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The College Catalogue

Mathematics – Courses

Introductory, Intermediate, and Advanced Courses

1050 {50} - MCSR. Quantitative Reasoning. Every semester. Eric Gaze.

Explores the ways and means by which we communicate with numbers; the everyday math we encounter on a regular basis. The fundamental quantitative skill set is covered in depth providing a firm foundation for further coursework in mathematics and the sciences. Topics include ratios, rates, percentages, units, descriptive statistics, linear and exponential modeling, correlation, logic, probability. A project-based course using Microsoft Excel, emphasizing conceptual understanding and application. Reading of current newspaper articles and exercises involving personal finance are incorporated to place the mathematics in real-world context.

1200 {155} a - MCSR. Introduction to Statistics and Data Analysis. Every fall. Fall 2013. Michael King.

A general introduction to statistics in which students learn to draw conclusions from data using statistical techniques. Examples are drawn from many different areas of application. The computer is used extensively. Topics include exploratory data analysis, planning and design of experiments, probability, one and two sample t-procedures, and simple linear regression. Not open to students who have credit for Mathematics 1300 {165}, Psychology 2520 {252}, or Economics 2557 {257}.

1300 {165} a - MCSR. Biostatistics. Every semester. Fall 2013. John O’Brien. Spring 2014. The Department.

An introduction to the statistical methods used in the life sciences. Emphasizes conceptual understanding and includes topics from exploratory data analysis, the planning and design of experiments, probability, and statistical inference. One and two sample t-procedures and their non-parametric analogs, one-way ANOVA, simple linear regression, goodness of fit tests, and the chi-square test for independence are discussed. An average of four to five hours of class meetings and computer laboratory sessions per week. Not open to students who have credit for or are concurrently enrolled in Mathematics 1200 {155}, Economics 2557 {257}, or Psychology 2520 {252}.

1600 {161} a - MCSR. Differential Calculus. Every semester. The Department.

Functions, including the trigonometric, exponential, and logarithmic functions; the derivative and the rules for differentiation; the anti-derivative; applications of the derivative and the anti-derivative. Four to five hours of class meetings and computer laboratory sessions per week, on average. Open to students who have taken at least three years of mathematics in secondary school.

Prerequisite: Placement in Mathematics 1600 {161}.

1700 {171} a - MCSR. Integral Calculus. Every semester. The Department.

The definite integral; the Fundamental theorems; improper integrals; applications of the definite integral; differential equations; and approximations including Taylor polynomials and Fourier series. An average of four to five hours of class meetings and computer laboratory sessions per week.

Prerequisite: Mathematics 1600 {161} or placement in Mathematics 1700 {171}.

1750 {172} a - MCSR. Integral Calculus, Advanced Section. Every fall. The Department.

A review of the exponential and logarithmic functions, techniques of integration, and numerical integration. Improper integrals. Approximations using Taylor polynomials and infinite series. Emphasis on differential equation models and their solutions. An average of four to five hours of class meetings and computer laboratory sessions per week. Open to students whose backgrounds include the equivalent of Mathematics 1600 {161} and the first half of Mathematics 1700 {171}. Designed for first-year students who have completed an AB Advanced Placement calculus course in their secondary schools.

Prerequisite: Placement in Mathematics 1750 {172}.

1800 {181} a - MCSR. Multivariate Calculus. Every semester. The Department.

Multivariate calculus in two and three dimensions. Vectors and curves in two and three dimensions; partial and directional derivatives; the gradient; the chain rule in higher dimensions; double and triple integration; polar, cylindrical, and spherical coordinates; line integration; conservative vector fields; and Green’s theorem. An average of four to five hours of class meetings and computer laboratory sessions per week.

Prerequisite: Mathematics 1700 {171} or 1750 {172}, or placement in Mathematics 1800 {181}.

2000 {201} a - MCSR. Linear Algebra. Every semester. Fall 2013. Thomas Pietraho, Manuel Reyes. Spring 2014. The Department.

Topics include vectors, matrices, vector spaces, inner product spaces, linear transformations, eigenvalues and eigenvectors, and quadratic forms. Applications to be chosen from linear equations, discrete dynamical systems, Markov chains, least-squares approximation, and Fourier series.

Prerequisite: Mathematics 1800 {181}, placement in Mathematics 2000 level, or permission of the instructor.

2020 {200} a - MCSR. Introduction to Mathematical Reasoning. Every semester. Fall 2013. Jennifer Taback. Spring 2014. The Department.

An introduction to logical deductive reasoning and mathematical proof through diverse topics in higher mathematics. Specific topics include set and function theory, modular arithmetic, proof by induction, and the cardinality of infinite sets. May also consider additional topics such as graph theory, number theory, and finite state automata.

Prerequisite: Mathematics 1800 {181}, placement in Mathematics 2000 level, or permission of the instructor.

2108 {204} a - MCSR. Biomathematics. Fall 2014. Mary Lou Zeeman.

A study of mathematical modeling in biology, with a focus on translating back and forth between biological questions and their mathematical representation. Biological questions are drawn from a broad range of topics, including disease, ecology, genetics, population dynamics, and neurobiology. Mathematical methods include discrete and continuous (ODE) models and simulation, box models, linearization, stability analysis, attractors, oscillations, limiting behavior, feedback, and multiple time-scales. Three hours of class meetings and 1.5 hours of computer laboratory sessions per week. Within the biology major, this course may count as the mathematics credit or as biology credit, but not both. Students are expected to have taken a year of high school or college biology prior to this course. (Same as Biology 1174 {174}.)

Prerequisite: Mathematics 1600 {161} or higher, placement in Mathematics 2108, or permission of the instructor.

2109 {229} a - MCSR. Optimization. Every other year. Fall 2013. Adam Levy.

A study of optimization problems arising in a variety of situations in the social and natural sciences. Analytic and numerical methods are used to study problems in mathematical programming, including linear models, but with an emphasis on modern nonlinear models. Issues of duality and sensitivity to data perturbations are covered, and there are extensive applications to real-world problems.

Prerequisite: Mathematics 1800 {181}, placement in Mathematics 2000 level, or permission of the instructor.

2206 {225} a - MCSR. Probability. Every semester. Fall 2013. Amanda Redlich. Spring 2014. The Department.

A study of the mathematical models used to formalize nondeterministic or “chance” phenomena. General topics include combinatorial models, probability spaces, conditional probability, discrete and continuous random variables, independence, and expected values. Specific probability densities, such as the binomial, Poisson, exponential, and normal, are discussed in depth.

Prerequisite: Mathematics 1800 {181}, placement in Mathematics 2000 level, or permission of the instructor.

2208 {224} a - MCSR. Ordinary Differential Equations. Fall 2013. Mohammad Tajdari.

A study of some of the ordinary differential equations that model a variety of systems in the physical, natural, and social sciences. Classical methods for solving differential equations with an emphasis on modern, qualitative techniques for studying the behavior of solutions to differential equations. Applications to the analysis of a broad set of topics, including population dynamics, oscillators, and economic markets. Computer software is used as an important tool, but no prior programming background is assumed.

Prerequisite: Mathematics 2000 {201} or permission of the instructor.

2209 {244} a - MCSR. Numerical Methods. Every other fall. Fall 2013. Adam Levy.

An introduction to the theory and application of numerical analysis. Topics include approximation theory, numerical integration and differentiation, iterative methods for solving equations, and numerical analysis of differential equations.

Prerequisite: Mathematics 2000 {201} or permission of the instructor.

2301 {231} a - MCSR. Intermediate Linear Algebra. Spring 2014. The Department.

A continuation of Linear Algebra focused on the interplay of algebra and geometry as well as mathematical theory and its applications. Topics include matrix decompositions, eigenvalues and spectral theory, vector and Hilbert spaces, norms and low-rank approximations. Applications to biology, computer science, economics, and statistics, including artificial learning and pattern recognition, principal component analysis, and stochastic systems. Course and laboratory work balanced between theory and application.

Prerequisite: Mathematics 2000 {201} and 2020 {200}, or permission of the instructor.

2302 {232} a - MCSR. Number Theory. Every other fall. Fall 2014. The Department.

A standard course in elementary number theory, which traces the historical development and includes the major contributions of Euclid, Fermat, Euler, Gauss, and Dirichlet. Prime numbers, factorization, and number-theoretic functions. Perfect numbers and Mersenne primes. Fermat’s theorem and its consequences. Congruences and the law of quadratic reciprocity. The problem of unique factorization in various number systems. Integer solutions to algebraic equations. Primes in arithmetic progressions. An effort is made to collect along the way a list of unsolved problems.

Prerequisite: Mathematics 2020 {200} or permission of the instructor.

2303 {233} a - MCSR. Functions of a Complex Variable. Every other fall. Fall 2013. Justin Marks.

The differential and integral calculus of functions of a complex variable. Cauchy’s theorem and Cauchy’s integral formula, power series, singularities, Taylor’s theorem, Laurent’s theorem, the residue calculus, harmonic functions, and conformal mapping.

Prerequisite: Mathematics 1800 {181}, placement in Mathematics 2000 level, or permission of the instructor.

2404 {247} a - MCSR. Geometry. Fall 2015. The Department.

A survey of modern approaches to Euclidean geometry in two dimensions. Axiomatic foundations of metric geometry. Transformational geometry: isometries and similarities. Klein’s Erlanger Programm. Symmetric figures. Other topics may be chosen from three-dimensional geometry, ornamental groups, area, volume, fractional dimension, and fractals.

Prerequisite: Mathematics 2020 {200} or permission of the instructor.

2501 {253} a - MCSR. Vector Calculus. Every other spring. Spring 2015. The Department.

A study of vector calculus based on linear algebra. The central unifying theme is the theory and application of differential forms. Topics include the derivative as a linear transformation between Euclidean spaces; the Inverse Function Theorem and the Implicit Function Theorem; multiple integration and the Change of Variables Theorem; vector fields, tenors, and differential forms; line and surface integration; integration of differential forms; the exterior derivative; closed and exact forms; the generalized Stokes’ Theorem; gradient, curl, divergence and the integral theorems of Green, Gauss, and Stokes; manifolds in Euclidean space; applications in the physical sciences.

Prerequisite: Mathematics 2000 {201}.

2502 {252} a - MCSR. Mathematical Cryptography. Fall 2013. Michael King.

Classical and modern methods of cryptography and cryptanalysis. Topics include public key cryptography and the RSA encryption algorithm, factoring techniques, and recently proposed cryptosystems based on group theory and graph theory.

Prerequisite: Mathematics 2020 {200} or permission of the instructor.

2601 {258} a - MCSR. Combinatorics and Graph Theory. Every other fall. Fall 2014. The Department.

An introduction to combinatorics and graph theory. Topics to be covered may include enumeration, matching theory, generating functions, partially ordered sets, Latin squares, designs, and graph algorithms.

Prerequisite: Mathematics 2020 {200} or permission of the instructor.

2602 {262} a - MCSR. Introduction to Algebraic Structures. Every year. Fall 2013. Jennifer Taback.

An introduction to the theory of finite and infinite groups, with examples ranging from symmetry groups to groups of polynomials and matrices. Properties of mappings that preserve algebraic structures are studied. Topics include cyclic groups, homomorphisms and isomorphisms, normal subgroups, factor groups, the structure of finite abelian groups, and Sylow theorems.

Prerequisite: Mathematics 2000 {201} and 2020 {200}, or permission of the instructor.

2603 {263} a - MCSR. Introduction to Analysis. Every year. Spring 2014. The Department.

Building on the theoretical underpinnings of calculus, develops the rudiments of mathematical analysis. Concepts such as limits and convergence from calculus are made rigorous and extended to other contexts, including spaces of functions. Specific topics include metric spaces, point-set topology, sequences and series, continuity, differentiability, the theory of Riemann integration, functional approximation, and convergence.

Prerequisite: Mathematics 2020 {200}. (A different mathematics course numbered 2000–2969 {200–289} may be used instead, with approval from the instructor.)

2604 {267} a - MCSR. Topology. Spring 2015. The Department.

Topology studies those properties of objects that are preserved under continuous changes. Examines abstract definition of a topology and examples of topological spaces, connectedness and compactness, countability and separation axioms, classification of surfaces, algebraic topology—including homotopy, the fundamental group, covering spaces, and introductory category theory.

Prerequisite: Mathematics 2020 {200} or permission of the instructor.

2606 {265} a - MCSR. Statistics. Every spring. Spring 2015. John O’Brien.

An introduction to the fundamentals of mathematical statistics. General topics include likelihood methods, point and interval estimation, and tests of significance. Applications include inference about binomial, Poisson, and exponential models, frequency data, and analysis of normal measurements.

Prerequisite: Mathematics 2206 {225} or permission of the instructor.

2970–2973 {291–294} a. Intermediate Independent Study in Mathematics.
The Department.

2999 {299} a. Intermediate Collaborative Study in Mathematics. The Department.

3108 {304 or 318} a. Advanced Topics in Modeling. Fall 2015. Mary Lou Zeeman.

Development, analysis, and simulation of mathematical models. Application topics drawn from a variety of disciplines such as biology, environmental sciences, earth and oceanographic sciences, climate, and sustainability. Analysis topics include oscillation, chaos, bistability, bifurcation, perturbation, resilience, and their consequences for prediction. Three hours of class meetings and 1.5 hours of computer laboratory sessions per week. Not open to students who have credit for Mathematics 304.

Prerequisite: Mathematics 2000 {201}, 2020 {200}, and 2208 {224}.

3109 a. Advanced Topics in Optimization. Spring 2014. Adam Levy.

A study of infinite-dimensional optimization, including calculus of variations and optimal control. Classical, analytic techniques are covered, as well as numerical methods for solving optimal control problems. Applications in many topic areas, including economics, biology, and robotics.

Prerequisite: Mathematics 2000 {201}, 2109 {229}, and 2208 {224}.

3208 a. Advanced Topics in Dynamical Systems. Spring 2015. Mary Lou Zeeman.

Selected topics in nonlinear dynamical systems with applications in the physical, natural, and social sciences. Material selected from continuous and discrete systems, local and global techniques, stability theory, oscillators, bifurcation theory, multiple time scales, and chaos.

Prerequisite: Mathematics 2000 {201}, 2020 {200}, 2208 {224}, and 2603 {263}.

3209 a. Advanced Topics in Numerical Methods. Fall 2014. Adam Levy.

A study of some of the partial differential equations that model a variety of systems in the natural and social sciences. Classical methods for solving partial differential equations are covered, as well as modern, numerical techniques for approximating solutions. Applications to the analysis of a broad set of topics, including air quality, traffic flow, and imaging. Computer software is used as an important tool. Not open to students who have credit for Mathematics 244.

Prerequisite: Mathematics 2000 {201}, 2208 {224}, and 2209 {244}.

3404 {307} a. Advanced Topics in Geometry. Every other fall. Fall 2013. William Barker.

A survey of affine, projective, and non-Euclidean geometries in two dimensions, unified by the transformational viewpoint of Klein’s Erlanger Programm. Special focus will be placed on conic sections, Cayley-Klein geometries, and projective embeddings. Additional topics as time permits: complex numbers in plane geometry, quaternions in three-dimensional geometry, and the geometry of four-dimensional space-time in special relativity. Mathematics 2404 {247} is helpful but not required.

Prerequisite: Mathematics 2000 {201} and 2020 {200}, or permission of the instructor.

3602 {302} a. Advanced Topics in Algebra. Every other spring. Spring 2014. The Department.

Introduction to rings and fields. Vector spaces over arbitrary fields. Additional topics may include Galois theory, algebraic number theory, finite fields, and symmetric functions.

Prerequisite: Mathematics 2602 {262} or permission of the instructor.

3603 {303} a. Advanced Topics in Analysis. Every other spring. Spring 2015. Thomas Pietraho.

One or more selected topics from advanced analysis. Possible topics include Lebesque measure and integration theory, Fourier analysis, Hilbert and Banach space theory, and stochastic calculus with applications to mathematical finance.

Prerequisite: Mathematics 2000 {201} and 2603 {263}, or permission of the instructor.

3606 {305} a. Advanced Topics in Probability and Statistics. Every other fall. Fall 2013. John O’Brien.

One or more specialized topics in probability and statistics. Possible topics include regression analysis, nonparametric statistics, logistic regression, and other linear and nonlinear approaches to modeling data. Emphasis is on the mathematical derivation of the statistical procedures and on the application of the statistical theory to real-life problems.

Prerequisite: Mathematics 2000 {201} and 2606 {265}, or permission of the instructor.

4000–4003 {401–404} a. Advanced Independent Study in Mathematics. The Department.

4029 {405} a. Advanced Collaborative Study in Mathematics. The Department.

4050–4051 a. Honors Project in Mathematics. The Department.

Online Catalogue content is current as of August 1, 2013. For most current course information, use the online course finder. Also see Addenda.