This semester I worked with Harvard student, Trevor Bass, to determine a combinatorial model for rectangular and octagonal figures embedded within the aztec diamond.
We were able to determine a recurrence relation to determine the number of tilings of Aztec octagons using Kuo's condensation.
A draft of the paper we are working can be read **here**

This semester I am doing research in Combinatorics at Harvard University with Professor James Propp under the supervision of the MIT Professor Richard Stanley. Professor Propp has started the REACH, Research Experiences in Algebraic Combinatorics, at Harvard to investigate various phenomena and patterns in Combinatorics, especially tilings . More information on his research can be found at: http://www.math.harvard.edu/~propp/reach/

I will be working on the p-adic properties of enumeration of domino tilings.

**History**

Mathematicians John, Sachs and Zernitz found a formula for the exponent of
two in the number of domino tilings of an m x n rectangular checkerboard based
on the prime factorization of that number. This work was continued in 1961
by In 1961 Temperley and Fisher, and Kasteleyn. Pachter later showed that
the number of tilings of a 2n x 2n square is always 2^n by an odd perfect
square. Using this information Cohn was able to show that the number of tilings
of a 2n x 2n board shows 2-adic continuity.

**Work Plan**

I will be working with a group to continue and extend Cohn's work. We shall
look at 3-adic properties of domino tilings of an m x n board and then we will
extend it to look at p-adic properties. The plan for this semester is:

- The first few weeks will be spent becoming familiar with the techniques needed to approach such a question. This includes learning about domino tilings, p-adic theory and reading Cohn's paper. Pachter and Kasteleyn's papers will be studied to grasp their methods in order to use them in our research.
- Then for the remainder of the semester, work will be done to attempt to prove Professor James Propp's conjecture that N(2n,2n) is divisible by 3 whenever n is congruent to 2 mod 5.
- Finally work will be done to show 2-adic continuity of N(n, 2n) as a function of m and to prove that N(n, 2n) is congruent to 1 modulo 4.

I am working with:

Siddique Khan

John Gonzales

Amanda Beeson

Trevor Bass

Our Reach research has slightly changed direction. Eric Kuo
wrote a math paper describing how condensation can be used to determine certain
properties of an Aztec diamond. We shall use Kuo's condensation, specifically,
how the octahedral recurrence, to determine the number
of tiling of an m x n rectangle embedded
in the Aztec Diamond. We shall also look at the relationship between the
weighting the faces of the diamond and weighting the edges of the diamond,
to determine why the both give the same answer even though they have different
starting conditions.

__Weekly summary of work done:__

Week 1: September 22nd - 28th (5.5 hrs) |
Week 2: Sept 29th - Oct 5th (4 hrs) |
Week 3: October 6th - 12th (2.5 hrs) |

Week 4: October 13th - 19th (7 hrs) |
Week 5: October 20th - 26th (7 hrs) |
Week 6: Oct 27th- Nov 2nd (7 hrs) |

Week 7: November 3rd - 9th (4 hrs) |
Week 8: November 10th -16th (8 hrs) |
Week9: November 17 - 23 (7 hrs) |

Week 10 November 24 - 30 (9 hrs) |
Week 11 December 1st -7th (9 hrs) |
Week 12 December 8th - 14th (4 hrs) |

Total hours to date: 74 hrs

kezia@mit.edu

617-225-8419

IM: keziac