# Courses and Requirements

*The goal of Hamilton’s Mathematics and Statistics Department is to enable students to analyze and organize information using quantitative and statistical tools, to reason and argue logically, to employ appropriate problem-solving strategies, and to communicate complex ideas clearly and efficiently.*

Concentrators may not take any course in the Mathematics and Statistics Department numbered 224 or higher on a CR/NC basis. Concentrators may apply at most one course taken on a CR/NC basis towards the concentration.

Beginning with the class of 2020, students concentrating in mathematics must satisfy the Social, Structural, and Institutional Hierarchies requirement by completing one of Math 152, Math 498, Econ 166, Hist 226 or, for those interested in pursuing a career in education, Educ 204, Educ 206, Educ 339 or Educ 415. Students who have not completed one of these courses may petition the department to accept a course in another department as fulfilling the SSIH requirement by providing a written rationale explaining how the proposed course will help them gain an understanding of social, structural, and institutional hierarchies that is relevant to mathematics and their post-graduation plans.

Students may earn departmental honors by completing courses that satisfy the concentration with an average of 3.6 or higher, by taking a fourth full-credit elective that is at the 300 level or higher, and by making a public presentation to the department on a mathematical or statistical topic during their junior or senior year.

A minor in mathematics consists of 116, 216, 224 and two electives. Minors in mathematics may not take any course in the Mathematics and Statistics Department numbered 224 or above on a CR/NC basis. Minors in mathematics may apply at most one course taken on a CR/NC basis towards the minor.

A minor in statistics consists of five courses: 116, 216, 152, 254 and 351. Students may substitute Econ 166, Govt 230 or Psych/Neuro 201 for 152. Students may use Advanced Placement, International Baccalaureate, or A-level courses to count for only one of the five courses. Students with two or more such courses, or who want to count Econ 166, Govt 230 or Psych/Neuro 201 towards a concentration or another minor can complete the minor in statistics by selecting from the following additional courses: 352, 355, 356, 503 or 551 for a minimum of 4 courses from the Mathematics and Statistics Department. A student with a concentration in mathematics may not minor in statistics. Minors in statistics may not take any course in the Mathematics and Statistics Department numbered 224 or above on a CR/NC basis. Minors in statistics may apply at most one course taken on a CR/NC basis towards the minor.

While all courses offered by the department satisfy the QSR requirement, students seeking an entry-level course only for this purpose are encouraged to consider COLEG 105S: A World of Impending Disaster.

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113 F,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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116 F,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. 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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152 F,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 statistical computer software and authentic data, with a focus on investigating issues of social, structural, and institutional hierarchies.
(Social, Structural, and Institutional Hierarchies.)
(Quantitative and Symbolic Reasoning.)
May not be taken by students who have taken Math 252 or 253, or have taken or are taking Econ 166, Econ 265, Psych 201, Neuro 201 or Govt 230.
Maximum enrollment, 25.
The Department.

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216 F,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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224 F,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, 18.
The Department.

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231 F
Linear Optimization.
**

An introduction to solving optimization problems involving linear functions subject to linear constraints (linear programming). Topics include the simplex method, duality theory, sensitivity analysis and integer programming. Features applications to business, economics, transportation, health care and other areas.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 224 is required; 216 is recommended.
Cockburn.

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234 S
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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235 F,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.)
(Speaking-Intensive.)
Prerequisite, 224.
Maximum enrollment, 20.
The Department.

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254 F,S
Statistical Modeling and Applications.
**

A second course in introductory statistics with a focus on applications to real data using the statistical software R. Topics include linear and logistic regression, analysis of variance, categorical data analysis, and selected topics in randomization and machine learning. Emphasis will be placed on interpretation and presentation of results.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 152, 252 or Math 253; Econ 166, Pysch/Neuro 201, or Govt 230. Students who have a score of 4 or 5 in AP Statistics may enroll with the instructor’s permission. May not be taken by students who have taken or are taking Econ 400.
Maximum enrollment, 25.
The Department.

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304 S
Topology.
**

An introduction to point set topology, a foundational topic for much of modern mathematics. We will cover topological spaces, separation axioms, quotient spaces, compactness, connectedness, path connectedness, and homotopy. In the last part of the course we will cover the fundamental group, the most basic algebraic topological invariant.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 314W.
Ceniceros.

**
314 F,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.)
(Quantitative and Symbolic Reasoning.)
Prerequisite, 116 or 216, and 224.
Maximum enrollment, 18.
The Department.

**
[315 S]
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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318 S
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, 216 and 314.
Kantrowitz.

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322 S
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.

**
[324 S]
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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325 F,S
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, 18.
The Department.

**
[335 S]
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-Bendixson theorem, strange attractors, discrete dynamical systems and symbolic dynamics.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 235 and 314.

**
[337 S]
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.

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351 F
Probability and Statistical Inference.
**

An introduction to set-theoretic probability with applications to mathematical statistics. Topics include probability spaces, discrete and continuous random variables, single and multivariate distributions, and limit theorems, leading into the mathematical theory of estimators, sampling distributions, confidence intervals, and hypothesis testing.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 216 and an introductory statistics course which may be one of Math 152/252/253, Econ 166, Pysch/Neuro 201, or Govt 230. Students who have a score of 4 or 5 in AP Statistics may enroll with the instructor’s permission.
Bowman.

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352 S
Statistical Theory and Computation.
**

Study of the mathematical theory underlying classical and modern techniques in statistics, including implementation and visualization in the statistical programming language R. Topics include linear models and regression, randomization and bootstrap, Monte Carlo methods and sampling, Bayesian statistics, and topics in applied statistical modeling.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 224 and 351.
The Department.

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[355 F]
Mathematics of Machine Learning.
**

An introduction to machine learning with a focus on the mathematics required to perform various algorithms. Topics include linear mappings, inner product spaces, orthogonality, matrix decompositions, gradients, supervised and unsupervised machine learning, principal components analysis, and artificial neural networks. Familiarity with rudimentary computer programming recommended.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 216 and Math 224.

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356 S
Statistical Methods in Machine Learning.
**

This course covers statistical methods in machine learning such as decision trees, random forests and support vector machines The course will use a project-based approach to give students hands-on experience using these techniques by analyzing large and complex real-world datasets. More importantly, they will learn the statistical principles behind these procedures, such as loss functions, maximum likelihood estimation and bias-variance trade-off as well as why these principles matter in real world settings.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 254 and Math 351.
Maximum enrollment, 25.
The Department.

**
[361 S]
Number Theory and Applications.
**

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. Applications will include cryptography, the practice of encrypting and decrypting messages, and cryptanalysis, the practice of developing secure encryption and decryption protocols and probing them for possible flaws.
(Quantitative and Symbolic Reasoning.)
(Speaking-Intensive.)
Prerequisite, 325 or consent of instructor.

**
[363 S]
Differential Geometry.
**

Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. This course is an introduction to the differential geometry of curves and surfaces in three dimensional Euclidean space. Topics include Frenet frames for curves, Gaussian and mean curvature, the first and second fundamental forms, geodesics, and Gauss’s Theorema Egregium.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 216 and Math 224W.

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498 S
Seminar on Mathematics in Social Context.
**

Examines the role mathematics plays in the construction and perpetuation of social stratification, as well as the influence of social categorization on the development of mathematics and mathematicians. Works such as *Hidden Figures* by Margot Lee Shetterly, *Weapons of Math Destruction* by Cathy O’Neil, and *Radical Equations: From Mississippi to the Algebra Project* by Bob Moses ‘56 may be included. One-half course credit. Open only to mathematics concentrators.
(Social, Structural, and Institutional Hierarchies.)
Maximum enrollment, 20.
The Department.

**
502 F
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.
Dykstra.

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503 F
Senior Seminar in Statistics.
**

The goal of this seminar is to dig deeper into 3 main areas in statistics: 1. Multivariate statistical methods; 2. Nonparametric statistics; 3. Machine/Statistical learning. Students will learn the theoretical foundations of these tools, compare and contrast these methods in data analysis projects. Emphasis will be given to analyzing very large (and messy) datasets.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 254 or Math 351 or Econ 266, or permission of instructor.
Maximum enrollment, 12.
Kuruwita.

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506 F
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.
Cockburn.

**
[508 F]
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.)
Prerequisite, Math 314W.
Maximum enrollment, 12.

**
[510 F]
Senior Seminar in Applied Network Analysis.
**

Our world is built of networks: the internet, social networks, transportation networks, communication networks, biological networks. Natural and useful question include "What makes a network robust?” "Can we predict where failures might occur?” “What can we do to slow propagation of viruses along a network?” This courses will cover abstract mathematical properties of networks that can help us answer these questions. These will be examined in the context of both theoretical and real world networks. Further, student groups will analyze and report on a real world network of their choice.
(Quantitative and Symbolic Reasoning.)
Prerequisite, 314 or 325.
Maximum enrollment, 12.

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[513 F]
Senior Seminar in 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, Math 325.
Maximum enrollment, 12.

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525 F
Senior Seminar in Computational Algebra.
**

An introduction to the intersection of algebra and computation. The course will feature applications to statistics, data analysis, and other fields. Sample advanced topics in algebra include commutative algebra, algebraic geometry, and homological algebra. In addition to presenting material to each other at the beginning of the course, students will complete multiple guided-inquiry projects throughout the semester. Some programming experience (or enthusiasm to gain programming experience) will be helpful for working through examples.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 325.
Maximum enrollment, 12.
Gibbons.

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551 F
Senior Seminar in Applied Probability.
**

An exploration of modern applications of probability theory, including topics such as stochastic approximation, random walks and Brownian motion, graphical models, information theory, and large deviations. Student groups will choose a topic to study in depth and apply probabilistic methods to solve original problems using the statistical programming language R.
(Quantitative and Symbolic Reasoning.)
Prerequisite, Math 351.
Maximum enrollment, 12.
Bowman.

(*from the Hamilton Course Catalogue*)