So there was a bit of a “math holiday” for me over the last few days as I took my lady and our son to Philadelphia for her birthday. We didn’t do anything too exciting but it was our first time visiting the city (and our first road trip since settling down in Morrisville a couple weeks ago) so it was good times.
And wouldn’t you know – today when I got home, I had a math book waiting in my mailbox! Always good to get an unexpected math haul!
I’ll probably get back into serious math tomorrow…hopefully, anyway…and at least part of my time will be spent studying so-called “locally metric spaces” with my bud L. I’m excited.
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 is prime. He later says that it follows immediately from Alexander’s Theorem as, and I quote: Every 2-sphere in bounds a 3-ball. And that’s it. Done.
Elsewhere, Hatcher expands his above statement: …every 2-sphere in bounds a ball on each side…[and h]ence 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 is the identity of the connected sum operation, that is irreducible (and that every irreducible manifold is prime), that one gets the trivial sum by splitting along a 2-sphere in which bounds a 3-ball in , 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.
Today is the fifth full day at our new place and things are finally starting to settle in. Until today, we’d been sleeping/sitting/otherwise living on the floor, for the most part. In particular:
A couple days ago, we got our Wifi connected so our internet access went from patchy and occasional to great and full-time.
After spending the first few days sleeping on the floor, we got a couple air mattresses on Monday. That came with some slight added comfort.
Today, our new couch came in. I can’t overstate how amazingly comfortable this fucking thing is, and believe me when I say: It’s completely changed my whole attitude to have a comfortable place to sit!
As a result of the added couch-induced comfort, I’m letting today be my first day transitioning to The Princeton Schedule of mathing all day and working (for a wage) at night. So far today, it’s been all 3-manifolds and foliations, particularly getting things I ought to already know typed into Mnemosyne so that I can make sure I know know them moving forward.
There’s so much math I should be better at; I’m really looking forward to using this year to bridge the gap from where I am to where I ought to be.
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!
One of the things I’ve always enjoyed about moving farther in mathematics is being able to understand/process old information from a variety of different contexts. Of course, I’m still no Benson Farb (if you’ve never heard Benson talk about the insolvability of the quintic using algebro-geometric machinery, you’re really missing out), but I really do enjoy learning new ways to visualize things I’ve known for a while.
For example, there’s a novel trapezoidal proof of the Pythagorean Theorem that makes sense if you know simple things like the area of a right trapezoid:
. Seeing stuff like that makes me happy, and it makes me realize:
It could be fun/worthwhile to begin a blog post series of Interesting Proofs which outlines various “interesting” proofs of fundamental(-ish) mathematical facts. I’m imagining a recurring series of blog posts connected only by the fact that they’re novel proofs of things that most people have known since adolescence.
You guys should chime in to tell me whether you’d be interested in anything like that. Pretends people actually read this nonsense
"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos