# Gracious Living

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.

Homomorphisms, Generators, and Cyclic Groups
November 12, 2010, 16:07
Filed under: Algebra, Math | Tags: , , , ,

Last time we saw the group $D_8$ of symmetries of the square represented in terms of a single flip and a single rotation.  This time, I plan on going more deeply into what this means, and what it does for group theory.  Actually, I don’t really have a plan — basically I’m going to keep talking about things that seem useful until the post is done.  Who’s with me?

Cardinality
November 1, 2010, 04:45
Filed under: Math, Set Theory | Tags: , ,

(This was originally a longer post that went through to countability and the continuum hypothesis. I’ve broken it in half, ending at the definition of infinite sets, in the interest of keeping things short, and also possibly of MaBloWriMo. The next half will be up soon!)

What we’re going to do today is measure sets.  I said that sets have no structure back in the first couple of posts, but this isn’t entirely true.  Finite sets, for example, have a definite number of elements.  Today, I show that you can extend this idea of number to infinite sets by considering a special class of function between sets that preserves number.  The strange properties of this will end up making topology and set theory a lot more interesting.

Relations
October 25, 2010, 13:38
Filed under: Math, Set Theory | Tags: , ,

A relation from $A$ to $B$ is just a subset of $A\times B$.  For example, $R=\{(a,b)\in\mathbb{R}\times\mathbb{R}:a^2=b^2\}$ is a relation from $\mathbb{R}$ to $\mathbb{R}$ that includes, say, $(0,0),(4,4),(4,-4),(-4,4),(3,-3)$, and so on.  We generally write $aRb$ instead of $(a,b)\in R$.  Except in very few circumstances, we actually only care about three kinds of relations.

Sets
October 24, 2010, 17:31
Filed under: Math, Set Theory | Tags: , ,

This will probably be review for a lot of readers, but I just want to make sure we’re all on the same page before we really start.  These terms get used a lot a lot in higher math.  In the 1900’s, sets were seen by luminaries like Bertrand Russell as being the salvation of math, since you could describe basically any mathematical structure in terms of sets.  Nowadays, mathematicians tend to believe that the “guts” of math really aren’t that important, and category theory, which instead describes the relations between structures, has become more foundational.  By the way, category theory is just about the coolest possible thing, but its coolness only really becomes apparent once you have a bedrock of examples to work with.  I’ll start talking about it much later.  For now, let’s talk about sets.