College Math Teaching

July 28, 2015

J. H. Conway, Terry Tao and avoiding work

Filed under: advanced mathematics, algebra, media — Tags: , , , , , — collegemathteaching @ 7:48 pm

The mainstream media recently had some excellent articles on two mathematical giants:

John Conway and Terrance Tao. I’ve never met Terry Tao though I do read (or try to follow) his blog.

I did meet John Conway when he visited the University of Texas. He is a friend of my dissertation advisor and gave some talks on knot diagram colorings.

I had a private conversation with him at a party, and he gave me some ideas which resulted in three papers for me! Here is one of them.

Yes, I am avoiding studying a book on the theory of interest; I am teaching that course this fall and need to get ahead of the game.

Unfortunately, when I don’t teach, my use of time becomes undisciplined.

July 13, 2015

Trolled by Newton’s Law of Cooling…

Filed under: calculus, differential equations, editorial — Tags: , — collegemathteaching @ 8:55 pm

From a humor website: there is a Facebook account called “customer service” who trolls customers making complaints. Though that isn’t a topic here, it is interesting to see Newton’s Cooling Law get mentioned:

newtonscoolinglaw

July 2, 2015

Knowledge that can’t be communicated is worthless

Filed under: editorial, pedagogy — Tags: — collegemathteaching @ 10:31 pm

arf

In the past, I’ve passed out this cartoon to my students. Too many times, I’ve heard “I understand how to do the problem, but I can’t do the problem on the exam.”

Well, I suppose that is a bit like saying:

“I know how to swim, but when I jump in the pool, I drown.”

“I know how to fly the plane, but when I try, I crash.”

July 1, 2015

Embarrassing gaps in my mathematical knowledge

Filed under: mathematician, topology — Tags: , — collegemathteaching @ 1:56 pm

Yes, mathematics is a huge, huge subject and no one knows everything. And, when I was a graduate student, I could only focus on 1-2 advanced courses at a time, and when I was working on my thesis, I almost had a “blinders on” approach to finishing that thing up. I think that I had to do that, given my intellectual limitations.

So, even in “my area”, my knowledge outside of a very narrow area was weak at best.

Add to this: 20+ years of teaching 3 courses per semester; I’ve even forgotten some of what I once knew well, though in return, I’ve picked up elementary knowledge in disciplines that I didn’t know before.

But, I have many gaps in my own “area”. One of these is in the area of hyperbolic geometry and the geometry of knot complements (think of this way: take a smooth simple closed curve in R^3 , add a point at infinity to get S^3 (a compact space), now take a solid torus product neighborhood of the knot (“thicken” the knot up into a sort of “rope”) then remove this “rope” from S^3 . What you have left over is a “knot complement” manifold.

Now these knot complements fall into one of 3 different types: they are torus knot complements (the knot can live on the “skin” of a torus),

torusknot

satellite knot complements (the knot can live inside the solid torus that is the product neighborhood of a different, mathematically inequivalent knot,

satelliteknot

or the knot complement is “hyperbolic”; it can be given a hyperbolic structure. At least for “most” knots of small “crossing number” (roughly: how many crossings the knot diagram has), are hyperbolic knots.

So it turns out that the complement of such knots can be filled with “horoballs”; roughly speaking, these are the interior of spheres which are “tangent to infinity”; infinity is the “missing stuff” that was removed when the knot was removed from S^3. And, I really never understood what was going on at all.

horo_fig8

I suppose that one can view the boundary of these balls (called “horospheres”) as one would view, say, the level planes z = k in R^3 ; those planes become spheres when the point at infinity is added. This is a horoball packing of the complement of the figure 8 knot; missing is the horosphere at z = 1 which can be thought of as a plane.

But the internet is a wonderful thing, and I found a lecture based on the work of Anastasiia Tsvietkova and Morwen Thistlethwaite (who generated the horoball packing photo above) and I’ll be trying to wrap my head around this.

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