TOTAL SO FAR (Spring Semester starting in early January): 31 hours

     Work for the Spring semester has consisted of finishing up the reciprocity theorem paper, working on extending Speyer's                   matrix method, and attending REACH meetings. Speyer's method does not work so seamlessly for non-perfect matchings so it may not generalize as was expected at the end of fall and the beginning of this semester. In that case the goal will be to get Speyer's work written up, incorporating such things as bow-tie graphs.    

 

 

 

 

TOTAL SO FAR (Fall Semester): 77 hours

WEEK 10 11/22 - 11/28

• I looked around (especially in the arXiv database) for articles on tilings to find out whether somebody has done matrix encodings, or worked on similar problems. I didn't find what I was looking for. I also used Speyers hooks idea to encode Kyle's type of graph but adjunction does not work. I will need to talk to members of my group so that they can explain what I am doing wrong.
• Total: 7 hours

  WEEK 9 11/15 - 11/21

• I have been learning Maple to write some code to test my interactions conjecture. I am also continuing to play with Speyer's idea. This was a pretty busy week for me.
• Total: 6 hours

  WEEK 8 11/8 - 11/14

• Jim said that the linear recurrences for the the m x n matchings increase exponentially as m increases but he was wandering whether the number of interaction increases exponentially as well. Obviously, if it doesn't my whole idea about the relation between the number of interactions and the sum of the absolute values of coefficients in the recurrence relations only works for the first few cases. The answer seems to be simple: an  interaction is defined as the number of matchings between the last two columns of the lattice. What we really have here, then, is a 2 x n lattice where n increases as m increases. But the number of matchings in the 2 x n lattice increases exponentially with n so the number of interactions increases exponentially with m (the height of the graph/lattice). In addition I have been looking at modifying Speyer's article on the matrix approach. I thought I found a way to encode graphs in matrices but it didn't work in practice. Maybe I should go back to Speyers approach with hooks. 
• Total: 12 hours

  WEEK 7 11/01-11/07

• Read Speyers article on a matrix approach to the perfect matchings problem. Trying to see if it can be generalized to less than perfect matchings and Kyle's idea about hedges, vedges, ect. I am realizing that if we adopt a matrix multiplication approach (with multiplication defined the usual linear algebra way) the result is not going to be as clean as in Speyers article where multiplying two matrices that each encode a graph yields a third matrix that correctly encodes the resulting adjoined graph. Speyer also indicates that taking inverses enables one to go backwards, getting the previous graph. I believe that we will need to represent adjunction in such a way that there is a constant (correction factor) matrix (possibly a vector) that will be added on to get the desired result.  
• Total: 8 hours

  WEEK 6 10/28 -10/31

• Kyle presented his combinatorial object and we looked at proving adjunction for it. I went through his preliminary write-up to get a better feel for what he came up with.
• Total: 3 hours

  WEEK 5 10/21 - 10/27

• Decided to work on combinatorial reciprocity for the monomer-dimer model on a k x n grid. Spent three hours rereading article on reciprocity of domino tilings. Spent two hours playing around with Kyle's 11 polynomials (I am trying to understand whether it is possible to simplify the model into fewer partitions). Spent four hours in REACH meetings and two more hours finally getting this page to work on the Brandeis server.
• Total: 11 hours
  
  WEEK 4 10/14 - 10/20
• Spent three hours reading Generatingfunctionology. Spent an hour looking over one page articles and starting to read Gabriel's article (this may take some time). Spent two hours in REACH meetings (missed one meeting).
• Total: 6 hours
  
  WEEK 3 10/7 - 10/13
• Came up with idea about the recurrence relations that govern the number of matchings in a m x n lattice where m is constant. Define an interaction to be a matching that involves only the last two columns of the lattice. The number of such interactions seems to equal the sum of the absolute values of the coefficients in the recurrence relation! I spent about two hours playing around with this idea. Read Jim's aricle (two hours). Spent four hours in REACH meetings.
• Total: 8 hours
  
  WEEK 2 9/30 - 10/6
• Spent two hours working on problem and a total of three hours reading Generatingfunctionology by H. Wilf. Spent four hours in REACH meetings.
• Total: 9 hours
  
  WEEK 1 9/23 - 9/29
• Spent two hours working on problem (but instead of proving recurrence made mistake of finding the generating function). Spent an hour reading Generatingfunctionology and four hours in REACH meetings.
• Total: 7 hours.