Courses in Mathematics

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.

The first in a three-semester calculus sequence, this course covers the basic ideas of differential calculus. Differential calculus is concerned primarily with the fundamental problem of determining instantaneous rates of change. In this course we will study instantaneous rates of change from both a qualitative geometric and a quantitative analytic perspective. We will cover in detail the underlying theory, techniques, and applications of the derivative. The problem of anti-differentiation, identifying quantities given their rates of change, will also be introduced. The course will conclude by relating the process of anti-differentiation to the problem of finding the area beneath curves, thus providing an intuitive link between differential calculus and integral calculus. Those who have had a year of high-school calculus but do not have advanced placement credit for MATH 111 should take the Calculus Placement Exam to determine whether they are ready for MATH 112. Students who have .5 unit of credit for calculus may not receive credit for MATH 111. Prerequisites: solid grounding in algebra, trigonometry, and elementary functions. Students who have credit for MATH 110Y-111Y may not take this course. Enrollment limited.

The second in a three-semester calculus sequence, this course is concerned primarily with the basic ideas of integral calculus and the Riemann sums that serve as its foundation. We will cover in detail the ideas of integral calculus, including integration and the fundamental theorem, techniques of integration, numerical methods, and applications of integration. Analysis of differential equations by separation of variables, Euler?s method, and slope fields will be a part of the course, as will the ideas of convergence related to improper integrals, sequences, series and Taylor Series. Prerequisite: MATH 111 or MATH 110Y-111Y, or permission of the instructor. Enrollment limited.

The third in a three-semester calculus sequence, this course examines differentiation and integration in three dimensions. Topics of study include functions of more than one variable, vectors and vector algebra, partial derivatives, optimization, and multiple integrals. Some of the following topics from vector calculus will also be covered as time permits: vector fields, line integrals, flux integrals, curl, and divergence. Prerequisite: MATH 112 or permission of the instructor.

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.) Prerequisite: MATH 213 or permission of instructor.

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. Prerequisite: MATH 213 or permission of instructor.

Discrete mathematics 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. Discrete 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. Prerequisite: Math 107 or Math 222 or permission of instructor

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 formed 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. Prerequisite: MATH 222 or permission of instructor.

The theory of dynamical systems is the study of the behavior of physical or mathematical systems that change over time according to specific rules. Dynamical systems have applications to many areas of science and social science-research, including models of population growth and decline, interspecies relationships, traffic-flow problems, battles, river meanders, weather patterns, heartbeat rates, chemical reactions, and financial markets. In this course we will study both discrete and continuous time models, presenting the two approaches in a unified manner. Upon completion of the course, students should comprehend the basic concepts and recent developments in the field of dynamical systems, including the stability theory of equilibria and the theory of transitions to chaos. Students will develop the ability to analyze simple nonlinear discrete and continuous dynamical systems and to chart parameter regions of stability, periodicity, and chaos. Further, students will gain an appreciation for the power as well as the limitations of dynamical systems theory and chaos when applied to realistic systems such as ecologies and financial markets. Rather than taking a formal theorem-proof style, the course will be taught in a manner that stresses the geometry, intuition, and appreciation of dynamical systems. Computer technology will be used extensively to perform simulations and experiments. Prerequisite: MATH 111. Co-requisite: MATH 112.

This course is a mathematical examination of the formal language most common in mathematics: predicate calculus. We will examine various definitions of meaning and proof for this language, and consider its strengths and inadequacies. We will develop some elementary computability theory en route to rigorous proofs of Godel?s Incompleteness Theorems. Prerequisite: Either MATH 222 or PHIL 120 or permission of the instructor.

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. Prerequisite: Math 224, or permission of the instructor.

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. Prerequisite or co-requisite: MATH 213.

The phrase ?abstract algebra? correctly suggests some sort of a generalization of a topic most of us learned in high school, though it goes very much beyond that, of course. Three of the most important structures in abstract algebra are groups, rings, and fields; all three are, in fact, abstractions of familiar objects--the integers form a group or ring, while the real numbers give us an example of a field. Each of these structures has the property that any two of the subjects in the system may be ?combined? in some way to produce a new object in the system. In the system of integers, for example, this ?combining? might be addition or multiplication. Groups and rings are fundamental tools for any mathematician and many scientists, but these concepts are beautiful and worthy of study in their own right--group theory and ring theory currently are both very active areas of mathematical research.

In this course, the student examines the basics of groups and rings, with emphasis on the many examples of these algebraic structures. A possible example might be a study of symmetry with the aid of group theory. Prerequisite: MATH 222 or permission of the instructor. Junior standing is usually recommended.

This course provides a mathematical introduction to probability. Topics include basic probability theory, random variables, discrete and continuous distributions, moments of random variables, functions of random variables, computer simulation, the central limit theorem and other asymptotic properties. Prerequisite: MATH 213.

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. Students will be asked to consider many functions that take on real values?that is, each object in our domain will be associated with a real number. For instance, every point in the plane can be associated with its distance from the origin. Two points in the plane give rise to a real number: the distance between them. The concept of distance will be a major theme of the course.

?Analysis? is one of the principle branches of mathematics. One often hears that analysis is the theoretical underpinnings of the calculus, but though this has a kernel of truth, it is an answer that misleads by oversimplifying. Certainly, analysis had its inception in the attempt to give a careful, mathematically sound explanation of the ideas of the calculus. But over the last century, analysis has grown out of its original packaging and is now much more than simply the theory of the calculus. Analysis is the mathematics of ?closeness??the mathematics of limiting processes. The idea of continuity can be phrased in terms of limits. Both derivatives and integrals are the end results of taking a limit. Compactness is a property of sets that underlies many of the most important theorems encountered in calculus. These and related ideas will be the subject of the course. Prerequisites: MATH 213 and MATH 222.

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. Prerequisites: MATH 213 and 234.

This course picks up where MATH 335 ends. In Math 435, however, the focus is on using the tools considered in Abstract Algebra I. Mathematicians and scientists apply the fundamental algebraic notions of group, ring, and field to a wide variety of mathematical areas and scientific disciplines; in MATH 435, the student explores these applications. The structure will be that of a topics course, the focus being on classical problems that can be solved (and historically were solved) using algebraic structures as tools.

Topics that may be considered include insolvability of a quintic polynomial, the factoring of polynomials (just as in high school, but over arbitrary rings rather than the real numbers), the classification of finite simple groups (something proven very recently), special cases of ?Fermat?s Last Theorem,? Eisenstein?s criterion for irreducibility, the beautiful subject of Galois theory, and more. The class may borrow knowledge from subjects including linear algebra, number theory, complex numbers, calculus, and computer programming, though all one needs to know about these subjects will be covered in class. Prerequisite: MATH 335.

This is an analysis course with variable content, depending on the needs and interests of the students. Prerequisite: MATH 341.

This course enables students to study a topic of special interest under the direction of a member of the mathematics department. Prerequisites: permission of instructor and department chair.

The content of this course is variable and adapted to the needs of senior candidates for honors in mathematics. Prerequisite: permission of department.