Majoring in Mathematics

This article is intended to provide a brief introduction to what college mathematics is all about. It is aimed at high school students who have enjoyed mathematics in high school, and are starting to think about what to study in college. This article is a work in progress -- please expand and leave suggestions at the talk page.

Here is an outline of what I hope to write.

What College Mathematics is Not

Before we look at the sorts of problems that actually are dealt with by mathematics in college, let's look at some that are not. Rest assured, many of the most painful aspects of math classes you've taken before aren't so bad in college!

1. Simply learning more advanced versions of mathematical ideas from high school. Tired of learning tricks for integration? Majoring in math won't just teach you new methods to integrate functions -- if you've taken an introductory calculus class, chances are you already know the important ones! What it will do is force you to think carefully about basic questions about what an integral is. Given some subset of space, what does it really mean to find its volume?
2. Doing lots of computations. Chances are you'll have to do a few, but most mathematics in college is based around "proofs" -- very careful and rigorous arguments which show that mathematical statements are true. Many proofs won't involve any computations at all!
3. Learning a bag of tricks to do Olympiad-type problems. If you've ever taken the AMC, Math League, or other similar competition tests, and not done well, don't worry! Much of these tests check nothing more than whether you're familiar with some collection of techniques for solving special problems. One math professor at Harvard keeps in his desk a copy of a math competition on which he scored a perfect 0/120. Mathematical theory goes far beyond these problems.

Instead, college mathematics is about developing new ways to rigorously approach a wide array of problems. More than aiming to find tricky techniques to solve certain problems, it is about building powerful approaches that solve many problems, and learning to think rigorously about them.

Major Topics of Study

Given that a math major won't just be getting more practice at the sorts of math encountered in high school, what is it about? Here's a list of some of the major fields that are the foundation of a mathematics education.

For each field, I hope give a very general description and to describe a few problems/theorems that are nice results and capture some of the essence of a subject. I'll aim for a few paragraphs about each one. Then I will give a problem or two that is a much more open-ended question that has motivated the development of large amounts of theory. For now I have no written about any of these -- please leave feedback and suggestions on the talk page and I will try to arrive at the best set of motivating problems!

Algebra

1. How many positions are there for a Rubik's cube. Brief discussion of group theory, and how a group acts on the Rubik's cube.
2. Bezout's theorem. Discuss intersection of conic sections, plane cubics, etc. Two conics intersect at 4 points, in generally.
3. Is the quintic solvable in radicals? And what in the world does this problem have to do with group theory?

A motivating question: failure of unique factorization in certain number fields (with explicit examples)

Geometry and Topology

1. Konigsberg bridge problem, with complete solution.
2. Ham sandwich theorem (easy to state, harder to motivate solution. Possibly there's something better)
3. Double-bubble conjecture (a nice result, easy to state and wave hands about "curvature", and with a connection to CP!)

A motivating problem: how can we prove that a sphere and a torus are different?

Probability and Statistics

1. Discussion of Benford's law, application to recognizing fraud.
2. Problem 2
3. Problem 3

Motivating problem:

Analysis

1. Riemann mapping theorem. State this in terms of conformal mappings of the unit disk and draw lots of pictures!
2. Sharkovsky's theorem. A cute result, and makes possible some discussion of fixed points.
3. Periodic orbits in billiards in polyhedra. I think it should be possible to say some interesting things about this but with some real content. Possibly there's a better idea.

Hard problem: What does an integral really mean? Babble a bit about integrating the characteristic function of Q.

Other fields

1. Logic: still need a good problem
2. Number theory: connected to the other fields above in many ways. Prime number theorem?
3. Set theory: some discussion of axioms, maybe talk about AoC.

Related fields

Many other subjects are often considered parts of mathematics as well. Depending on specific undergraduate programs, a math major may or may not be required to take courses in these areas, but many of the same ideas are used in these subjects as in the others.

1. Computer science: something about algorithmic complexity: maybe the fact that primality testing may be done in polynomial time?

...

Interplay

None of these fields exists in a vacuum, and there is rich interplay between them. If you insert "algebraic" or "differential" before the name of just about any branch of math, you get a more specialized, but important field. Motivate some of these connections:

1. Algebra+topology: discussion of fundamental group.
2. Analysis+topology: discuss conservative fields, and hint at de Rham cohomology without actually saying those words.
3. Algebra+analysis: find a good simple example of a Lie group showing up in the study of some differential equations.

So What?

After finishing a major in mathematics, most people don't keep working in Pure Math. But the ways of thinking and skills that the study of mathematics provide make available a wide range of career opportunities.