SSL Notes for meeting #13 (10/21)
Today's note-taker: Carl
Next Time: Emilie
Today's snack-provider: Martin
Next Time: Carl
In B107 lab, we more or less jump into things (no real administrative stuff
off the bat). People get 5 minutes each to run the show -- Jim wants to fade
into woodwork. [Short discussion over who gets to go last and who doesn't
want to go first]
ABBY anyone get through any articles?
HAL looked through Kuo, understands it mostly. and was of course interested
in the Aztec diamonds -- the numbers seem to work out in special cases
ABBY spent time on Itsara
HAL proof entails _building_
MARTIN (talks about Kuo) bipartite color...
ABBY what's bipartite mean?
MARTIN can divide it into two pieces:
have vertices which are your points, and connections (edges)
divide vertices in graph into two sections where no edges
lie wholly in one half
o----o (circles the left half and circles the right half)
\ /
\/
/\
/ \
o o
\ /
\/
/\
/ \
o o
JIM the reasons snakes are bipartite is: colorings (colors dots on left
half of the above graph green, right half blue), it's not so important
that you can draw a nice circle around each half.
[demonstrates this with snake graph, the 'shaded / unshaded' scheme.]
*--o--*--o
| | | |
o--*--o--*
SAM Itsara paper: observations of snake graphs in general
_
_|_| _ _
|_|_| <--> |_()_| have the same # of matchings
This seems clear, but what about something like:
_ _
_|_|_|
_ _|_|_| ?
|_|_|_| This doesn't seem quite as intuitive...
HAL (answers collapsing question)
_ ___ ___
It should turn into this: |_|_((|))_|
Which has an edge repeated 5 times, instead of just doubling it.
(5 min rule comes into effect)
PAUL wants Hal to talk about splitting cubes into two halves
(Jim intercedes, asks Hal for clarification)
HAL takes to board.
A E' B
+---+ +---+ A & B are graphs, E' is a set of edges connecting A & B
| | | | such that no two edges in E' share a common vertex.
| *---------* |
| | | | Phi is any subset of E'
| *---------* |
| | | | M(A,Phi) is the matchings of A with Phi removed
| *---------* |
| | | | For the total matchings of the graph G,
| *---------* |
| | | | M(G) = SUM_{Phi in E'} M(A,Phi) * M(B,Phi)
+---+ +---+
... in 3D it probably won't be as nice cut to divide vertices though..
CARL asks Hal for clarification about E' and Phi.
asks Jim to finish the collapsing snake stuff:
JIM the idea is replacing a chain of 2 edges with a chain of 0 edges.
First consider going from 3 to 1:
(A*--*--*--*B) --> (A*--*B)
and 2 to 0 is similar: (A*--*--*B) --> (A*B)
The thing to keep in mind is that the "combined vertex" inherits all
connectivity, so:
_
_|_| _ _ _ _
|_|_| -/-> |_|_|, but rather --> |_()_|
MARTIN last time promised to write program: takes transfer matrix and
vectors, turns into rational generating function, enumerates
matrices of given size that could represent transfer matrices.
(goes to board to explain what transfer matrix is)
it's a little bit hard for me to explain exactly what a transfer
matrix is... it can be applied to snake graphs and domino tilings.
let's say we're only looking at straight snakes:
+----+
| +--|--+
| |. | .| . .
| +--|--+
| +--|--+
| |. | .| . .
| +--|--+
+----+
which is the same as:
+----+----+----+----+
| +--|--+ | | |
| | | | | | |
| +--|--+ | | |
+----+----+----+----+
| +--|--+ | | |
| | | | | | |
| +--|--+ | | |
+----+----+----+----+
we want to specify what we do in one column by what we've done in the last
transfer matrix for this looks like [1 1] (ie, the A marix we've seen)
[1 0]
EMILIE looked at papers, had question on Itsara paper. Can anyone explain
the U and V thing? (on 2nd page)
JIM takes the board:
*---*---D
/ \ / \ /
*---B---C---* Want to label all segments according to their slope:
/ \ / \ / v for /, w for \, and u for ---
A---*---*
Use the Ptolemy relation for weight from A to B:
f(AB) = (u^2 + v^2)/w = f(CD)
so, f(AC) = ... = ... = (v^2 + (u^2)(w + (v^2 + u^2)/u)/v)/((u^2 + w^2)/v)
= ... equalllllls, well, ok, does anyone have maple up?
MARTIN yeah, I do. (lets maple try.) It doesn't look very simple though.
JIM hmmm... well, f(AC) * f(BD) = f(BC) * f(AD)
Anyway, this is neat because it has geometric meaning for hyperbolic
geometry, and has combinatorial significance also...
STEVEN I'd like to see someone demonstrate Kuo's 4 vertex thm
HAL I vaguely understand it, I could show you if you want. (ok, yes)
a--*--*--b
G = | | | |
d--*--*--c
ok, we want to superimpose: M(G) and M(G-{a,b,c,d})
*--*--* * *--* * *
e.g. | | or () () |
*--*--* * *--* * *
are examples of H in H*, where H is a double graph,
and H* is the set of all double graphs
HAL wants to talk about other stuff, goes to computer
(the gasket / circle radius triples thing)
messes around in maple,
w:= (x,y,z) |--> simplify(xyz/(xy + yz + zx (xyz(x+y+z))^(1/2)))
simplifies surprisingly well. (all this stuff is on Hal's SSL site,
which he will continue to update as he finds stuff)
says: when I asked Jim 'how you define cluster algebra' he said,
'need to use formal variables.' when I try this in maple,
it doesn't look so nice when it tries to simplify.
JIM says 'well, ... it should' ... we could still use Descartes thm to be
smarter than maple. there is a better way to write these in that it
looks nicer.
STEVEN why not replace the radical with another formal variable?
just pretend it's independent of the first three curvatures.
Jim seems to think this is worth trying.
Yeah, let's eat. (Thanks Martin for the cookies and lemonade!)
-- After Break --
Steven and Jim are stuck in B107, everyone else goes log wild in B105
(just kidding, we talked more about cluster algebras than logarithms)
Hal leads discussion on n-cluster algebras from article, Sam and Paul briefly
collaborate on top secret math project, which will not be included in minutes
What is k-regular? every vertex has k vertices coming into it.
EMILIE are these real geometric connections or just abstract 'connections'?
HAL what's the difference?
Hal draws two examples of 3-regular objects:
#1: a cube
#2:
*
/ | \
/ | \
*---* | *
// \ * /| (somebody) What _is_ that?
/ / \ / \* |
*--* *-* | |
\ \ / \ /* | PAUL it's an arrowhead!
\\ / * \|
*---* | *
\ | /
\ | /
*
We reconvene:
Links to relevant articles by Fomin and Zelevinsky:
Cluster algebras I: Foundations
http://front.math.ucdavis.edu/math.RT/0104151
The Laurent phenomenon
http://front.math.ucdavis.edu/math.CO/0104241
JIM This is useful framework, for understanding variables.
If you want to know where the numbers come from, you want to look at
the polynomials -- then replace the variables by 1's...
(this can offer insight)
Look at Fibonacci polynomials and whether they fit into cluster algebra