I have now been to two weeks worth of all of my classes, and remain excited about all of them, so this is a good sign. This is a math post. I have decided that bringing my laptop to school every day is a little ridiculous, and the keyboards on the computers here are British and thus strange and I don't want to type long things on them, so I should come up with some sort of schedule. My current plan is thus to have a non-mathy post on Monday/Tuesday and a mathy post on Thursday/Friday which of the two days in each case depending on things like what other plans/hw I have ;-).
Geometric Graph Theory
A few days ago we gave three proofs of the same theorem ( If G is a three connected planar graph, S is a nonempty subset of V(G) and f0 is a function that maps elements of S to the real plane, R, then there is a unique function f:V->R, such that f(i) = f0(i) for i in S and it is harmonic at all other vertices. ) The first proof of this involved random walks and probability theory. The second had us view the graph as an electrical circuit and used physics to obtain a proof. The third started out with ideal rubber bands that obey Hookes law from these we defined an energy function such that minimizing it gave the desired configuration. It was very interesting seeing these three very different approaches - and the resulting equivalences between statements about physics, math and probability theory. I really like math that plays in the areas between the defined subjects :-). Also, I want my shelf of textbooks that I have used in former classes. There are not that many on it yet, but it would be really nice to be able to look up the details of theorems I mostly remember...
Topics in Geometry
This class is also very good at getting into the area's between subjects! Proofs have a tendency to start out as a geometry problem, involve using algebra properties to define isometries, manipulating the isometries geometrically - playing with things like distance functions, and then drawing them back in to the group context to use the techniques that became so familiar in Abstract. It is really interesting seeing geometric interpretations of group theoretic ideas that previously did not seem like things to be visualized (granted, when the geometry gets into higher dimensions it also does not really get visualized.) Additionally, when we don't have questions during the "office hour" the prof does cool geometry stuff, like giving a geometric proof of the Cauchy-Schwarz inequality.
Conjecture and Proof
This class is definitely the three credit version of Putnam seminar. On Tuesdays the prof goes over cool proofs and points out various useful ideas and techniques. On Thursdays he collects the HW and then we go over it, having people present the different proofs they have come up with. It is really cool to see the sometimes drastically different approaches people take to problems. Problem solving is not exactly my favorite type of math, but the practice and being exposed to different techniques and ways of thinking is good for me and the proofs are definitely exciting.
We seem to be down to 4 people in Set Theory, which means that it will happen, but that it will be turned into a reading class where we are expected to teach more of it to ourselves and it will only meet once a week. *sigh* Ah well, it will still be fun. It will also teach me to pay very, very close attention to proofs in class. The prof has a tendency to go through a proof, look like he is ready to move on and then say "Oh, as a foot note, that was not a proof and the theorem is false as currently stated" and then we stare at the board for a minute or so until we figure out what the problem is. Today we had to figure out why a0<=a and b0<=b does not imply a0^(b0)<=a^b. In set theory, 0^0 = 1 and 0^a=0 for all a not equal to 1, so if b and b0 are both 0 it is false. We also looked at why the proof of the fact that the countable union of countable sets is countable requires the axiom of choice. It is a fun class :-)