Gracious Living

Limit Point and Sequential Compactness
December 30, 2010, 02:51
Filed under: Math, Topology | Tags: , , ,

Back to topology.  The interesting thing about compactness, as I see it, is that its definition isn’t very intuitive.  We want to talk about what are basically “closed and bounded” sets without really using closedness, which doesn’t behave well with subspaces, or boundedness, which doesn’t behave well with anything.  At the time this idea came about in the early part of this century, the mathematicians ssewho invented it (Borel, Weierstrass, and Bolzano, mainly, I think) messed around with a couple other definitions but they turned out to rely on intuition from Euclidean space, and so not be nearly as useful.  In this post, we’ll study those other definitions.

I debated with myself for a long time about what to do first: this, the separation axioms (which measure how good the topology is at distinguishing disjoint sets), or the countability axioms (which measure, well, how countable the topology is).  In a sense, each one depends on the other two for examples and theorems, and in this case, though it’s easy to show that all the properties I’m listing are different, we’ll need to refer to the other properties to discover when they are the same.  So I expect this post to be brief; after we have all the other properties defined, it’ll be easier to talk about what implies what.

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Subgroups of Free Groups are Free
December 28, 2010, 07:32
Filed under: Algebra, Math | Tags: , , , , ,

Okay, first post for a while.  As I promised quite a while back, let’s prove together that subgroups of free groups are free.  It’s surprising that this is nontrivial to prove: just try to come up with some subgroups of F_2 and you’ll see what I mean.  In fact, using only basic algebraic topology and a bit of graph theory, we can come up with a really simple argument that replaces this one.  Perhaps that’s an argument in favor of algebraic topology.  But I think this angle is sort of interesting, and it should be a fresh experience for me, at least.

The proof is due to Jean-Pierre “Duh Bear” Serre in his book Trees.  A heads up if you track this down — Serre has a really weird way of defining graphs.  Fortunately, for this proof at least, a little bit of work translates things into the same language of graphs and digraphs that we saw when talking about Cayley graphs.  I review that below the fold.  It takes a while to set up the machinery, though the proof itself isn’t too long.  To recompense, I’ve left out a couple minor details, which you’re probably able to fill in.  If some step doesn’t make sense, work it out — or try to disprove it!

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Symmetric Groups
December 18, 2010, 11:36
Filed under: Algebra, Math, Uncategorized | Tags: , , , ,

We’ve seen symmetric groups before.  The symmetric group on an arbitrary set, S_X or {\rm Sym}(X), is the group of bijections from the set to itself.  As usual, we’re only interested in the finite case S_n, which we call the symmetric group on n symbols.  These are pretty important finite groups, and so I hope you’ll accept my apology for writing a post just about their internal structure.  The language we use to talk about symmetric groups ends up popping up all the time.

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Differential Geometry and the Sphere Theorem
December 16, 2010, 20:38
Filed under: Topology, Uncategorized | Tags: , , , ,

I just wrote a paper about the Sphere Theorem for my differential geometry class.  Since I can’t really get it out of my head, I thought it’d be fun to use to give a brief overview of differential/Riemannian geometry, in more or less layman’s terms.  This is somewhat of a departure from my normal posting style: though you’ll get more out of this if you’ve understood the topology posts so far, I’ll try to write in broad enough strokes that calculus is the only prerequisite.  Really, the most complicated idea behind this is the idea of a map from $\mathbb{R}^m$ to $\mathbb{R}^n$ being differentiable.  I also refer to vectors, but I think that that’s a pretty intuitive concept, in general.

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December 11, 2010, 19:32
Filed under: Math, personal | Tags: , , ,

I’ve been really busy with finals and haven’t had time to finish a proper post, but wow is this amazing.  The suggestion (not new, but very eloquently expressed) is to stop using \pi, the ratio of a circle’s circumference to its diameter, and start using \tau=2\pi, the ratio of its circumference to its radius.  (Spelled “tau”, rhymes with “cow”: do you know your Greek alphabet?)  Pretty much every important use of \pi is actually a use of 2\pi: the circumference formula, integration in polar coordinates or around a circle, finding roots of unity, but also Gaussian/normal distribution stuff, Fourier transforms, and zeta function identities.  Where there are exceptions, the insistence on using \pi instead of 2\pi obscures the nature of the equation: writing the area of a circle as \frac{1}{2}\tau r^2 instead of \pi r^2 highlights the fact that it’s the output of an integral, and though e^{i\pi}+1=0 looks beautiful, it obscures its own meaning: complex exponentiation by \pi corresponds to a rotation by 180 degrees or \pi radians.  If you write e^{i\tau}=1 (or, as the author suggests, e^{i\tau}=1+0 to preserve the “beauty”), you immediately see what the formula’s getting at: x\mapsto e^{ix} is a periodic function with period \tau corresponding to a rotation of the complex plane.

And if that weren’t enough, look at this piece of beauty:

A diagram of important angles on the circle, measured in multiples of tau

See how easy things could be?

Although \pi has been around for a long time, changes in terminology do happen and math moves right along. I don’t know what we would do if we still had to do group theory with Galois’ original wordings of “substitutions” and “arrangements.” So I think that among mathematicians, this could catch on pretty easily.  It’s easy to spread the meme, too — when you’re presenting something, just start out with “let \tau=2\pi” and watch the daylight glimmer on people’s faces.

But I’m not so sure that this could catch on in schools, which is where it is most needed.  I’m a math major and I still take a few seconds to remember those godforsaken radian angle measures.  It’s annoying and breaks one’s train of thought.  Why do we expect school-age kids to have to jump through this hoop every time they do a trig calculation, which is pretty much all they’re graded on? (another stupid thing about education, but I digress.)  All it teaches them is that math is about memorization of arbitrary things.  With \tau, there’s nothing to memorize: one-third of the circle is \tau/3.

I just feel that there’s considerable inertia in changing math curricula, especially over something as basic as this.  The only reason we learn math at all is because Eisenhower and Kennedy decided to put a man on the moon.  And because Egypt was troubled by the horrible asp.  Putting \tau into the curriculum would require schools to pay for new versions of these stupid things called “curriculum planners” and “textbooks.”  Quite frankly, that’s not what American schools need to be spending their money on, and it’s by no means even the biggest problem with math education.  The author of the page I linked suggests that teachers introduce \tau to teach kids critical thinking and show them that the best way to do something isn’t always the way you’re first taught.  This sounds awesome — an actually interesting debate in a math classroom?! — but it seems like a teacher that did that would meet interference from higher up, on the grounds that it stifles kids’ ability to communicate effectively (all math education being about communication, to people as diverse as cashiers and accountants).

On the other hand, the textbook-publishing lobby would love an excuse to publish a new edition.  In addition, if \tau becomes accepted in the math community, it’s only a matter of time before education starts changing as well.  Especially if the curriculum started to change so that it had more to do with actual math… wouldn’t that be a treat.

Cardinal Arithmetic
December 8, 2010, 03:23
Filed under: Math, Set Theory | Tags: , , , ,

Let’s get tricky!  Unlike ordinal arithmetic, the arithmetic of cardinals isn’t really an extension of Peano arithmetic.  Rather, it’s a consequence of the set model of ordinals, with the sum of two ordinals being the order type of their disjoint union, and their product being the order type of their Cartesian product.

Part of the reason for this is that cardinals have a vastly different notion of “successor.”  See, the ordinals are well-ordered, so any subset has a least element, and thus we can identify each cardinal with the least ordinal of its cardinality.  So we’ll say \aleph_0=\omega,\aleph_1=\omega_1,\dotsc.  (Exercise: these are all limit ordinals.)  Then the cardinals, as a subset of the ordinals, are also well-ordered, and thus we can define the successor of a cardinal as the least cardinal greater than it, and the limit of a set of cardinals as their least upper bound.

Since the cardinals, as aleph numbers, are also indexed by the ordinals, we have \aleph_{S\alpha}=S\aleph_\alpha,\aleph_{\lim\alpha}=\lim\aleph_\alpha.  We can treat the map \alpha\mapsto\aleph_\alpha as a function \aleph:\mathbf{On}\rightarrow\mathbf{On}; then this function is normal.  So in particular it has a fixed point (indeed, an unbounded class of them): this is an ordinal \alpha such that \omega_\alpha=\alpha.  Even though the cardinals increase a lot “faster” than the ordinals, this number is simultaneously the \alphath cardinal and the \alphath ordinal!  The first such ordinal is \omega_{\omega_{\omega_{\dotsc}}}.

This fills me with the greatest glee.

We won’t be assuming the generalized continuum hypothesis for the first part of this, so we won’t mention the beth numbers (which are \beth_0=\aleph_0,\beth_{S\alpha}=2^{\beth_\alpha},\beth_{\lim\alpha}=\lim\beth_\alpha).  If you do assume it, the beth numbers are the aleph numbers; if you don’t, it’s difficult to say anything about them arithmetic-wise.

Like Devlin, I’ll use \kappa,\lambda,\mu for cardinals, \alpha,\beta,\gamma for ordinals.

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Ordinal Arithmetic
December 7, 2010, 21:03
Filed under: Math, Set Theory | Tags: , , ,

Freeing yourself of the bounds of Peano arithmetic, you come upon a number of different routes to take.  You want the addition and multiplication operations you defined to make sense in a larger scope, but what this scope is varies depending on your needs.  The standard thing to do is to start “closing” the ordered semiring \mathbb{N} in a number of different ways: group closure by adding negatives and reciprocals, metric closure by adding limits, or algebraic closure by adding solutions to polynomials.  But instead, we could just model the elements of \mathbb{N} as the finite ordinals and use transfinite recursion to extend the operations to operations on ordinals.  Transfinite recursion is like induction on cagefighting elephant steroids.

Once you’ve done this, you lose some of your semiring properties, but you retain the order and you win the exciting new property of it being closed under limits.  This is ordinal arithmetic.  Building it will let us understand the structure of the ordinals more rigorously than we had before: there was an ordinal we called \omega^2+1, but what does this actually mean?

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