## Thursday, September 13, 2007

### Math!

I have now been to all of the classes I think I am planning to take at least once, so this post is going to be my first impressions. For those of you who are not as interested in the math, at the bottom there are also a few pictures ;-)

Geometric Graph Theory

Last spring I took Mudd's Graph Theory course. Throughout the class, we made a point out of the fact that the specific drawing, embedding, of a graph was not important. We focused on properties inherent to the graph regardless of drawings. Drawing were useful for seeing what was going on and obtaining intuition, but did not tend to play a crucial role in the proofs. I get the feeling that this semester the most important part of most of the proofs is going to be finding the correct embedding of a graph (sometimes in 2-space, sometimes in 3-space). It is going to involve things like observing that every three connected planar graph can be drawn as the skeleton of a convex polytope. In turn, given the skeleton of a convex polytope we can prove that it is a 3-connected planar graph.

We first providing a method of projection that gives us a straight line embedding in the plane with no crossings - pick any face of the polytope and choose a point "close" to that face, where close means that if we extend the plane of any other face of the polytope that point and the polytope are in the same half space, and project onto the plane of the original face via that point. To prove that it is three connected we need to show that removing any two arbitrary points a and b leaves the graph connected. Select any 4 points a,b,c,d in the polytope. Either these 4 points are in a plane or they form a tetrahedron. If they form a tetrahedron, then we can find some plane that will separate a and b from c and d. Consider the portion of the polytope in the half space containing c and d. If c and d are adjacent, then we are done, otherwise follow a path from c and a path from d that always moves away from the dividing plane. Eventually, we will not be able to move farther away (since we are on a convex polytope). At this point, the paths will intersect, the final verticies of the paths will be the endpoints of an edge of the polytope and we can connect paths entirely in that "half" of the polytope, or the final vertices of the paths will be two of the vertices bounding a face of the polytope that is parallel to the dividing plane - in which case they are part of a cycle and we can connect the path. In all of these cases we have shown that c and d must be connected in the "half" of the polytope that does not contain a and b and so removing a and b does not disconnect the graph. If all 4 points are coplanar, we use that plane to divide the polytope and then restrict ourselves to stay on one side of the plane (it does not matter which unless they are all part of the same face, in which case we obviously take the side of the plane that contains the rest of the polytope) and use the same argument.

I find it very exciting that we can use phrases like "always moving away from the specified plane" when proving things about graphs! I am going to learn a completely new way of thinking about graph problems :-). The class is being cotaught. One of the professors is amazing, the other is slightly harder to follow, but the material is exciting enough that it is definitely going to be worth the work.

Topics in Geometry

The first day was mostly just outlining what we are going to be covering and the approach we are going to take, but I am still excitied. We are going to start with Euclidean Geometry, but instead of using the axiomatic approach (which the professor deemed "stupid") we are going to study geometry via transformations. In particular, we are going to spend a lot of time studying the isometry group of R^n. (Anyone know of any nice way to write math in a blog?) I really like seeing the connections between subjects, so studying Geometry via Algebra should be fun :-). After Euclidean, we are going to take a similar approach to studying the spherical, projective, and hyperbolic geometries. It will be nice to formally study them for more than a week long class at Mathcamp.

Conjecture and Proof

The best description I have come up with so far is that that this class looks like it is going to be the equivalent of turning the Putnam Seminar into a 3 credit class (and then forbidding collaboration on most of the problems...). It is full of very shiny proofs - the sort that are wonderfully simple and elegant and make you wonder how on earth anyone every found them. One of the things I really like about it is the same thing I really like about the Putnam Seminar - it is going to make me stop and think about all sorts of different areas of mathematics. It is far too easy to learn material one semester and then let it drift to the back of my brain and get rusty from disuse. I suspect C&P will make me use most of the math I know at some point or other :-). The fact that collaboration is mostly forbidden makes me sad though. Mudd has me very used to sitting with friends and bouncing ideas around...

Set Theory

Everything is a set. For instance, the ordered pair (a,b) can be thought of as the set {{a},{a,b}}. The basic axioms of set theory allow us to easily prove that {{a},{a,b}} = {{c},{c,d}} iff a=c and b=d, so it does indeed act like an ordered pair. Set Theory looks like it is going to be all about looking at familiar things in an unfamiliar way. The professor has a good, though subtle, sense of humor. Overall it looks like it will be a good class. The only problem is that there were exactly 6 of us that went yesterday, at least 2 of which I suspect are not going to actually take the class. If a class drops under 6, they turn it into a reading class rather than a normal class (as long as there are still at least 3 of us, which I think there will be). I would rather have it as a normal class, but a reading class will be fine too if it comes to that.

And now the non-math.

Some of the flowers that are around the building most of our classes are in:

Near the giant hour glass that I posted about earlier, there is a large parking lot. Currently, this parking lot appears to be some sort of advertisement show - it has billboards set up all over it. In addition to the billboards, there are animals made out of trash and oddly decorated phone booths.

Last night several of us hung out at Christina and Chelsey's place and had görögdinnye and tea. It is important to note that this was not organized by either of the mathcampers that are here this semester (though we were both there).