Minutes for 10/24 (notetaker: Anna)

Administrative stuff:

- The next note-taker is Rui.
- Nick volunteered to burn three more Math 192 CD's.
- Jim's grant administrator is leaving, so everyone is encouraged to 
suggest questions to ask him. 
      One question suggested : how to make direct deposit?

We are at the point now where we have brocken up into groups working on the 
one-pagers.  Some of the one-pagers haven't been picked up by anyone.  
Here's more information on them:

Combinatorial reciprocity for domino tilings:

- on 2n-by-2n projective planes:
  This problem might be amenable to Kasteleyn method, which is a form 
of discrete Fourier analysis.

The relevant articles are
"Dimers and Dominoes" jamespropp.org/domino.ps.gz
  which demonstrates Kasteleyn method;
"Trees and Matchings" www.combinatorics.org/Volume_7/Abstracts/v7i1r25.html
applications of the Kasteleyn method and of the
   matrix-tree method, both of which involve discrete Fourier theory on 
   graphs.  One of these examples has interesting 3-adic properties;
Article by Lu and Wu, who used this method to prove reciprocity on 
cyllinder and mobius strips.

- on cylinders and mobius strips:
  Is there a more direct way of seeing it then by using Kasteleyn method?

Relevant article
last year's work by David Speyer 
www.math.harvard.edu/~propp/reach/speyer/


Gale-Robinson polynomials and their combinatorial meaning:

Their recurrence is closely related to cube recurrence, so groves can be 
used in finding the combinatorial objects corresponding to the G-R 
polynomials.  There have been several combinatorial objects proposed for 
these polynomials in certain cases : Pine-cone graphs and Crosses & 
wrenches.  It would be interesting to find their connections to groves.
(Groves arise from solutions to the general 4-term G-R recurrence;
pine-cone graphs and Crosses-and-wrenches arise from solutions to
the general 3-term G-R recurrence.)

Related to groves:
Do symmetric reduced groves have a recurrence?  This problem can be looked 
at algebraically, by using the one-to-one correspondence between groves and 
monomials gotten from the cube recurrence.  The symmetric groves can be 
picked out by setting appropriate weights to the variables.


Related to Somos-4 and Somos-5 sequences:
Last year, REACH participants came up with combinatorial objects 
corresponding to these sequences.  It would be interesting to know if a 
restricted version of those objects also satisfies some Somos sequence.


The rest of the meeting was spent working in subgroups.