and Symbolic Reasoning.) (Oral Presentations.) Prerequisite, 224.

450F,S Senior Research.
A project for senior concentrators in mathematics, in addition to participation in the Senior Seminar. Prerequisite, consent of department. The Department.

Seminars offered in recent years

437F Senior Seminar in Mathematics.
Study of a major topic through literature, student presentations and group discussions, with an emphasis on student presentations of student-generated results. Choice of topic to be determined by the department in consultation with its senior concentrators. The Department.

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437-01F Senior Seminar in Algebra.
An introduction to group theory with an emphasis on group actions. Students are responsible for providing examples, counterexamples, and proofs of theorems and regularly present their work in class. The course concludes with the students researching a topic in group theory of their own choosing. The topics covered in the course will include symmetric groups, dihedral groups, (subgroups of) general linear groups, projective linear groups, Lagrange's theorem, Orbit-Stabilizer theorem, Cauchy's theorem, and Sylow's theorems. (Quantitative and Symbolic Reasoning.) (Oral Presentations.) Prerequisite, 325. Maximum enrollment, 12. Wiscons, J.

[437-03F] Senior Seminar in Mathematical Modeling.
The description of biological, physical and social phenomena using the language of mathematics. Focuses on the construction of software-based mathematical models and on the analysis and critique of such models. Prerequisite, Math 235 and 253, or consent of the instructor. Maximum enrollment, 12.

437-04F Senior Seminar in Statistics.
A continuation of studies in mathematical statistics and the analysis of data. Topics include maximum likelihood estimation, regression, analysis of variance and design of experiments. Prerequisite, 251 or 351, and 253 or 352. Maximum enrollment, 12. Kuruwita.

[437-05] Senior Seminar in Topology.
Students jointly produce a textbook based on an outline provided. Topics include topological spaces, continuity of maps and homeomorphism. Spaces are described as connected and Hausdorff. The fundamental group is computed and used to classify various spaces. Prerequisite, Math 314W or Math 325W. Maximum enrollment, 12.

[437-08] Senior Seminar in Graph Symmetries.
Focuses on symmetries of simple and directed graphs. Graphs studied include the integer lattice, Kneser graphs, hypercubes, Cayley graphs. Given an outline containing definitions, theorems, and conjectures, students find examples, proofs and counterexamples, and create a course text with their results. No prior knowledge of graph theory is needed. Prerequisite, 325. Maximum enrollment, 12.

437-09F Senior Seminar in Philosophical Foundations of Mathematics.
The first half of this seminar focuses on the set theoretical foundations of mathematics, including ordered sets, ordinal and cardinal numbers, and the classic set paradoxes. Students will be given definitions for which they must find examples and theorems for which they must find proofs. Readings includes classic papers in the philosophy of mathematics by such authors as Bertrand Russell, Kurt Gödel, David Hilbert, A. J. Ayer and Henri Poincaré. Final paper required. Prerequisite, 314. Maximum enrollment, 12. Cockburn.

[437-10F] Senior Seminar in the History of Mathematics.
Survey of the history of mathematics through the nineteenth century, including the mathematics of ancient civilizations and the roots of fundamental concepts. (Oral Presentations.) Prerequisite, 224. Maximum enrollment, 12.

437-11F Senior Seminar in Dynamics.
Various topics from discrete dynamics are explored by working through a series of exploratory modules. Students work in teams and present their findings to the class. Topics include fixed points and their classifications, cycles and their classifications, fractal sets, sensitive dependence and chaos, symbolic dynamics and Sharkovskii’s periodic point theorem. (Quantitative and Symbolic Reasoning.) (Oral Presentations.) Prerequisite, Math 314W. Maximum enrollment, 12. Dykstra.

437-12F Senior Seminar in Applied Statistics.
An exploration of the analysis of data using techniques from Math 253 and beyond, with particular emphasis on regression and the general linear model. Students will be expected to do some independent explorations. Prerequisites, Math 224, Math 253, or permission of the instructor. (Quantitative and Symbolic Reasoning.) Prerequisite, Math 224, Math 253, or permission of the instructor. Maximum enrollment, 12. Kelly.

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