Week of Dec 1 Total hours: Running total:

Dec 3:

- Went to REACH meeting. (2 hrs)

Week of Nov 24 Total hours: 3 Running total: 64.5

Nov 26:

- Started writing up the proof of locality of matchings on crosses and wrenches graphs. (1 hr)
- Went to REACH meeting. (2 hrs)

Week of Nov 17 Total hours: 5.5 Running total: 61.5

Nov 19:

- Read about Kuo condensation and thought a bit about applying graphical condensation to groves. (1.5 hrs)
- Went to REACH meeting. (2 hrs)

- Went to REACH meeting. (2 hrs)

Week of Nov 10 Total hours: 5.5 Running total: 56

Nov 12:

- I've been trying, without luck to come up with a height function on crosses and wrenches graphs. The obvious one doesn't work, but I don't think it should be too hard to tweek it into something that does. (1.5 hrs)
- Went to REACH meeting. (2 hrs)

- Went to REACH meeting. (2 hrs)

Week of Nov 3 Total hours: 6 hrs Running total: 50.5

Nov 5:

- Went to REACH meeting. (2 hrs)

- I was thinking about local moves on groves. I'm pretty convinced that we're going to have to look at height functions on matchings and push that through the correspondence and generalize. What we're doing now doesn't seem to be getting us anywhere. (2 hrs)

- Went to REACH meeting. (2 hrs)

Week of Oct 27 Total hours: 5.5 hrs Running total: 44.5

Oct 29:

- Went to REACH meeting. (2 hrs)

- Went to REACH meeting. (2 hrs)
- Thought about how to show that we can get from any perfect matching of a crosses and wrenches graph to any other by rotating faces one at a time. I think this shouldn't be too hard, but I have to work out the details. (1.5 hr)

Week of Oct 20 Total hours: 7 Running total: 39

Oct 22:

- Went to REACH meeting. (2 hrs)
- I thought of a natural generalization of Markov triples to
*Markov n-tuples*. An n-tuple of integers (x_1, x_2, ... , x_n) is a*Markov n-tuple*if it satisfies (x_1)^2 + (x_2)^2 + ... + (x_n)^2 = n(x_1)(x_2)...(x_n). Everything about Markov numbers seems to generalize. I'm working on a pdf that shows all of it. Its really messy at first (sorry) and currently unfinished, but here it is: Markov n-tuples along with the LaTeX stuff that goes with it. Everything is pretty much following the case where n=3. Look at my Oct 15 entry. It would be interesting to find a combinatorial explaination for Markov n-tuples. Perhaps Rui's snakes generalize. (3 hrs).

- Went to REACH meeting. (2 hrs)

- Some more thoughts on Markov numbers. First of all, I found that
I was thinking about the wrong question a bit. The tree where each vertex
is of degree three isn't quite as interesting as I thought. It is clear
that the subset of the tree in which there is a 5, say, in the first position
is
__not__connected. It is in fact in two parts. Here are paths to the two parts:

(1,1,1) --> (1,2,1) --> (5,2,1)

The real tree that we are interested in is where all permutations of the same numbers are identified. And the real question, I think, is "For a given Markov number n, are there unique numbers k, l less than n such that (l,k,n) is a Markov triple?" I thought about this for a while, but I can't say that I made much progress. I did, however, get a lot of different ways of thinking of the Markov numbers in my head. (3 hrs)

Week of Oct 13 Total hours: 11 Running total: 32

Oct 15:

- Read through the "first page"s of possible articles. Thought
about Markov numbers. The graph that Jim showed last meeting is the
Cayley graph of
**Z**/2***Z**/2***Z**/2=G (the free product of**Z**/2with itself three times), so you get a map from G to the "Markov triples group" M (with induced composition from G: each ordered triple is a path in the graph of G and composition is just concatination of pathes). By the argument that Gabriel gave (by infinite descent), this map is surjective (which I actually needed to define composition in M). I don't think it should be very hard to show that it is injective. If (x, y, z) is a Markov triple with z the largest of the three, then if we apply the transformation z'z=x^2 +y^2 to get the triple (x, y, z'), we will get z' < z (this was the argument that Gabriel gave, I think), but if we apply the transformation to x or y, we should get x' > z or y' > z (see details below). This means that as you get further from the center of the graph, the point identified with (1, 1, 1), the numbers can only get bigger. In particular, you can't get back to (1, 1, 1). This means that the map from G to M has trivial kernel, so it is an isomorphism. Hence each ordered Markov triple occurs exactly once in the graphical representation that we gave. There is still the question of whether a given number, say 5, can occur in a particular slot, say the first one, in only one way. That is, is the set {(5, y, z) in M} connected when you look at its image in G. Again, I haven't actually proven much, but I'm having lots of interesting ideas. I talked to Kyle breifly about the (not neccisarily perfect) matchings of the mxn grid. It may be possible to think of a matching of the mxn grid as a perfect matching of the mxnx2 grid, but this is not a bijection, so it is probably not too profitable. (2 hrs)

similarly, y' > z

for the infinite descent argument, we need to show that z^2 > x^2 + y^2

(.5 hrs)

- Went to REACH meeting. Took notes. (2 hrs)
- Poured my heart an soul into making beautiful ASCII pictures with excellent explainations in the notes. (2.5 hrs)

- Went to REACH meeting. (2 hrs)
- Talked to Anna and thought about some general ideas. One thing that it seems we're doing is taking interesting recurrences and finding what they are counting. This ensures that the objects you find (such as groves) are interesting, so you can study them. Then we can take related interesting recurrences and do the same and see how the objects relate. Another approach would be to take related objects and see how their recurrences relate. For example, it may be possible to generalize groves so that they are not neccisarily triangular, but also square, hexagonal, etc. One such generalization might arrise as follows: a grove arrises from a tiling of a triangular polyhex by rhombuses; we could consider regular tilings by rhombuses of hyperbolic regions that have similar properties. For a grove, the rhombuses are 6 to a vertex and 3 to a vertex. We could take a tiling where they are 7 to a vertex and 3 to a vertex. (1 hr)

- Read over Rui's stuff on Markov numbers. I'm still a bit fuzzy on some of it, I think, but I like the ideas in it. I'll think about it some more and ask Rui about anything that is still fuzzy. (1 hr)

Week of Oct 6 Total hours: 8

Oct 6:

- I now have access to Maple, Mathematica and MATLAB on the computer I normally work with. Yahoo. Read some tutorials about how to use Maple and MATLAB. (2 hrs)

- Read the papers on Integrality and Symmetry. Thought about running back the recurence of the 1xn grid (non-perfect) matchings and had some thoughts on the mxn case. (2 hrs)

- Went to REACH meeting. (2 hrs)

- Went to REACH meeting. (2 hrs)

Week of Sept 29 Total hours: 6.5

Sept 29:

- Thought about second homework: how many matchings of a 3xn grid are there? (2.5 hrs) I'm trying to do it with generating functions like Jim did for 2xn case. I have an interesting transition matrix that I think can give me a nice result, I need to diagonalize it though. I'll work it out when I find someone who can show me how to make a computer do it for me. I also did it like we did the 2xn case. I got a system of linear recurences in 5 different functions. I still have to collapse it down to one recurrance for a_n.

- Worked on homework problem with matchings of 3xn grid. Tried doing stuff with the transformation matrix with Kyle. (1.5 hrs) Worked out the generating functions with Kyle on maple. They're ugly. (.5 hrs)

- Went to REACH meeting. (2 hrs)

- Went to REACH meeting. (2 hrs)

Week of Sept 22 Total hours: 6.5

Sept 24:

- Went to first REACH meeting (2 hrs). Started learning how to make web page.

- Learned about how to make web page. Made this page (1 hr). I think this is a temporary format . . . I've been told that its much easier to build and maintain a site through blogger, so I think I'll end up doing that.
- Thought about the first homework problem: how many matchings of a 2xn grid are there (1 hr).

- Went to second REACH meeting (2 hrs).

- Brandeis offers grants for undergrads doing research. It looks like its only for students doing individual research, but I'm meeting with Dean Hahn on monday at 3:30 to check out the options.

- Thought about the second homework problem: how many matchings of a 3xn grid are there. (.5 hrs)