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.

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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!

Interesting proofs series?

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:

http://eigenstuff.com/post/128236655616

. 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

Week 3, Day 1 or Properties of Lie Brackets

Today is the first day of the third week of the semester. I know that counting down like this is going to make it seem longer than it already seems, but it seems so long that I can’t seem to help remaining conscious of the precise time frame I’m dealing with.

Such is life, I suppose.

I’ve noticed an amusing trend in my page views involving solutions from Dr. Hatcher’s book, namely that I’ve been receiving an abnormally-high level of page views lately, almost all of which seem to center on those solutions. I guess that means that the semester has started elsewhere too and that people find topology as difficult and frustrating as I do.

For those of you who fit this bill and who are reading this right now: My plan is to start doing more problems ASAP, so that page might get its first update in quite a while.

Today, though, I want (read: need) to talk about differential geometry. In particular, we spent some time in class last week discussing the Lie bracket and its properties, and because we have a derivation of one particular property, I wanted to take the time to put that here for my own benefit.

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One week in

So, I’ve officially survived the first week of my (second) second year as a Ph.D. student. It’s hard to imagine sometimes that I’ve been a grad student for over three years.

I’m such an old man.

It’s been even less easy than I’d anticipated, too, which is nicht spaß; sadly, too, this is the first week where teaching responsibilities are actually a thing (we get week 1 off from such things), so now it’s difficult and time-consuming. Jajajajaja.

Rather than making this entry all about Yours Truly, I figured I’d pit stop in and use some of my (*gasp*) unoccupied time write up a little spiel I saw for the first time in my Riemannian geometry class last week.

In topology, there’s an idea called invariance of dimension which can be stated in many different contexts, situations, etc. This can be modified in the case of manifolds with differential structures, and because the idea of the proof seemed a bit cool, I decided to throw it up here for you guys. Throughout, the notation M^m denotes a manifold M of dimension m with an associated differential structure.

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Function Algebras, and a resounding NO! to the Peak Point Conjecture

A former professor of mine (who I’ve discussed here previously) still to this day does some of the best work of anyone I’ve ever known. When he and I were colleagues (we existed at very-near geographical locations), I didn’t know the capacity of his amazing work, rather I just knew his work was amazing and that he for all intents and purposes should’ve been at a bigger, more prominent school.

Now, I know a bit more.

One of the things for which he was pretty famous (subjectively, natürlich) was proving a 50-ish year old unsolved problem in manifold theory; another of his fortes, though, is in the study of function algebras. That’s where this little journey takes us.

A function algebra is a family \Lambda of continuous functions defined on a compact set X which (i) is closed with respect to pointwise multiplication and addition, (ii) contains the constant functions and separates points of X, and (iii) is closed as a subspace of C(X) where, here, C(X) denotes the space of continuous functions defined on X equipped with the sup norm: \|f\|=\sup_{x\in X}|f(x)|. Associated to such an A is the collection M=\mathcal{M}_A of all nonzero homomorphisms \varphi:A\to\mathbb{C}; one easily verifies that every maximal ideal of A is the kernel of some element of M and vice versa, whereby the space \mathcal{M}_A is called the maximal ideal space associated to A. Also:

Definition: A point p in X is said to be a peak point of A provided there exists a function f\in A so that f(p)=1 and |f|<1 on X\setminus\{p\}.

One problem of importance in the realm of function algebras is to characterize C(X) with respect to such algebras A of X. To quote Anderson and Izzo:

A central problem in the subject of uniform algebras is to characterize C(X) among the uniform algebras on X.

One attempt at satisfying this necessity was the so-called peak point conjecture, which was strongly believed to be true until it was shown to be definitively untrue. The purpose of this entry is to focus a little on topics related thereto including the conjecture itself, the counterexample and its construction, and the related results (including work done by Izzo and various collaborators).

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Revisiting, and something light

Well, today was the fourth day of my official employment with Wolfram. This job is absolutely amazing; I couldn’t be more stoked. It’s saddening, of course, that I’m not spending my days engulfed in the books I’d been looking at earlier in the summer; it’s also a bit saddening that I have less time to spend with you beautiful people. Regardless, things are pretty amazing and overall, I couldn’t be happier.

I wanted to take some time to swing by here and say something, though, and fortunately for me, my TA duties this semester have yielded me something of precisely the right balance of depth (or lack thereof) and length (or brevity) to be fitting for tonight’s (this morning’s) pit stop.

On Wednesday (July 3), I was sitting in a precalculus class, doing Wolfram stuff and vaguely listening to what the instructor was talking about at the time. The topic? Logarithms. As someone who’s solo-taught precalculus before, I know precisely how little students understand – or like – or care – about logarithms. I also know how much we try to convince them to believe without their understanding which – among others – has to be a primary cause for their confusion and disdain.

One thing we try to get them to believe? The change of base formula. The change of base formula says that given a base b>0, b\neq 1, the quantity \log_b(x) is equal to the quantity

\log_b(x)=\displaystyle\frac{\log_c(x)}{\log_c(b)} where c>0, c\neq 1.

This information is shared with students at that level largely so they can feel comfortable evaluating an expression like \log_{15}(31) in their calculators given only the capacity to utilize \log(x)=\log_{10}(x) and \ln(x)=\log_e(x) functionality. Surely, they never really need to know it.

And then I realized…

In all my years in mathematics, I’ve never actually seen this rule proven before. That, of course, sparked my interest, and so I went back to my office and jotted the (surprisingly simple) proof on my whiteboard just to appease my curiosity. Here’s the way that goes:

Proof of The Change of Base Formula.
Let y=\log_b(x) so that b^y=x. In particular, then, it follows that for c>0, c\neq 1,

\displaystyle\frac{\log_c(x)}{\log_c(b)} = \frac{\log_c\left(b^y\right)}{\log_c(b)}=\frac{y\cdot\log_c(b)}{\log_c(b)}=y=\log_b(x). \square

I think I may force my next round of precalculus students to know that. It keeps ’em fresh, on their toes, where they gotta baayayaeeee….

Did anyone just catch my reference to ‘Heat’? Or, rather, my reference to Aries Spears’ reference to ‘Heat’?

I hope everyone’s 4th was safe and that there were only minimal injuries due to inebriation, explosives, and general tomfoolery.

Until next time….