SSL Minutes Thursday, March 25 2004

Notetaker today: Sam
Snacks today: Emilie

Notetaker for next time: Hal
Snacks for next time: Sam

ANNOUNCEMENT: There will be a party after Monday's Math Club meeting on account 
of the high praise received regarding the NSF meeting.
It will be after Jesse Beder's talk on Braid Groups and all are invited.

Paul asks a question about edge variables and lifting recurrences into higher 
dimensional contexts.
Jim: For example, in the following recurrence we want to find suitable 
B,C,D,E so that

Ba(n)a(n-k) = Ca(n-i)a(n-k+i) + Da(n-j) + Ea(n-k+j)

This will help us understand combinatorial data of the recurrence which is 
hidden (by the fact that certain "edges" have weight 1).

Sam: I found the algebraic identity which forms the last piece of the puzzle 
in the formulas for P_n and Q_n.

Jim: Now that we have a finished proof, we want to work on writing this up. 
The next order of business may involve looking at 1) connection with the 
continued fraction algorithm, 2) combinatorial proof, and 3) q-analogs.
In fact, during the rest of the term, we should shift our focus towards 
documenting the results we have found. People should work together, read 
each other's write-ups, and unify their results.

Paul asks about what combinatorial data we can glean once we have faithful 
polynomials.
Jim: It turns out there are situations where not all combinatorial properties 
can be deduced from the faithful polynomials straightforwardly.

Example given is a straight snake graph, where all the vertical edges have 
weights, and the horizontal ones are "weightless," (i.e. weight 1). 
By putting in extra variables, we can get more clues about the combinatorial 
picture, vis-a-vis the revealing of these hidden edges.
My previous e-mail fully delineates this idea for a specific sequence.

There are some LaTex questions floated about.
Carl asks about greatest integer symbol (which is \lfloor or \rfloor), and 
we also have \lceil and \rceil).

Jim points out the utility of \eqnarray for lining up a bunch of equations.
Jon answers Hal's questions about limits of summation in a fractional 
expression: trick is to use \sum\limits forcing them to be on top.

Martin discusses some of his computer results regarding the Markoff mod p 
graphs. He has written programs to calculate the number of triples which 
solve the Markoff relation and the order of the automorphism group of the 
graph.
For the Markoff relation, |Aut(G)| = 24 regardless of p (once p is large
enough).

He has also studied the Markoff brothers, Chico (1,1,2,4) and Groucho 
(1,2,3,6).  He has found that whenever the number of triples solving 
either the Chico or Groucho relation is the same as the number of triples 
which solve the original Markoff relation, then the order of the 
automorphism group is 24. Otherwise, its order is 8.  Not sure what this 
result entails exactly.

One thing to notice is that in the original Markoff relation (1,1,1,3), we 
can permute the variables x,y,z. So we can get 3! = 6 symmetries of the graph 
this way. Also, we can change the sign or any *two* variables (either (x,y), 
(x,z), (y,z)) or none at all. This gives us 4 more symmetries. So we have 
6*4 = 24, the order of the automorphism group.

Jim: Initially we were interested in studying possible expander properties of 
these graphs. Here we might want to consider the spectrum, particularly the 
"spectral gap": the difference between the two greatest eigenvalues.

Martin will see what he and Magma can do in terms of programming, and will 
do some research regarding an explicit algorithm for the expander property.

Paul and Emilie will be working on writing up some of their results regarding
the generalized recurrences exhibiting Laurentness.
And Paul will also work on writing up some results regarding cube snakes.

(BREAK)

We reconvene in B107 to work on our group projects. Hal, Carl, Jon, and Sam 
discuss aspects of writing up Newton's method and the perspective the authors
should adopt.
Jim, Emilie, and Paul work together on the recurrences.