Gracious Living


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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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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Banach-Tarski part 2
November 21, 2010, 23:30
Filed under: Algebra, Math | Tags: , , , , ,

So I sort of left you hanging last time.  We talked about equidecomposability, showed that F_2 was paradoxical under its own action on itself, and embedded F_2 into SO(3).  From here, it just becomes a matter of putting all the steps together: first the sphere, then the ball minus its center, then the whole ball.

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Banach-Tarski part 1
November 21, 2010, 06:33
Filed under: Algebra, Math | Tags: , , , , , ,

Okay, here’s the moment you’ve been waiting for: the proof of the Banach-Tarski Paradox.  Here’s what the paradox says:

Theorem (Banach-Tarski).  There are a finite number of disjoint subsets of \mathbb{R}^3 whose union is the unit ball, and such that we can apply an isometry to each of them and wind up with disjoint sets whose union is a pair of unit balls.

Or “we can cut a unit ball up into a finite number of pieces, rearrange them, and put them back together to make two balls.”

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Isometries of Euclidean Space
November 18, 2010, 21:06
Filed under: Algebra, Math | Tags: , , , , ,

Finite-dimensional vector spaces \mathbb{R}^n come packed with something extra: an inner product.  An inner product is a map that multiplies two vectors and gives you a scalar.  It’s usually written with a dot, or with angle brackets.  For real vector spaces, we define it to be a map V\times V\rightarrow\mathbb{R} with the following properties:

  • Symmetry: \langle x,y\rangle=\langle y,x\rangle
  • Bilinearity: \langle ax+bx^\prime,y\rangle=a\langle x,y\rangle+b\langle x^\prime,y\rangle, where a,b are scalars and x^\prime is another vector, and the same for the second coordinate
  • Positive-definiteness: \langle x,x\rangle\ge 0, and it is only equal to 0 when x=0.

(I’m going to stop using boldface for vectors, since it’s usually clear what’s a vector and what’s not.)  One of the uses of an inner product is to define the length of a vector: just set \|x\|=\sqrt{\langle x,x\rangle}.  This is only 0 if x is, and otherwise it’s always real and positive because the inner product is positive definite.  Another use is to define the angle between two nonzero vectors: set \langle \cos\theta=\frac{\langle x,y\rangle}{\|x\|\|y\|}.  In particular, \langle \theta is right iff \langle x,y\rangle=0.  In this case, we say x and y are orthogonal.

In Euclidean space, the inner product is the dot product: \langle (x_1,x_2,\dotsc,x_n),(y_1,y_2,\dotsc,y_n)=x_1y_1+x_2y_2+\dotsb+x_ny_n.  This is primarily what we’re concerned with today, so we’ll return to abstract inner products another day.

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Vector Spaces
November 17, 2010, 17:30
Filed under: Algebra, Math | Tags: , , ,

Okay, it’s time to make a big leap forwards in terms of concreteness.  The Banach-Tarski paradox makes a strong statement about \mathbb{R}^3 that isn’t true about \mathbb{R}^2 or \mathbb{R}.  Now, we still don’t really know what \mathbb{R} is, but if we pretend we know what it is, we can say stuff about \mathbb{R}^3.  Certainly, \mathbb{R}^3 has the product topology of \mathbb{R}\times\mathbb{R}\times\mathbb{R} — but it has much more than this.  It has an origin, for instance, and a distance function, and a way to measure angles.  The distance function, in turn, allows us to define spheres and isometries (i. e. distance-preserving maps), which are both part of the statement of Banach-Tarski.  All of these are summarized by saying that \mathbb{R}^3 is a vector space.

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