The Basics

Calculus is the study of change: if you know a function describing a situation, what does this tell about how the situation is changing? Conversely, if you know how it is changing, can you describe it in absolute terms?

Calculus is one of the classical topics in mathematics, dating back to the seventeenth century. It is useful both in scientific fields and in applied studies from engineering to business. The courses you will take emphasize computational and problem-solving skills rather than theory.

If you like Calculus, consider taking: Differential Equations ( Math 2310 , 3310 ) or Vector Calculus ( Math 4400 ) to learn more methods; or take Mathematical Analysis ( Math 4200 ) to study the theory behind Calculus.

Linear Algebra is the study of linear relationships: solutions to systems of linear equations, lines and vectors in multidimensional spaces, and how we use matrices to study these things. It is also considered a classical topic, although it was developed a hundred and fifty years later than calculus. Linear algebra is used in such diverse fields as computer science, economics, and population dynamics.

If you like linear algebra, consider taking Linear Algebra and Matrix Theory ( Math 4500 ) to learn more about matrices and their uses.

Other Recommendations. A programming course is required for the math major, and usually students take Cosc 1010. The programming requirement can also be satisfied by any 4 hour computer course which includes learning a high-level programming language. Programming is an increasingly important skill in both applied and theoretical mathematics, and you should consider taking further courses, or even getting a computer science minor.

We recommend that you take two semesters of physics - in particular, one of the two sequences which requires calculus (Phys 1210, 1220 or 1310, 1320). Much of mathematics developed from and is motivated by physics. If you want to be a teacher, this will help you understand the connections. A physics background will also be helpful if you decide to work in industry and need to interact with engineers or scientists.

Proof, what's a proof?

Most upper-division ( 3000-4000 level ) courses are more abstract and theoretical; they rely heavily on proofs. There are two types of courses that are designed to help you get ready to deal with this: the seminar courses, and the foundations courses.

Seminar Courses. If you've ever wondered if there is more to mathematics than calculus, this is the place to start finding out. The two seminar courses are the Mathematics Major Seminar (2800, offered in the Spring), and the Putnam Team Seminar (3800, offered in the Fall). You are required to take one of them; which one you choose depends on your interests. Try to take one as early as possible - these courses are fun, and a chance to meet and work with other math majors. Both courses are graded S/U, and evaluation is based on participation.

In the Mathematics Major Seminar ( Math 2800 ), we try to give students an appreciation of the many faces of mathematics: classical problems, recent discoveries, mathematical people. Students typically explore problems together in small groups, and have the chance to discuss ideas and strategies.

Are you fond of mathematical puzzles and problems? Then take the Putnam Team Seminar ( Math 2850 ). This is intended partly to serve as preparation for the William Lowell Putnam Exam, a mathematics competition given yearly by the Mathematics Association of America and taken by thousands of students nationwide. The seminar focuses on problem-solving, using past Putnam problems and problems from other math contests, and will give you the chance to pit your skills against those of other students.

Foundations courses. Here again you have a choice of two courses: Fundamental Concepts of Mathematics (3000, offered in the Fall), or Polynomials (3200, offered in the Spring). Both courses emphasize reading and writing proofs.

The foundations courses are not required, but we strongly recommend that you take one of them. They are a prerequisite for many of the 4000 level courses, and not taking either one will severely limit your choices.

In the seminar course, we encourage an informal approach and try to show you the joy of discovering new mathematical ideas. This is the way that new mathematics is developed, and is an important step in understanding. However, modern mathematics has exacting standards and uses the framework of axioms, carefully phrased definitions, theorems, and proofs to guarantee correctness.

The foundations courses develop an understanding and appreciation for these formal methods, and provide the background required for most upper-division courses. One of them is usually taken by a math major by the beginning of their junior year.

Fundamental Concepts of Mathematics ( Math 3000 ) emphasizes set theory and basic logic. In the process of studying these, students learn basic proof methods and critical thinking. You will practice ways to find a proof for a mathematical statement, and how to write it using standard mathematical conventions.

In Polynomials ( Math 3200 ), you will also spend a lot of time finding and writing proofs, but here the subject of your proofs will be properties of polynomials and their roots. You may think you know all about polynomials from high school; in this course you will discover both the theory behind the facts you learned there, and many new and useful properties.

These classes are the beginning of a new phase in your mathematical education: abstract thinking. In either class, the important thing for you to take away is an appreciation of proofs, and the ability to produce them. You will use this in most upper-division classes, and learning it well will be an immense help. However, because this is something new, most students struggle with these courses. Consider working together with other students, and don't forget that your instructor is ready to help.

Electives - What To Take?

If you don't declare a concentration, you have nine upper-division math electives to choose. Many of the upper-division courses can be grouped into various topic areas, and each is briefly described below.

Algebra. High school algebra emphasizes the manipulation and solving of polynomial equations; linear algebra uses matrices to manipulate and solve systems of linear equations. Modern abstract algebra focuses on the structures and theory underlying these manipulations and investigates the context in which we can generalize to something other than the real numbers. In these courses, you will further develop your abilities to construct proofs and deal with abstract theory. There are two courses in abstract algebra: Applied Algebra ( Math 3500 ) and Introduction to Abstract Algebra ( Math 3550 ).

There are a surprising number of applications of abstract algebra to modern technology, and Applied Algebra focuses on some of these. On the internet, we want to communicate information securely and reliably; we can use group theory to construct codes which allow us to do this. This course will explore both the applications and the theory behind it.

The other upper-division algebra course, Introduction to Abstract Algebra, covers a wider variety of algebraic structures: groups, rings, fields, and their substructures. The emphasis is developing the relevant theory and the connections to the algebra that you already know.

Another course which fits under the broad topic of algebra is Linear Algebra and Matrix Theory ( Math 4500 ). This is a continuation of Linear Algebra (Math 2250); it studies in detail the structure of matrices. Math 2250 deals with the cases where matrices have nice structure; in this course, we consider what to do when they don't.

Analysis. By your second year, you have already run into this area of mathematics: calculus is considered to be part of analysis. The upper-division courses take this in two directions. First, we can examine rigorously the theory behind calculus. Second, we can extend the theory to other number systems, in particular the complex numbers.

Complex Analysis ( Math 4230 ) is an introduction to complex numbers, complex-valued functions, and calculus applied to such functions. It gives a new perspective on the usual (freshman) calculus. Complex variables are used in solving differential equations and have wide application in engineering and the sciences, as well as being an important part of analysis.

Math 4400 is called Vector Calculus, and can be thought of as calculus of several variables; these ideas are first introduced in Calculus III. The course emphasizes using theory and geometry as an aid to calculation, and is more theorem/proof oriented than Calculus III, but less so than 4200 and 4205.

The two semester sequence called Mathematical Analysis ( Math 4200 and Math 4205 ) focuses on the theory of limits and the real number system, the basis on which the theory of calculus rests. We want to develop these ideas with rigor and precision, so the emphasis is on proof and counterexamples (and one of the foundations courses is required as a prerequisite). You may find it helpful to take Math 4400 before tackling 4200.

Applied Mathematics. Differential equations is the area in mathematics which is most often applied to problems, and so the basic applied math courses are the differential equations courses 2310, 3310, and 4440. Math 2310 develops techniques for solving ordinary differential equations, with an emphasis on computation and problem-solving; this is continued and combined with linear algebra in 3310 to solve systems of ODE's. If you continue on to 4440, you will extend some of the theory to more than one variable, and learn to solve partial differential equations.

Introduction to Mathematical Modeling ( Math 4300 ) is a course which directly deals with applying mathematics to real-world problems. Differential equations are one mathematical tool which you will use in this course, but you will also learn to use other types of mathematics such as sequences and matrices.

Numerical Analysis ( Math 4340 ) involves a third type of applied mathematics: using computers to solve mathematical problems. It's easy to treat machine computation as the answer to everything, but someone had to write the program to tell it what to do, and we need to know what kinds of error the computer introduces into the solution.

Probability and Statistics. The two courses Math 4250 (Mathematical Theory of Probability) and Stat 4260 (Introduction to the Theory of Statistics) form a two-semester sequence which will introduce you to the study of randomness and how to get information from it. Understanding basic principles of statistics is a part of many jobs, and these courses are highly recommended if you want to work in industry. The emphasis is practical rather than theoretical. We suggest taking 4260 immediately after taking 4250 - don't wait a year in between.

Stat 4260 is part of the applied mathematics concentration, and is recommended for any math major; however it does not count as an upper-division math elective.

Other Topics. There are a number of courses that don't fit neatly into any of these categories. These include interesting topics which will broaden your view and relate to other things that you have learned.

The History of Mathematics ( Math 4000 ) shows you the roots of the subject and the people who were obsessed with it. It will give you the chance to see how disparate topics developed together. (If you like this kind of thing, you should also consider taking Phil 3140, Philosophy of Science, and Phil 3500, History of Science. Each is a C1 course.)

The Theory of Numbers ( Math 4550 ) incorporates techniques from many areas of mathematics to study what seems like a simple topic: the integers, and number systems developed from them. This is a subject which historically has provided puzzles for mathematicians and non-mathematicians alike; it is also an important element in modern cryptography. Recommended for the intellectually curious.

Foundations of Geometry ( Math 4600 ) gives a careful treatment of the axioms of geometry - you may have seen some of this in high school, but you now have a much greater level of mathematical sophistication from which to view it. This leads to considering other axiom systems, giving us non-Euclidean geometry. The course is intended mainly for prospective high school teachers.

Math 4800, Seminar in Mathematics is a catchall course for special topics. If this appears in the schedule you should check to see if you are interested in the particular topic for that semester.

The Basics

Quick Links