Mathematics
Note: This page contains all of the regular courses taught by this department. Not all courses are offered every year. Check the searchable schedule to see which courses are being offered in the upcoming semester.
MATH 105 Surprises at Infinity
Credit: 0.5 QR
Our intuitions about sets, numbers, shapes, and logic all break down in the realm of the infinite. Seemingly paradoxical facts about infinity are the subject of this course. We will discuss what infinity is, how it has been viewed through history, why some infinities are bigger than others, how a finite shape can have an infinite perimeter, and why some mathematical statements can be neither proved nor disproved. This will very likely be quite different from any mathematics course you have ever taken. Surprises at Infinity focuses on ideas and reasoning rather than algebraic manipulation, though some algebraic work will be required to clarify big ideas. The class will be a mixture of lecture and discussion, based on selected readings. You can expect essay tests, frequent homework, and writing assignments. No prerequisites. Offered typically every one to two years.
MATH 106 Elements of Statistics
Credit: 0.5 QR
This is a basic course in statistics. The topics to be covered are the nature of statistical reasoning, graphical and descriptive statistical methods, design of experiments, sampling methods, probability, probability distributions, sampling distributions, estimation, and statistical inference. Confidence intervals and hypothesis tests for means and proportions will be studied in the one- and two-sample settings. Minitab, a statistical software package, will be used, and students will be engaged in a wide variety of hands-on projects. Offered every semester.
MATH 108 Models of Life
Credit: 0.5 QR
This course will explore various areas of mathematics involved in modeling the growth and form of biological organisms and populations. In particular, we will ask such questions as: How do you model the growth of a population of animals? How can you model the growth of a tree? How do sunflowers and seashells grow? How do mathematicians quantify symmetry? The course will be a "hands-on" course and will make extensive use of the graphical capabilities of the computer software package Maple. The course will not involve significant amounts of symbolic manipulation. Rather, assignments will usually involve readings, papers, and computer projects. The course will rely on ideas from a wide range of mathematical fields, including geometry, linear algebra, mathematical modeling, and computer graphics. Offered every two to three years.
MATH 110Y Calculus/Elementary Functions
Credit: 0.5 QR
This course is a year-long introduction to calculus that integrates an extensive review of algebra and elementary functions with the topics taught in Calculus A (MATH 111). The course is intended for students who need to strengthen their quantitative and algebraic precalculus skills in order to learn calculus more effectively. Topics include functions and their properties (including exponential, logarithmic, and trigonometric functions), limits and continuity, and a thorough introduction to the study of rates of change, called differential calculus. The course will end with a brief introduction to integral calculus (the problem of finding areas) and the connection between integral and differential calculus. Students who have credit for MATH 111 may not take this course. MATH 110Y is offered every fall and MATH 111Y is offered every spring.
MATH 118 Introduction to Programming
Credit: 0.5 QR
This course presents an introduction to computer programming intended both for those who plan to take further courses in which a strong background in computation is desirable and for those who are interested in learning basic programming principles. The course will expose the student to a variety of applications where an algorithmic approach is natural and will include both numerical and non-numerical computation. The principles of program structure and style will be emphasized. Offered every semester.
MATH 206 Data Analysis
Credit: 0.5 QR
This course follows MATH 106 and focuses on (1) additional topics in statistics, including linear regression, nonparametric methods, discrete data analysis, and analysis of variance; (2) efficient use of statistical software in data analysis and statistical inference; and (3) writing and presenting statistical reports, including graphics. The MATH 106-206 sequence provides a foundation for statistical work in applied fields such as econometrics, psychology, and biology. It also serves as preparation for study of theoretical probability and statistics.
MATH 216 Nonparametric Statistics
Credit: 0.5 QR
This course will focus on nonparametric and distribution-free statistical procedures. These procedures will rely heavily on counting and ranking techniques. In the one and two sample settings, the sign, signed-rank, and Mann-Whitney-Wilcoxon procedures will be discussed. Correlation and one-way analysis of variance techniques will also be investigated. A variety of special topics will be used to wrap up the course, including bootstrapping, censored data, contingency tables, and the two-way layout. The primary emphasis will be on data analysis and the intuitive nature of nonparametric statistics. Illustrations will be from real data sets, and students will be asked to locate an interesting data set and prepare a report detailing an appropriate nonparametric analysis.
MATH 218 Data Structures and Program Design
Credit: 0.5 QR
This course is intended as a second course in programming, as well as an introduction to the concept of computational complexity and the major abstract data structures (such as arrays, stacks, queues, link lists, graphs, and trees), their implementation and application, and the role they play in the design of efficient algorithms. Students will be required to write a number of programs using a high-level language.
MATH 222 Foundations
Credit: 0.5 QR
This course introduces students to mathematical reasoning and rigor in the context of set-theoretic questions. The course will cover basic logic and set theory, relations--including orderings, functions, and equivalence relations--and the fundamental aspects of cardinality. Emphasis will be placed on helping students in reading, writing, and understanding mathematical reasoning. Students will be actively engaged in creative work in mathematics. Students interested in majoring in mathematics should take this course no later than the spring semester of their sophomore year. Advanced first-year students interested in mathematics are encouraged to consider taking this course in their first year. (Please see a member of the mathematics faculty if you think you might want to do this.)
MATH 224 Linear Algebra
Credit: 0.5 QR
Linear algebra grew out of the study of the problem of organizing and solving systems of equations. Today, ideas from linear algebra are highly useful in most areas of higher-level mathematics. Moreover, there are numerous uses of linear algebra in other disciplines, including computer science, physics, chemistry, biology, and economics. This course involves the study of vector spaces and matrices, which may be thought of as functions between vector spaces. In the past, linear algebra involved tedious calculations. Now we have computers to do this work for us, allowing us to spend more time on concepts and intuition. A computer algebra system such as Maple will likely be used.
MATH 226 Design and Analysis of Experiments
Credit: 0.5 QR
This course will focus on standard methods of designing and analyzing experiments. Simple comparative designs, factorial designs, block designs, and appropriate post-hoc comparisons will be discussed. These techniques are commonly used by statisticians and experimental scientists in a wide variety of fields. Statistical software will be introduced and heavily used throughout the course. No prior experience with the software is necessary. Each student will be asked to design an experiment, conduct the experiment, and collect and analyze the appropriate data.
MATH 227 Combinatorics
Credit: 0.5 QR
Combinatorics is, broadly speaking, the study of finite sets and finite mathematical structures. A great many mathematical topics are included in this description, including graph theory, combinatorial designs, partially ordered sets, networks, lattices and Boolean algebra, and combinatorial methods of counting, including combinations and permutations, partitions, generating functions, the principle of inclusion and exclusion, and the Stirling and Catalan numbers. This course will cover a selection of these topics. Combinatorics mathematics has applications in a wide variety of non-mathematical areas, including computer science (both in algorithms and hardware design), chemistry, sociology, government, and urban planning, and this course may be especially appropriate for students interested in the mathematics related to one of these fields.
MATH 230 Euclidean and Non-Euclidean Geometry
Credit: 0.5 QR
The Elements of Euclid, written over two thousand years ago, is a stunning achievement. The Elements and the non-Euclidean geometries discovered by Bolyai and Lobachevsky in the nineteenth century form the basis of modern geometry. From this start, our view of what constitutes geometry has grown considerably. This is due in part to many new theorems that have been proved in Euclidean and non-Euclidean geometry but also to the many ways in which geometry and other branches of mathematics have come to influence one another over time. Geometric ideas have widespread use in analysis, linear algebra, differential equations, topology, graph theory, and computer science, to name just a few areas. These fields, in turn, affect the way that geometers think about their subject. Students in MATH 230 will consider Euclidean geometry from an advanced standpoint, but will also have the opportunity to learn about several non-Euclidean geometries such as (possibly) the Poincare plane, geometries relevant to special relativity, or the geometries of Bolyai and Lobachevsky. In addition, the course may take up topics in differential geometry, topology, vector space geometry, mechanics, or other areas, depending on the interests of the students and the instructor.
MATH 232 Vector Calculus
Credit: 0.5 QR
Physical and natural phenomena depend on a complex array of factors, and to analyze these factors requires the understanding of geometry in two and three (or more) dimensions. This course will continue the study of multivariable calculus begun in MATH 213. Topics of study will include vector fields, line and surface integrals, potential functions, classical vector analysis, and Fourier Series. Computer labs will be incorporated throughout the course, and physical applications will be plentiful.
MATH 236 Random Structures
Credit: 0.5
This course will explore the theory, structure, applications, and interesting consequences when probability is introduced to mathematical objects. Some of the core topics will be random graphs, random walks and Markov processes, as well as randomness applied to sets, permutations, polynomials, functions, integer partitions, and codes. Previous study of all of these mathematical objects is not a prerequisite, as essential background will be covered during the course. In addition to studying the random structures themselves, a concurrent focus of the course will be the development of mathematical tools to analyze them, such as combinatorial concepts, indicator variables, generating functions, discrete distributions, laws of large numbers, asymptotic theory, and computer simulation. Prerequisite: MATH 112 or permission of the instructor.
Instructor: Jones
MATH 258 Mathematical Biology
Credit: 0.5 QR
In biological sciences, mathematical models are becoming increasingly important as tools for turning biological assumptions into quantitative predictions. In this course, students will learn how to fashion and use these tools to explore questions ranging across the biological sciences. We will survey a variety of dynamic modeling techniques, including both discrete and continuous approaches. Biological applications may include population dynamics, molecular evolution, ecosystem stability, epidemic spread, nerve impulses, sex allocation, and cellular transport processes. The course is appropriate both for math majors interested in biological applications, and for biology majors who want the mathematical tools necessary to address complex, contemporary questions. As science is becoming an increasingly collaborative effort, biology and math majors will be encouraged to work together on many aspects of the course. Coursework will include homework problem-solving exercises and short computational projects. Final independent projects will require the development and extension of an existing biological model selected from the primary literature, using mathematical software like Mathematica, Matlab, R, or Maple. Students will make a poster presentation of their results. Prerequisites: This course will build on (but not be limited by) an introductory-level knowledge base in both subjects, including MATH 111 and either BIOL 112 or BIOL 113. Interested biology and math majors lacking one of the prerequisites are encouraged to consult with the instructor. Offered every other year.
MATH 324 Linear Algebra II
Credit: 0.5 QR
This course deepens the studies begun in MATH 224. Topics will vary depending on the needs and interests of the students. However, the topics are likely to include some of the following: abstract vector spaces, linear mappings and canonical forms, linear models and eigenvector analysis, inner product spaces.
MATH 327 Number Theory Seminar
Credit: 0.5 QR
Patterns within the set of natural numbers have enticed mathematicians for well over two millennia, making number theory one of the oldest branches of mathematics. Rich with problems that are easy to state but fiendishly difficult to solve, the subject continues to fascinate professionals and amateurs alike. In this course, we will get a glimpse at both the old and the new. In the first two-thirds of the semester, we will study topics from classical number theory, focusing primarily on divisibility, congruences, arithmetic functions, sums of squares, and the distribution of primes. In the final weeks we will explore some of the current questions and applications of number theory. We will study the famous RSA cryptosystem, and students will be reading and presenting some current (carefully chosen) research papers.
MATH 328 An Introduction to Coding Theory and Cryptography
Credit: 0.5
Coding theory, or the theory of error-correcting codes, and cryptography are two recent applications of algebra and discrete mathematics to information and communications systems. The goals of this course are to introduce students to these subjects and to understand some of the basic mathematical tools used. While coding theory is concerned with the reliability of communication, the main problem of cryptography is the security and privacy of communication. Applications of coding theory range from enabling the clear transmission of pictures from distant planets to quality of sound in compact disks. Cryptography is a key technology in electronic security systems. Topics likely to be covered include basics of block coding, encoding and decoding, linear codes, perfect codes, cyclic codes, BCH and Reed-Solomon codes, and classical and public-key cryptography. Other topics may be included depending on the availability of time and the background and interests of the students. Other than some basic linear algebra, the necessary mathematical background (mostly abstract algebra) will be covered within the course.
MATH 333 Differential Equations
Credit: 0.5 QR
Differential equations arise naturally to model dynamical systems such as occur in physics, biology, and economics, and have given major impetus to other fields in mathematics, such as topology and the theory of chaos. This course covers basic analytic, numerical, and qualitative methods for the solution and understanding of ordinary differential equations. Computer-based technology will be used.
MATH 335 Abstract Algebra I
Credit: 0.5 QR
Abstract algebra is the study of algebraic structures that describe common properties and patterns exhibited by seemingly disparate mathematical objects. The phrase "abstract algebra" refers to the fact that some of these structures are generalizations of the material from high school algebra relating to algebraic equations and their methods of solution. In Abstract Algebra I, we focus entirely on group theory. A group is an algebraic structure that allows one to describe symmetry in a rigorous way. The theory has many applications in physics and chemistry. Since mathematical objects exhibit pattern and symmetry as well, group theory is an essential tool for the mathematician. Furthermore, group theory is the starting point in defining many other more elaborate algebraic structures including rings, fields, and vector spaces. In this course, we will cover the basics of groups, including the classification of finitely generated abelian groups, factor groups, the three isomorphism theorems, and group actions. The course culminates in a study of Sylow theory. Throughout the semester there will be an emphasis on examples, many of them coming from calculus, linear algebra, discrete math, and elementary number theory. There will also be a couple of projects illustrating how a formal algebraic structure can empower one to tackle seemingly difficult questions about concrete objects (e.g., the Rubik's cube or the card game SET). Finally, there will be a heavy emphasis on the reading and writing of mathematical proofs.
MATH 336 Probability
Credit: 0.5 QR
This course provides a mathematical introduction to probability. Topics include basic probability theory, random variables, discrete and continuous distributions, mathematical expectation, functions of random variables, and asymptotic theory.
MATH 341 Real Analysis I
Credit: 0.5 QR
This course is a first introduction to real analysis. "Real" refers to the real numbers. Much of our work will revolve around the real number system. We will start by carefully considering the axioms that describe it. "Analysis" is the branch of mathematics that deals with limiting processes. Thus the concept of distance will also be a major theme of the course. In the context of a general metric space (a space in which we can measure distances), we will consider open and closed sets, limits of sequences, limits of functions, continuity, completeness, compactness, and connectedness. Other topics may be included, if time permits.
MATH 347 Mathematical Models
Credit: 0.5 QR
This course introduces students to the concepts, techniques, and power of mathematical modeling. Both deterministic and probabilistic models will be explored, with examples taken from the social, physical, and life sciences. Students engage cooperatively and individually in the formulation of mathematical models and in learning mathematical techniques used to investigate those models. Prerequisites: MATH 106 and 224 or 258 or permission of instructor. Offered every two to three years.
MATH 352 Complex Functions
Credit: 0.5 QR
The course starts with an introduction to the complex numbers and the complex plane. Next students are asked to consider what it might mean to say that a complex function is differentiable (or analytic, as it is called in this context). For a complex function that takes a complex number z to f(z), it is easy to write down (and make sense of) the statement that f is analytic at z if
exists. In the course we will study the amazing results that come from making such a seemingly innocent assumption. Differentiability for functions of one complex variable turns out to be a very different thing from differentiability in functions of one real variable. Topics covered will include analyticity and the Cauchy- Riemann equations, complex integration, Cauchy's theorem and its consequences, connections to power series, and the residue theorem and its applications.
MATH 398 Junior Honors
Credit: 0.5 QR
The goal of the junior honors seminar is twofold: to develop a greater understanding of a broad selection of mathematical topics, and to gain the experience of independent exploration in mathematics. Students will work under the close supervision of a faculty member on three areas of interest. Topics of study will be chosen by the student. As a culmination of the course, each student will write a proposal describing his or her plan of study for senior honors.
MATH 416 Linear Regression Models
Credit: 0.5 QR
This course will focus on linear regression models. Simple linear regression with one predictor variable will serve as the starting point. Models, inferences, diagnostics, and remedial measures for dealing with invalid assumptions will be examined. The matrix approach to simple linear regression will be presented and used to develop more general multiple regression models. Building and evaluating models for real data will be the ultimate goal of the course. Time series models, nonlinear regression models, and logistic regression models may also be studied if time permits.
MATH 435 Abstract Algebra II
Credit: 0.5 QR
Abstract Algebra II picks up where MATH 335 ends, focusing primarily on rings and fields. Serving as a good generalization of the structure and properties exhibited by the integers, a ring is an algebraic structure consisting of a set together with two operations--addition and multiplication. If a ring has the additional property that division is well-defined, one gets a field. Fields provide a useful generalization of many familiar number systems: the rational numbers, the real numbers, and the complex numbers. Topics to be covered include: polynomial rings; ideals; homomorphisms and ring quotients; Euclidean domains, principal ideal domains, unique factorization domains; the Gaussian integers; factorization techniques and irreducibility criteria. The final block of the semester will serve as an introduction to field theory, covering algebraic field extensions, symbolic adjunction of roots; construction with ruler and compass; and finite fields. Throughout the semester there will be an emphasis on examples, many of them coming from calculus, linear algebra, discrete math, and elementary number theory. There will also be a heavy emphasis on the reading and writing of mathematical proofs.
MATH 436 Mathematical Statistics
Credit: 0.5 QR
This course follows MATH 336 and introduces the mathematical theory of statistics. Topics include sampling distributions, order statistics, point estimation, maximum likelihood estimation, methods for comparing estimators, interval estimation, moment generating functions, bivariate transformations, likelihood ratio tests, and hypothesis testing. Computer simulations will accompany and corroborate many of the theoretical results. Course methods will often be applied to real data sets.
MATH 441 Real Analysis II
Credit: 0.5 QR
As the name suggests, this course is a successor to Real Analysis I. The course will include a study differentiation and (Riemann) integration of functions of one variable, sequences and series of functions, power series and their properties, iteration and fixed points. Other topics may be included as time permits. For example: a discussion of Newton's method or other numerical techniques; differentiation and integration of functions of several variables; spaces of continuous functions; the implicit function theorem; and everywhere continuous, nowhere differentiable functions.
MATH 460 Topology
Credit: 0.5 QR
Topology is a relatively new branch of geometry that studies very general properties of geometric objects, how these objects can be modified, and the relations between them. Three key concepts in topology are compactness, connectedness, and continuity, and the mathematics associated with these concepts is the focus of the course. Compactness is a general idea helping us to more fully understand the concept of limit, whether of numbers, functions, or even geometric objects. For example, the fact that a closed interval (or square, or cube, or n-dimensional ball) is compact is required for basic theorems of calculus. Connectedness is a concept generalizing the intuitive idea that an object is in one piece: the most famous of all the fractals, the Mandelbrot Set, is connected, even though its best computer-graphics representation might make this seem doubtful. Continuous functions are studied in calculus, and the general concept can be thought of as a way by which functions permit us to compare properties of different spaces or as a way of modifying one space so that it has the shape or properties of another. Economics, chemistry, and physics are among the subjects that find topology useful. The course will touch on selected topics that are used in applications.
MATH 493 Individual Study
Credit: 0.25-0.5
Individual Study in Mathematics is a privilege reserved for students who want to pursue a course of reading or complete a research project on a topic not regularly offered in the department?s curriculum. Because Individual Study is one option in a rich and varied mathematics curriculum, it is intended to supplement, not take the place of, coursework. For that reason, Individual Study cannot normally be used to fulfill requirements for the major. Typically, an IS will earn the student .5 units or .25 units of credit. To qualify to enroll in an Individual Study, a student must identify a member of the mathematics faculty willing to direct the project. The professor directing the Independent Study, in consultation with the student, will then determine a tentative syllabus for the course and describe in some detail the methods of assessment (e.g. problem sets to be submitted for evaluation biweekly; a twenty-page research paper submitted at the course's end, with rough drafts due at given intervals, etc.) Only after these goals and methods of assessment have been determined will the instructor sign off on the course. Additionally, Independent Study requires the approval of the department chair. At a minimum, the department expects the student to meet regularly with his or her instructor for at least one hour per week, or the equivalent at the discretion of the instructor. Prerequisites: permission of instructor and department chair.