Minutes for 2/11 (from Kezia Charles' notes)

Note taker for Thursday is Jennifer Krishnan. 
Next Tuesday is demo day. Please let Jim know if there are any useful
websites and programs he should show to the new people. 

We went around the group and said our name as well as a real
mathematician with a striking or silly name

Open discussion:
Your ideas for making REACH better? 
Jim suggested that we spend less time on the meeting talking about
possible projects. He is in the process of writing a document with all
the upcoming projects. The paper should be done within the next 2 weeks. 

Logistical questions? 
Jen asked if there was a solution to non-Harvard students logging in to
the computers. For right now, we will have to continue with the present
method. 

We worked on the homework problem and Jim broadened the question to look
at triangular grids of height two. We looked at the general problem of
"guessing the right denominators", because some of the terms may be
canceled, so we do not always know all the correct terms. E.g. 1/x = 
y/xy; combinatorially the y may be important.

Jim suggested that the reasons not all the research went smoothly last
year may have been:
-       The problems weren't well suited for the group
-       There was not much communication between groups
-       Groups consisting of people from different schools tended 
        not to meet outside of Reach hours. 

GROUPS:

Monomer: Kyle, Nick, John and Ilya
They wrote a paper, which will be submitted to the DMCS conference this
weekend. We can find it on Kyle's page. 

Two-dimensional Recurrences: Reid and Greta
There is the frieze recurrence e = (bd + 1)/ a 
(see http://jamespropp.org/bilinear/antifrieze)
and the variant e = (bd + c^2)/a which has also been studied.
So the group looked at the recurrence e = (bd + c)/a 
("number fences") and found that one gets 
Laurent polynomial with all coefficient equal to 1. 
(David Speyer also looked at this recurrence; see his article
http://www.math.harvard.edu/~propp/reach/speyer/stuff.{tex,dvi,ps,pdf} )
Combinatorially, these Laurent polynomials are counting paths 
on a triangular grid of height 2.  There is a preliminary
write up on Reid's web page. 
The number fence recurrence should also give us combinatorial
information about the recurrence S(n) S(n-4) = S(n-1) S(n-3) + S(n-2) .

Domino tilings of a mobius strip 
No group worked on this project last semester. However David Speyer had
done some research on it. Jim has linked his work to the website. 

P-adic properties of Domino tilings: Trevor, Kezia, Amanda, John and Siddique
This group was supposed to look at the p-adic properties of domino
tilings and try to prove Jim's conjecture that if n is 2 mod 5 the number of
tilings of a 2n x 2n grid is divisible by three. However, the group did
not have enough background in algebraic number theory and p-adic analysis.
So the project was changed to researching the properties of domino tilings 
in an Aztec diamond graph using Kuo's condensation. They have to start 
working on their paper.

Hexagon tilings group: Jen, Dan, Dave and Adri
Given a region of a hexagonal grid and a set of hexagonal tiles, the
group worked on determining an algorithm to tile the grid. However, they
are now working on an algorithm to tile a triangular grid instead of a
hexagonal one, since the methods that have already been developed on a
square grid seem more likely to be applicable.

Gale-Robinson:
No group eventually ended up working on the Gale Robinson recurrence.
The Somos 6 numbers and Somos 7 numbers which are given by recurrences
of the form AH = BG + CF + DE have been looked at by Gabriel and David. 
However there is still a lot of work to be done on Somos 4 and Somos 5 
numbers which are given by recurrences of the form AF = BE + CD. There 
is still research to be done in terms of crosses and wrenches. Jim will 
talk some more about it. 

Groves:  Gabriel, Inna, Ananda, Anton and Anna
This group determined that there are some local moves, which allowed you
to go from one grove to another. They tried to use Kuo's condensation to
show the enumeration of the groves. Inna is working on writing a Java
applet to show the local moves on a grove. Kyle wrote Maple code to
produce big groves, and is looking for someone who knows Java to convert
it. 

Markov numbers: Rui and Andy
They were able to find a combinatorial model for the Markov numbers
based on a perfect matching of certain graphs. They want to look at
using Kuo's condensation to cut and reshuffle the graph. There is a
preliminary write up of their work on Andy's website. 

Jim said that the work of the Markov numbers group is related to the
recurrence: S(n) S(n-3) = (S(n-1))^2 + (S(n-2))^2 (due to Dana Scott).
There are variants of this recurrence, like S(n) S(n-4) = 
(S(n-1))^2 + (S(n-2))^2 + (S(n-3))^2 , which might also have a
combinatorial interpretation.

We looked at Section A of the Research that Jim would like us to do this
semester. It is on the website. 
Reciprocity: Using the Temperleyan of the graph. 

1.      The Temperleyan of m and n T(m,n). One notices that if he runs the
recurrence backwards in m then n to get to T (m', n'), and runs the
recurrence backwards in n then m to get the same T(m', n'), one gets a
different answer. The research is to determine why this happens and come
up with a combinatorics explanation. 
2.      (n C k) = (n C n-k) and (n C k) = (-1)^k (-n+k-1 C k) where 
(n C k) = (n(n-1)...(n-k+1))/k! However this gives that (n C k) = -(n C k).
The research would be to determine a reciprocity picture to determine
why the reciprocity operations do not commute. 

3.      The "last frontier" for exact counting of perfects on a square
grid involves having the square grid live on a cross cap. If there is an 
mxn square grid on a cross cap, find a formula for the number of matchings. 
Does it work running the recurrence backwards as it does running it forward?

4.      Determining a random tiling of a large region with a ribbon tile. 

(More information can be found on section A on the website)