Filed under: Algebra, Math | Tags: algebra, geometry, graph theory, group theory, Math, serre

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 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!

Filed under: Topology, Uncategorized | Tags: differential geometry, elementary, geometry, Math, topology

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.

Filed under: Algebra, Math | Tags: algebra, geometry, group theory, linear algebra, MaBloWriMo, Math

So I sort of left you hanging last time. We talked about equidecomposability, showed that was paradoxical under its own action on itself, and embedded into . 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.

Filed under: Algebra, Math | Tags: algebra, geometry, group theory, linear algebra, MaBloWriMo, Math, topology

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 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.”

Filed under: Algebra, Math | Tags: algebra, geometry, group theory, MaBloWriMo, Math, topology

Finite-dimensional vector spaces 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 with the following properties:

- Symmetry:
- Bilinearity: , where are scalars and is another vector, and the same for the second coordinate
- Positive-definiteness: , and it is only equal to when .

(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 . This is only if 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 . In particular, is right iff . In this case, we say and are **orthogonal**.

In Euclidean space, the inner product is the **dot product**: . This is primarily what we’re concerned with today, so we’ll return to abstract inner products another day.

Okay, it’s time to make a big leap forwards in terms of concreteness. The Banach-Tarski paradox makes a strong statement about that *isn’t* true about or . Now, we still don’t really know what is, but if we pretend we know what it is, we can say stuff about . Certainly, has the product topology of — 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 is a **vector space**.