Minutes for 4/4/02 by Seth Kleinerman:

Reminders:

-Roberto will talk about the Hirota equation on Tuesday.
-Jim will present some ideas about discovering quadratic recurrence 
relations, as an analogue of unearthing linear recurrence relations, 
on Tuesday as well.

Meeting Summary:

For the first hour, we worked in the computer lab. Some dead links were
fixed, and perfect-matching-counting programs were stared at in
frustration. Trevor's compilation of the vaxmaple program appears to
work, except run it at a prompt so that when it quits you can port the
output to maple.

Gabriel presented the freehand style of drawing Somos-4 graphs. He
explained how one might get the four slopes +/- 1, 3 without already
knowing the graphs.

Jim said that Federico suggested that the top variables and the bottom
variables in the cube recurrence's three sheets of variables were
similar in their restricted range of values, but distinct from the 
middle sheet of variables.

We went over the link between ASMs and rhombus tilings of hexagons. Jim
raised the question of demonstrating a correspondence between tilings
invariant under all 12 symmetry operations and ASMs of order n, where
the hexagon had side length 2n. Perhaps it can be proven with perfect
matchings? The correspondence is an open question.

Gabriel suggested a sort of "Kuo condensation" of ASMs, where the small
matrix would be added in the center of the large one, then split up into
two medium ones.

The method of producing Hankel or Toeplitz matrices to discover linear
recurrence relations was explained. Since the determinant of the matrix
is zero uniformly wherever one is in the sequence, there is a linear
relation between the rows, and therefore between n entries of the
sequence, where the matrix had order n.