Trevor's REACH Page

Time Spent through mid-January:  122 hours

Vacation, through 1/10
Time this week (as of 12/4): 8     Cumulative time: 122

Over the vacation I just worked a little on trying to extend and prove a particular result derived from the octagon work.  I didn’t have very much time as I was attempting to find a thesis topic and thinking about unrelated problems in combinatorics, in particular problems in topological graph theory and the construction of expanders.

Week ending 12/20
Time this week (as of 12/4): 12     Cumulative time: 114

This week I just worked on trying to extend and prove a particular result derived from the octagon work, and in particular applied some of the methods Jim and Henry Cohn suggested.

Week ending 12/13
Time this week: 18     Cumulative time: 102

This week I worked on some results derived from the Aztec octagon work, and finally finished a draft of the aforementioned 1 mod 4 proof.

Week ending 12/6
Time this week (as of 12/4): 8     Cumulative time: 84

This week I continued to play around with approaches to the Aztec octagon stuff, and am now continuing some work I did last week to prove algebraically that the face- and edge-variables approaches to determining the number of domino tilings of a 2n-by-2n square via summing the weights of a suitably-weighted Aztec diamond are equivalent.  I have also, finally, begun writing up a proof that the number of domino tilings of the n-by-2n rectangle is congruent to 1 mod 4.  I hope to have a draft finished by the end of the week, so that I can edit it over Christmas break.

Week ending 11/29
Time this week: 12     Cumulative time: 76

This week I played around with several approaches to the Aztec octagon stuff, and did a few long octagon recurrence computations by hand to gain a feeling for what’s going on.  I also started thinking about how to write up some results from last year.

Week ending 11/22
Time this week: 8     Cumulative time: 64

This week I played around with the Aztec octagon stuff.

Week ending 11/15
Time this week: 6     Cumulative time: 56

This week I attended meetings, and continued to work on some combinatorial approaches toward the p-adic stuff. I did some related work on tilings of hexagon graphs, but need to gather more data before continuing thinking about this. This coming week I am planning on finally getting to writing up some of the work I did last semester. (However, for the past three weeks I have been extremely busy with fellowship and grad school applications, and catching up on the resulting backed-up course assisting. I hope to return to spending a lot of time on REACH, and I will make up missed time over Christmas break, when I will have nothing else to do.)

Week ending 11/8
Time this week: 6     Cumulative time: 50

This week I attended meetings, in which I worked on some combinatorial approaches toward the p-adic stuff. I also spelunked in a possible cave (because of possible irrelevance) for a couple of hours, but I plan on spending another couple of hours this weekend and if anything comes of it I'll mention it in next week's entry.

Week ending 11/1
Time this week: 5     Cumulative time: 44

This week I attended meetings and worked slightly more on coming up with general strategies for the p-adic group's work.

Week ending 10/25
Time this week: 8     Cumulative time: 39

I attended meetings, during which I explained some of my past research on computing thenumber of domino tilings of rectangles with various size constraints, and continuedconjecturing about the number of domino tilings of k-by-n rectangles.  I began reading a paper of Richard Stanley's on the latter conjectures.  Ialso played with computing the number of domino tilings of rectangles with various sizeconstraints via a combination of the exchange principle and linear algebra, but did notmake much progress.  Oh yes, I'd like to mention at this point that the next twoweeks will be extremely busy for me (as was this past week) with fellowship applications,graduate school searching, getting my thesis off the ground, and midterms, but I'm going to try to dedicate as much time to REACH as I can.

Week Ending 10/18
Time this week: 11     Cumulative time: 31

I attended meetings, and worked on finding a recurrence relation for the number of tilingsof the 2n-by-2n square, unsuccessfully, by inducting over a region of fixedwidth about the diagonal and by attempting a bijection between domino tilings of the 2n-by-2nsquare and sequences of (0,1) matrices. I also worked on computing recurrence relationsand coming up with conjectures for the number of tilings of k-by-nrectangles.

Week ending 10/11
Time this week: 9     Cumulative time: 20

This week I attended meetings, did the homework reading, and continued reading on p-adicanalysis in Robert and also in: F. Q. Gouvea, p-adic Numbers, Berlin:Springer-Verlag, 1993. I read: L. Pachter, Combinatorial Approaches and Conjectures for2-Divisibility Problems Concerning Domino Tilings of Polyominoes, The ElectronicJournal of Combinatorics 4 (1997), #R29, and began attempting to compute the numberof tilings of 2n-by-2n squares by extending its methods. I also beganreading: H. Cohn, 2-adic Behavior of Numbers of Domino Tilings,arXiv:math.CO/0008222 v1, 30 Aug 2000.

Week ending 10/4
Time this week: 6     Cumulative time: 11

This week I attended meetings, worked on homework problems, and tried to generalize.

Week ending 9/27
Time this week: 5     Cumulative time: 5

This week I attended meetings, worked on homework problems, and started reading up on p-adicanalysis in: A. M. Robert, A Course in p-adic Analysis, New York: Springer,2000.