Minutes from REACH Meeting, 12 March 2002, 3-5 PM.

1. Minutes: it was agreed that it made the most sense for the minutes to be
more of a record of what was discussed than an exposition. Hence, it will be
of use for links to people's posted write-ups, and the minute-taker has
responsibilities shifted to primary nagger of those doing the write-ups.

2. Either the bracketed remarks in laurent.html are to be written up at some
point, or they are to be replaced with a explanation to the effect that
REACH has better things to do.

3. All discussions of logistics ought to be in the minutes. For example,
cookie-buying will be covered from now on, as REACH is expected to supply
its own cookies.

4. The future of REACH: (a) it may be at Brandeis. In this case, public
transportation would take about 1 hour each way, and meetings would need to
be scheduled around the rail timetable. (b) it may be at MIT. In this case,
the unofficial acronym would be CRAM IT: Combinatorics Research At MIT. (c)
It may be at Harvard, at least partially. In this case, a room and computers
would need to be secured from the math department here.

5. Alternating sign matrices as a possible field of research next semester;
development of unusual symmetry conjecture to prove ASM theorem.

6. Kuo's proof seems to work for three-term Somos sequences. David Speyer
wrote up some results for Somos-3 (see below), and a pair of
combinatorialists have gotten similar results with Somos-4.

7. There may be a relation between Chris Douglas's work and REACH results on
Somos-4, but the diagrams are not the same, at any rate.

8. Gabriel is working out (or has already worked out) a change of variables
in the Somos-3 explanation.

9. Jim is planning to fix up the buggy face-variables/edge-variables program
he showed us last week.

10. Recurrences in which every other term comes from a separate recurrence
were discussed. It may be possible to integrate Somos sequences in this way,
using different starting conditions (because if one starts with -1s instead
of some 1s, the polynomial remains Laurent and integers are still produced).
Perhaps an algebraic structure can be placed on nonlinear recurrences
satisfied by the sequences. The problem is that quadratic recurrences are
difficult, in that there would be two different notions of multiplication:
multiplication of values and composition of operators. Lionel and David may
be thinking about this.

11. When given the choice to discuss tilted initial conditions, Somos-3, and
hypergraphs, the group decided to discuss Somos-3. The other two topics were
not discussed at this meeting, and remain open.

12. David gave a comprehensive presentation about the progress towards a
Somos-3 solution that has been made thus far. Notes can be be found on the
web, although perhaps not at the time of the writing of these minutes. It is
the minute-writer's task to pester him.

13. David is charged to fix several typos on his diagram. Specifically, the
0,-6,-8 in the lower left that should be 0,-8,-8, and some others that he
has marked down, relating to lapses in the constancy of the second
edge-variable coordinate.

14. The group discovered the exponent for the hexagon faces in the
face-variables version of the diagram: it corresponds to 2 minus the number
of matched edges of the hexagon.

15. A general note on converting edge-variable polynomials to graphs: there
was some initial difficulty based on the fact that hexagons were perceived
to be squares, but when they were reinterpreted as hexagons everything came
out all right. A question to ponder: how can one tell when a set of
seemingly paired variables do in fact correspond to edges on a square?

16. The meeting was adjourned.