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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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….

Paradoxes, paradoxically

So I was able – fortunately – to wake up early and to do some legit reading, despite having only a handful of sleep hours (4-ish?). That’s a definite positive. Right now, I’m about 30 minutes away from a forced obligation (that’s a definite negative), but I wanted to use the 30 minutes I have to still do something constructive. Rather than spend this time wracking my brain with really difficult, hard-to-understand reading that would leave me mentally exhausted for the aforementioned obligation, I decided to come here and write a little exposition regarding something mathematical.

In particular, I’m going to talk about the so-called Richard’s Paradox (see here).

Of course, the fact that I’m avoiding theoretical math to postpone mental exhaustion while using the time to come here and talk about theoretical math is a bit of a paradox as well, so I’ll basically be expositing, paradoxically, about paradoxes.

You have no idea how much I crack myself up.

The ideology that birthed Richard’s paradox is intimately tied to the idea of metamathematics, that is, the study of metatheories – theories about mathematical theories – using mathematical ideas and quantification. I’m not going to get too deeply involved in the discussion on that particular topic; the interested reader, of course, can scope out more here.

To begin, we let \mathbb{N} denote the set of nonzero positive integers (aka, the natural numbers) and we investigate the collection of all “formal English language statements of finite length” which define a number n of \mathbb{N}. For example, The first prime number, The smallest perfect number, and The cube of the first odd number larger than five are such statements, as they verbally describe the numbers 2, 6, and 73=343, respectively. On the other hand, statements like The number larger than all other numbers and Scotland is a place I’d like to visit fail to make the list due to the fact that the first doesn’t describe a number in \mathbb{N} and the second doesn’t describe a number at all. Let \mathcal{A}_n denote the collection of all so-called qualifying statements, that is, statements that do describe elements n\in\mathbb{N}.

Note, first, that the collection \mathcal{A}_n is infinite due to the fact that the statements The ith natural number is a qualifying statement for all i=1,2,\ldots. It’s also countable: Only a countable number of words exist in the English language, and each statement in \mathcal{A}_n consists of a finite union of these countably many words. This fact, along with obvious language considerations, says that \mathcal{A}_n can actually be given an ordering.

Indeed, consider a two-part ordering: First, organize the statements in \mathcal{A}_n by length so that the shortest statements appear first, and then organize statements of the same length by standard lexicographical (dictionary) ordering. The result is an ordered version of the countably infinite collection \mathcal{A}_n which we’ll again denote by \mathcal{A}_n.

As of now, almost nothing has been done. Continue reading