The end result is that I did very little in terms of wage earning, and in particular that our savings is down to approximately \$0 and if I don’t start earning pay soon we’re going to starve. Even so, the math library here…?

Tomorrow is the first day of the year’s first directed workshop-thing on 3-manifolds (http://www.math.ias.edu/wgso3m/agenda) and I’m indescribably excited about that. I’ve also gotten to a point where I have a schedule in place to earn a livable wage between all that (yay no starvation!) and will hopefully be able to parlay some of the awesome math I’ve been absorbing into things to post here…

…but today is not that day. ::wink::

Yours in math….

# S^3 (the most basic prime manifold) is prime

So a while ago, I was reading Hatcher’s notes on 3-manifolds. In there, he defines what it means for a manifold to be prime and states, casually, that the 3-sphere $S^3$ is prime. He later says that it follows immediately from Alexander’s Theorem as, and I quote: Every 2-sphere in $S^3$ bounds a 3-ball. And that’s it. Done.

Wait, what?!

Elsewhere, Hatcher expands his above statement: …every 2-sphere in $S^3$ bounds a ball on each side…[and h]ence $S^3$ is prime. Again, though, it isn’t accompanied by anything, and while this is clearly a trivial result, I just couldn’t see it for the longest time…I knew that it followed from a number of things, e.g. the fact that $S^3$ is the identity of the connected sum operation, that $S^3$ is irreducible (and that every irreducible manifold is prime), that one gets the trivial sum $M\# S^3=M$ by splitting along a 2-sphere $S$ in $M^3$ which bounds a 3-ball in $M$, etc. Even so, I didn’t want to leverage some enormous machinery to deduce the smallest of results and what I really wanted was for someone to tell me what I was missing. So I never stopped thinking about this, even after moving forward, until finally – it just clicked!

I figure other people who are as visualization-impaired as I may benefit from seeing this explained in greater depth, so in lieu of typing a blog post containing something new and attention-worthy, I figure I’d share this instead. Details after the break.

# Implementation for Interesting Proofs (Framework)

Okay, so previously, I blogged about potentially implementing a series on interesting proofs. Unsurprisingly, nobody read that post and/or cared, so I decided to go ahead with it anyway because I’m a loner, Dottie, a real rebel…

…anyway, the framework for that is now in place. It’s a barren landscape presently but I have the content necessary to add one proof with (hopefully) more to come!

# Foliation Theory Observation I

Observation I. Every article in this field is remarkably long.

I’ve currently got articles whose lengths are 60, 58, 50, 39, 59, 56, 58, 76, and 50 pages long in my “reading queue”. Of the others, there are none that are shorter than 25 pages with the exception of Gabai’s second article (18 pages, though sandwiched between a 60 page original and a 58 page triquel).

One would think that longer automatically implies “more detailed” (i.e., less terse, less difficult to read, etc.), but this isn’t necessarily the case; in particular, Gabai’s articles are ridiculously complex and brilliant and amazing, and even legitimate 3-manifold topologists specializing in foliation theory confess that it takes forever (literally not literally) to make it through even one of them.

My prediction is that when my son’s in college, the average math Ph.D. will take 10 years. Give or take.

So, to summarize the direction of my most recent mathematical endeavors: I woke up and decided that part of my aspiration was to become a geometric topologist, and I did that despite the fact that topology is (far and away) my worst subject.

That sounds precisely as terrible as it probably is.

# And, it’s Sunday….

Tomorrow is Day 1 of Week 7.

You thought I’d stopped counting, didn’t you?

Today, I posted a couple more solutions from Hatcher. I also got the okay from Dr. Petersen on my poster for Math Fun Day. I’ll probably post a .pdf image of the poster as the big day gets closer.

For now, I’m going to come study some Riemannian Geometry: I have to (very soon) pick a topic for a presentation in that class, and so it’s getting more and more necessary that make sure I know what’s going on now. Maybe I’ll surprise myself and know a lot.

I don’t expect to surprise myself. 😛

# Update, briefly

I’m about to have to get ready to meet some friends, but just FYI: I’ve added a couple new Hatcher solutions from section 2.1. I’ve got several more written down, but this surely isn’t a speedy venture. Just FYI.

Woot, progress!