# Mathematics

# Courses and Requirements

*The goal of Hamilton's Mathematics Department is to provide all students with opportunities to hone logical thinking and statistical reasoning skills, while offering concentrators an understanding of fundamental areas of pure and applied mathematics and what constitutes legitimate, rigorous argument through the analysis and construction of written proofs.*

For students matriculating in 2013 or later:

A concentration in mathematics consists of the courses 116, 216, 224, 314, 325, a Senior Seminar, and three electives, of which at least one must be at the 300 level or higher. Concentrators fulfill the Senior Program requirement by taking a Senior Seminar. 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.

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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. For students matriculating in 2013 or later, this course may not be counted toward the concentration or minor.
(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.

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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.

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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 216 or consent of instructor.
Maximum enrollment, 20.
The Department.

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231F 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.

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234S Counting and Codes.
**

Topics include enumeration and error correcting codes. Enumeration methods are used to count objects with a given description (used to compute probabilities and to estimate computer program running times). Error correcting codes are used to identify and fix small transmission errors (used in MP3 players, DVDs, cable TV). For each topic we will look at the big ideas, and apply them to small cases.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 224. Boutin.

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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.
(Quantitative and Symbolic Reasoning.)
Prerequisite, For students matriculating before 2013: 114, 115 or 215, and 224. For students matriculating in 2013 or later: 224 .
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.

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254F,S Statistical Modeling and Applications.
**

A continuation and extension of the study of statistics begun in 253. Topics include simple and multiple regression, analysis of variance, categorical data analysis, logistic regression, and nonparametric methods designed to test hypotheses and construct confidence intervals.
(Quantitative and Symbolic Reasoning.)
Prerequisite, a full-year AP course in statistics, or Math 253, or any introductory statistics course at Hamilton, or departmental permission.
Maximum enrollment, 25.
Kelly (Fall), Kuruwita (Spring).

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314F 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.)
(Quantitative and Symbolic Reasoning.)
Prerequisite, 116 or 216, and 224.
Maximum enrollment, 20.
Kantrowitz, Dykstra.

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[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.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 314 or consent of instructor.

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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.
(Quantitative and Symbolic Reasoning.)
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 or CS 123. Cockburn.

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[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.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 224.

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325S Modern Algebra.
**

An introduction to the three fundamental structures of abstract algebra: groups, rings and fields.
(Writing-intensive.)
(Quantitative and Symbolic Reasoning.)
Prerequisite, 224.
Maximum enrollment, 20.
Boutin, Cockburn.

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[335S]
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.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 235 and 314.

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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.

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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.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 116 or 216, and 224. 224 may be taken concurrently. Kelly.

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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.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 351. Kuruwita.

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[361S]
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.

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[501F]
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.)
Prerequisite, 325.
Maximum enrollment, 12.

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502F Senior Seminar in Mathematical Modeling.
**

The description of biological, physical, and social phenomena using the language of mathematics. Focuses on the construction, analysis, and critique of mathematical models using a broad range of techniques.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 235 or consent of instructor.
Maximum enrollment, 12.
.

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503F 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.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 351 and (253 or 254 or 352) or permission of instructor.
Maximum enrollment, 12.
Kuruwita.

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504F Senior Seminar in Topology.
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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.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 314W or Math 325W.
Maximum enrollment, 12.
.

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[505F]
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.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 325W.
Maximum enrollment, 12.

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506F 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, Math 314W.
Maximum enrollment, 12.
.

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[508F]
Senior Seminar in Dynamics.
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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.)
Prerequisite, Math 314W.
Maximum enrollment, 12.

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509F Senior Seminar in Applied Statistics.
**

An exploration into the analysis of data, using techniques from Math 253 and building from there. The seminar will undertake to formulate questions of interest and then use appropriate statistical techniques to answer the questions (and to learn the appropriate techniques as necessary). An emphasis on survey data is anticipated. Much of the work will be done in teams, and much of class time will be devoted to student presentations of student work. The seminar will be particularly relevant to students who anticipate needing to both understand and produce data-based analyses in applied areas.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 224W and (253 or 254) or permission of instructor.
Maximum enrollment, 12.
L Knop.

(*from the Hamilton Course Catalogue*)