Introductory, Intermediate, and Advanced Courses

1050 {50} - MCSR. Quantitative Reasoning. Every semester. Fall 2014. Eric Gaze and Miriam Logan. Spring 2015. Miriam Logan.

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, and 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.

Prerequisite: Placement in Mathematics 1050 {50} and permission of Director of Quantitative Reasoning.

1200 {155} a - MCSR. Introduction to Statistics and Data Analysis. Every fall. Fall 2014. Leon Harkleroad.

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}.

Prerequisite: Placement in Mathematics 1200 {155}.

1300 {165} a - MCSR. Biostatistics. Every semester. Fall 2014. John O’Brien. Spring 2015. 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}.

Prerequisite: Placement in Mathematics 1300 {165}.

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 semester. 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 2014. William Barker and Miriam Logan. Spring 2015. 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 courses, or permission of the instructor.

2020 {200} a - MCSR. Introduction to Mathematical Reasoning. Every semester. Fall 2014. Michael King. Spring 2015. 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 courses, 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, or permission of the instructor.

2109 {229} a - MCSR. Optimization. Every other spring. Spring 2015. 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 2000 {201}, or permission of the instructor.

2206 {225} a - MCSR. Probability. Every semester. Fall 2014. Justin Marks. Spring 2015. 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 courses, or permission of the instructor.

2208 {224} a - MCSR. Ordinary Differential Equations. Fall 2014. Michael King. Spring 2015. The Department.

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 spring. Spring 2016. 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. Every other fall. Fall 2015. 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.

2303 {233} a - MCSR. Functions of a Complex Variable. Every other fall. Fall 2015. The Department.

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}, or permission of the instructor.

2404 {247} a - MCSR. Geometry. Every other fall. 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} or permission of the instructor.

2502 {252} a - MCSR. Number Theory and Cryptography. Every other spring. Spring 2015. The Department.

A survey of number theory from Euclid’s proof that there are infinitely many primes through Wiles’s proof of Fermat’s Last Theorem in 1994. Prime numbers, unique prime factorization, and results on counting primes. The structure of modular number systems. Continued fractions and “best” approximations to irrational numbers. Investigation of the Gaussian integers and other generalizations. Squares, sums of squares, and the law of quadratic reciprocity. Applications to modern methods of cryptography, including public key cryptography and RSA encryption.

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

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

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. Group Theory. Every other fall. Fall 2014. Jennifer Taback.

An introduction to algebraic structures based on the study of groups. Finite and infinite groups, rings, and fields, with examples ranging from symmetry groups to polynomials and matrices. Studies properties of mappings that preserve algebraic structures. Topics include cyclic groups, isomorphisms and homomorphisms, normal subgroups, quotient groups, and the structure of finite abelian groups.

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

2603 {263} a - MCSR. Introduction to Analysis. Every year. Fall 2014. Thomas Pietraho.

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}, or permission of the instructor.

2604 {267} a - MCSR. Topology. Every other spring. Spring 2016. 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.

2702 a - MCSR. Rings and Fields. Every other fall. Fall 2015. Manuel Reyes.

An introduction to algebraic structures based on the study of rings and fields. Structure of groups, rings, and fields, with an emphasis on examples. Fundamental topics include: homomorphisms, ideals, quotient rings, integral domains, polynomial rings, field extensions. Further topics may include: unique factorization domains, rings of fractions, finite fields, vector spaces over arbitrary fields, and modules. Mathematics 2502 is helpful but not required.

Prerequisite: Mathematics 2000 {201} and 2020 {200}, 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. Every other spring. Spring 2016. The Department.

A study of mathematical modeling, with emphasis on how to identify scientific questions appropriate for modeling, how to develop a model appropriate for a given scientific question, and how to interpret model predictions. Applications drawn from the natural, physical, environmental, and sustainability sciences. Model analysis uses a combination of computer simulation and theoretical methods and focuses on predictive capacity of a model. Three hours of class meetings and 1.5 hours of computer laboratory sessions per week.

Prerequisite: Mathematics 2000 {201}, 2020 {200}, and 2208 {224}, or permission of the instructor.

3109 a. Optimal Control. Every other fall. Fall 2015. The Department.

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}, 2020 {200}, and 2208 {224}, or permission of the instructor.

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

A study of nonlinear dynamical systems arising in applications, with emphasis on modern geometric, topological, and analytical techniques to determine global system behavior, from which predictions can be made. Topics chosen from local stability theory and invariant manifolds, limit cycles and oscillation, global phase portraits, bifurcation and resilience, multiple time scales, and chaos. Theoretical methods supported by simulations. Applications drawn from across the sciences.

Prerequisite: Mathematics 2000 {201}, 2020 {200}, 2208 {224}, and 2603 {263}, or permission of the instructor.

3209 a. Partial Differential Equations. Every other fall. 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 264.

Prerequisite: Mathematics 2000 {201}, 2020 {200}, and 2208 {224}, or permission of the instructor.

3404 {307} a. Projective and Non-Euclidean Geometries. Every other fall. Fall 2014. 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 Group Theory. Every other spring. Spring 2015. The Department.

The study of group actions on geometric objects; understanding finite and discrete groups via generators and presentations. Applications to geometry, topology, and linear algebra, focusing on certain families of groups. Topics may include Cayley graphs, the word problem, growth of groups, and group representations.

Prerequisite: Mathematics 2602 {262} or 2702 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 2014. 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.

3702 a. Advanced Topics in Rings and Number Theory. Every other fall. Fall 2015. Manuel Reyes.

Advanced topics in modern algebra based on rings and fields. Possible topics include: Galois theory with applications to geometric constructions and (in)solvability of polynomial equations; algebraic number theory and number fields such as the p-adic number system; commutative algebra; algebraic geometry; and solutions to systems of polynomial equations.

Prerequisite: Mathematics 2602 {262} or 2702, 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.

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