College Math Teaching

February 19, 2013

A Message to Undergraduate Math Majors

Filed under: advanced mathematics, editorial, mathematics education, pedagogy, topology — collegemathteaching @ 8:08 pm

Ok, you are taking, maybe an analysis class or perhaps your first abstract algebra class. You are learning how a proof works. Of course, you might be studying a proof of, say, one of the Sylow Theorems in group theory, or perhaps a convergence proof in analysis.

That proof is elegant and to the point, isn’t it? But here are some things to remember:

1. You are seeing “what worked”; you aren’t seeing the scores of attempts that failed.

Example: one of my papers contains a counterexample to something I thought “for sure” was true; in fact I spent 2 years trying to “prove” the conjecture that I ended up publishing the counterexample for!

2. You are seeing a polished proof.

Example: right now, I am finishing up a paper on wild knots (simple closed curves in 3 space that are not deformable to smooth simple closed curves). I spend 3-4 days on one step of a construction, only to realize that not only were my steps not convincing, they WEREN’T at all necessary!

Here is what lead me to realize I was headed toward a dead end: I was proving something that directly depended on specific properties of the type of knots that I was studying, yet my construction was not using those properties. I was doomed to fail if I kept on this path!

For the record, here is the mistake that I was making:

Suppose you have an annulus in the plane; example: $A = ((x,y,0) | \frac{1}{4} \le x^2+y^2 \le 1)$. Now suppose you take another annulus $B$ in the region above the plane and attach it to $A$ along its two boundary circles. You get a torus $T$. But is $T$ necessarily unknotted in 3 space? Hint: we knot that in $S^3 = R^3 \cup {\infty}$, $T$ bounds at least ONE solid torus, but does it bound two of them?