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!

# Frustration, or Somebody’s got a case of the Tuesdays

This is going to be an entry about my algebraic topology class.

My previous topology classes were taught by someone who’s amazing <i>as a mathematician</i>. Most people in the class would agree, however, that this person was someone who was terrible as an instructor: Somehow, I made it through a graduate sequence of coursework despite receiving terrible grades throughout. This happened despite my spending 50+ hours a week on every homework assignment and slaving until I was on the verge of breakdown week in and week out. Somehow, I got a B.

My topology experience thus far is certainly not one of my finest achievements.

Fast forward to now and I’m in an algebraic topology class taught by someone who’s amazing. Amazing. Not amazing <i>insert quantifier here</i>, no – this person is simply amazing. And this topic is beautiful. And this class is hard.

This class is hard, too, despite the fact that we have no continual responsibilities. Indeed, we have zero homework whatsoever: Not required problems to turn in, not required problems to keep, not even suggested problems for our benefit. We simply have <i>zero homework</i> in this class. That’s a huge relief after last semester.

What we <i>do have</i>, though, are exams. We have three of them, and I have zero doubt now (nor have I had doubt at any point this semester) that my ass will be kicked by each and every one.

As a result, I’m working hard.

A week-ish ago, I spent some time going through the preliminary parts of the stuff we’re talking about (homology theory). I did examples, I spent lots of time drawing pictures, and I didn’t stop until I got it.

That’s right: A week-ish ago, I <i>got it</i>.

Today, however, I’m sitting in my office, frustrated and almost-defeated, blogging to you all and mourning the fact that a lot has apparently changed in the last week-ish.

Today, I just don’t get it.

If I were to make a list here cataloging the number of screw-ups I made trying to solve one problem over the course of about 20 hours, I’d be (a) making a really long list and (b) really <i>really</i> embarrassed.

I’m really <i>really</i> embarrassed right now.

Finally, after re-reading and re-re-reading Hatcher, I found source 1 of my confusion. Later, after consulting the online resources of mathematicians greater than myself (case in point here), I found the remaining sources of my confusion.

The upside is that now I’m no longer confused. On the other hand, the fact that I was as confused as I was (and about such basic material as that happened to be) makes me really <i>really</i> uneasy moving forward.

I need an intervention.

In the meantime, I’m going to try to dust myself off, hit the salt mines yet again, and lose my frustrations in the never-ending cycle of Lana del Rey that’s been permeating through my office for the past couple hours.

3 weeks, 2 days.

# 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.

# 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).