# John Gonzalez's REACH Research Webpage

## Research Experiences in Algebraic Combinatorics at Harvard, Fall 2002

• Nov 3 -- I believe I've found a proof that the number of domino tilings of a 2n by 2n board is divisible by 3 when n is congruent to 2 mod 5. I'm not certain of my proof so I am preparing it for next REACH meeting. I used p-adic methods which I've never used, so I'll need to check with someone before I actually post my work on this page. (3 hours)
• My total hours thus far is 51.
• Oct 31 -- Continued to read through Gouvea's section on field extensions, and worked some of the problems to understand it thoroughly. I was then able to show that the 3-adic absolute value of the product is at most one, using the nonarchimedean property and ideas in my reading. I worked for a few more hours to try to use these ideas in a more clever way to show that it was at most 1/3 but was unsuccessful. We also had a reach meeting today in which our group discussed Cohn's article and we all agreed to try to thoroughly understand it before our next meeting. (4 hours)
• Oct 30 -- My idea seemed promising but I don't understand how the p-adic metric extends to a finite extension of Q. I'm trying to take the p-adic valuation of the cosines by using the infinite series representation but i'm not sure if that makes sense because there are pi's and the series might not even converge p-adically. I'm reading Gouvea's section on vector spaces and field extensions.(3 hours)
• Oct 29 -- Reach Meeting. I'm now trying to apply p-adic methods to look at the product TT(1+a_ij) 3-adically. We're claiming that its 3-adic absolute value is at most 1/3. My idea is to look at the a_ij similar to Cohn's except with a 10k+5 root of unity, this corresponds to letting n = 5k+2. I believe it should be possible to use the nonarchimedean property of the absolute value to show that this product is at most 1/3. I plan to try a few examples. (3 hours)
• Oct 27 -- I'm reading Henry Cohn's article and trying to plow through some of the algebraic gaps not discussed. I spent lots of time reading Gouvea's P-adic Numbers book and my algebra book. I have a somewhat better understanding of the ideas Cohn uses. (3 hours)
• Oct 24 -- I worked on the domino tiling problem again for another couple of hours, this time trying to derive a recursion from which we can see some integer properties. I found two ways to look at this and neither of them made the problem easier. That is we can either use induction by considering a 2n x 2n square and then adding ten blocks to the bottom and the top dimensions, or we can add five blocks to every dimension to get the new 2n+10 x 2n+10 grid. In both cases, it's not easy to find the number of tilings of the given region. So I went to the REACH meeting and discussed the problem with people and concluded that I should probably read Cohn's article. 5 hours
• Oct 22 -- I worked on the domino tiling problem try to play with the nasty formula for the number of tilings and seeing if any nice integer properties could derive from it. This failed after about an hour. Then I went to the REACH meeting. Total 5 hours.
• Oct 16 -- I worked on the hexagon tiling problem for about three hours, mostly from scratch trying to see which tilings might be possible with trihexes and realizing that any triangular hexagonal region can be tiled with the two types of trihexes shown in class but not with solely the straight ones. This was Conway and Lagarias's result. Now I still haven't seen how they associate words to the regions but I'm trying to invent ways in which words can be used to describe tileable regions. We can describe a region as a path through a hexagonal grid where our possible our possible directions after coming to an intersection are left and right. Then any closed path through a hexagonal grid will have the property that there are six more lefts than rights. This is proven easily by induction. However it is not true that any sequence of lefts and rights which has six more lefts is a closed path. So I don't see how this representation is going to help much right now. I've also thought of using the representation of north, south, northeast, northwest, southeast, southwest, but this is probably too complicated because there are many things to worry about when creating a path that will associate to closed region. So I'm stuck right now with trying to use words to solve this problem. Perhaps I'll just try to find the paper by Conway and Lagarias and see what they did. (3 hours) I also read over everybody's webpages and tried to learn some html (1 hour)
• Oct 15 -- reach meeting (2 hours)
• Oct 14 - I read over all of the one page articles and briefly worked on a few of them (3 hours)
• Oct 10 -- reach meeting (2 hours) worked on website (1 hr)
• Oct 8 -- reach meeting (2 hours)
• Oct 6 -- I looked at Gabriel's paper but quickly became lost. So I looked at the other material on cube recurrences written by Jim. (3 hours)
• October 4 -- I looked at the website on Somos sequences, reviewed a thesis and some papers on there (3 hours). We also had the reach meeting today.
• Oct 2 -- reach meeting(2 hours)
• October 1 -- I tried the 3 * n matching problem with labeling each vertex of the graph with a letter / number according to what connections it has, and tried to generalize this to k * n, ultimately looking for the Cayley Hamilton solution that we discussed but I never found it (2 hours). I also worked a little on this webpage(1 hour).
• Sept 27 -- I worked on the 2 * n problem of matchings and found the generating function by looking at recursions. Then I looked at the 3 * n problem and realized that a solution with recurrences would be pretty messy, and even more messy for k * n(1 hour).
• Sept 26 -- reach meeting (2 hours)
• Sept 24 -- first reach meeting (2 hours)