Friday, September 28, 2007

"It can be larger than larger than infinite!"

Geometric Graph Theory

This class is making me glad I have the physics background I do. One of our proofs involved replacing some of the ideal rubber bands connecting the points of our graph with rigid bars so that we could make the "forces" at all of the vertices sum to zero while still forcing them not coincide. All of this is to prove that 3-connected planar graphs can be viewed as the skeleton of a convex polyhedron. I think this class is going to prove to be the most challenging, but it is fun :-). I'm definitely glad for my background with graph theory and knowing what sorts of techniques are useful for approaching graph theory problems. It made the first HW set much more manageable for me than it seemed to be for a lot of my classmates...

Topics in Geometry

It is a great deal of fun to do proofs that effectively involve staring until you understand the geometry of what is going on, translating it into Algebra so that you can actually explain what is going on, working through the Algebra until it gets stuck, translating back into the language of geometry to get past a few sticky parts and then going back to Algebra to finish the proof. I love that math is so wonderfully interconnected! It also highly amuses me that several times questions, such as "Is the empty set an affine subspace?" have gotten the response, "Well, some elementary text books say (that/it is), but I do not think that is a good idea."

Set Theory this week, aka Playing with Infinity

When playing with cardinals, we start out with all of the natural numbers. We say the number of natural numbers is aleph naught. If we add aleph naught copies of 1 together, we get aleph naught. If we add aleph naught copies of 2 together, we get 2*aleph naught, which is just aleph naught. In fact, if we take the sum of all of the natural numbers, then we get Aleph naught. However, if we multiply them all together, we get c (the size of the set of real numbers). Now we have c*c=c and c^(aleph naught)=c and for that matter c^c =c. However, 2^c>c and we call it c_1. In fact we can define countably many c_i's with the formula c_(n+1) = 2^c_n. If we add all of these c_i together, we get a cardinal that is strictly larger than any of the c_n, so we call it d. We can then follow the same procedure with d to create e and so on. Once we have countably many of these countable sets of levels of infinity, we can add all of them together to create an even larger infinity, and so on and so forth. It is exciting.

Conjecture and Proof

The number of different possible approaches to problems often amazes me. We had one this week about tiling a rectangle. Various solutions included counting different kinds of dominoes in certain parts of the board mod n, looking at what happens if you roll a sphere around in the rectangle, and two very different colorings of the board. Over all it continues to be a fun class full of really cool proofs.

1 comment:

Eric said...

Sake! I've been bad and have been forgetting to read this!

Set theory is indeed fun :). I'd like to keep hearing what you're doing in the class.

A while ago you asked whether there was a good way to display math (but I only just read it now). I know wordpress.com can display latex, but I'm not sure how that works or whether you could get that here. I usually just write in pseudo-tex, pretty much how you've been doing (but you can write "aleph_0" instead of typing "aleph naught" every time!).