College Math Teaching

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.

June 19, 2015

Scientific American article about finite simple groups

Filed under: advanced mathematics, algebra, mathematician — Tags: , — collegemathteaching @ 2:42 pm

For those of you who are a bit rusty: a finite group is a group that has a finite number of elements. A simple group is one that has no proper non-trivial normal subgroups (that is, only the identity and the whole group are normal subgroups).

It is a theorem that if G is a finite simple group then G falls into one of the following categories:

1. Cyclic (of prime order, of course)
2. Alternating (and not isomorphic to A_4 of course)
3. A member of a subclass of Lie Groups
4. One of 26 other groups that don’t fall into 1, 2 or 3.

Scientific American has a nice article about this theorem and the effort to get it written down and understood; the problem is that the proof of such a theorem is far from simple; it spans literally hundreds of research articles and would take thousands of pages to be complete. And, those who have an understanding of this result are aging and won’t be with us forever.

Here is a link to the preview of the article; if you don’t subscribe to SA it is probably in your library.

November 19, 2014

Tension between practitioners and theoretical mathematicians…

Filed under: academia, applied mathematics, mathematician, research — Tags: — collegemathteaching @ 2:01 am

I follow Schneier’s Security Blog. Today, he alerted his readers to this post about an NSA member’s take on the cryptography session of a mathematics conference. The whole post is worth reading, but these comments really drive home some of the tension between those of us in academia :

Alfredo DeSantis … spoke on β€œGraph decompositions and secret-sharing schemes,” a silly topic which brings joy to combinatorists and yawns to everyone else. […]

Perhaps it is beneficial to be attacked, for you can easily augment your publication list by offering a modification.

[…]

This result has no cryptanalytic application, but it serves to answer a question which someone with nothing else to think about might have asked.

[…]

I think I have hammered home my point often enough that I shall regard it as proved (by emphatic enunciation): the tendency at IACR meetings is for academic scientists (mathematicians, computer scientists, engineers, and philosophers masquerading as theoretical computer scientists) to present commendable research papers (in their own areas) which might affect cryptology at some future time or (more likely) in some other world. Naturally this is not anathema to us.

I freely admit this: when I do research, I attack problems that…interests me. I don’t worry if someone else finds them interesting or not; when I solve such a problem I submit it and see if someone else finds it interesting. If I solved the problem correctly and someone else finds it interesting: it gets published. If my solution is wrong, I attempt to fix the error. If no one else finds it interesting, I work on something else. πŸ™‚

August 26, 2014

How some mathematical definitions are made

I love what Brad Osgood says at 47:37.

The context: one is showing that the Fourier transform of the convolution of two functions is the product of the Fourier transforms (very similar to what happens in the Laplace transform); that is \mathcal{F}(f*g) = F(s)G(s) where f*g = \int^{\infty}_{-\infty} f(x-t)g(t) dt

July 17, 2014

I am going to celebrate this…

Filed under: mathematical ability, mathematician, research — Tags: — collegemathteaching @ 8:10 pm

This marks the second summer in a row I got news that a paper of mine has been accepted for publication. Last year, it was the College Mathematics Journal; this year it is the Journal of Knot Theory and its Ramifications.

Sure, that is a big “yawn”, “so what”, or “is that all?” for faculty at Division I research universities. But I teach at a 11-12 hour load institution which also has committee requirements.

And, to be blunt: I got my Ph. D. in 1991 and has a somewhat long slump in publication; I was beginning to wonder if my intellect had atrophied with time.

Ok, it has, in the sense that I don’t pick up material as quickly as I once did. But to counter that, the years of teaching across the curriculum (from business calculus to operations research to numerical analysis to differential equations) and the years of attending talks and attempting to learn new things has given me a bit more perspective. I make fewer “hidden assumptions” now.

So, I am going to celebrate this one…and then get back to work on spin-off ideas.

April 15, 2013

Google Doodle 15 April 2013

Filed under: mathematician, mathematics education — Tags: — collegemathteaching @ 1:06 pm

Screen shot 2013-04-15 at 8.03.05 AM

Which famous mathematician is being honored? πŸ™‚

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