SSL Minutes

3 May 2001

Boytcho mentions that he's extended Rachel's theorem concerning lack of 2-distal hex-FPLs to semi-regular hexagons.

Jim says that he now officially believes my proof of the Baxter permutation theorem. I asked him why now and not several weeks ago, especially since he hasn't yet read my paper. So everyone is rewarded with exactly one hour of pay. Of course, I'm already up against the 20hour/week limit.

Charlie the post-doc logician is visiting. That brought up the subject of how logic relates to tilings. Jim started talking about a theorem: There is no algorithm that will say if an arbitrary set of simply connected polyominos can tile the infinite plane. More precisely, there is an algorithm that can show that a set of polynomials can't tile the infinite plane, but it may take an arbitrary length of time to find a counterexample. So the theorem is co-CE.

Non-trivial theorem: If, for each N, you can tile the NxN square, allowing for overlaps over the boundary, then you can tile the infinite region. This gives the co-CE algorithm. (Proof to follow later.)

Rachel has been bad about filling in timesheets. She also has been cautious about not including hours spent thinking without writing. I mentioned that I am not a good accountant, even though I come from a long line of accountants. Then Rachel said something about coming from a long line of women. Then we discussed genetics for a while. Something about X and Y chromosomes and mitochondrial DNA. I really forget where that discussion was going.

Hal is taking notes.

Going around the room

Rachel is not as sure of her proof about 3-distal hex-pairings as much as she was a few days ago, but she's working on it.

The t-shirt has yet to be compiled.

Boytcho has what might be a proof of how 2-distal hex pairings can not exist.

But first, something for the glossary:

  • ASM FPL - square fully packed loop-state generated from an ASM. See notes from 20 Feb 2001 for complicated definition. Easy definition: color the terminals blue and red, alternating around the edges. start drawing the inside from left to right, up to down. if there is a zero in the ASM, make the line bend, and otherwise, make the lines go straight.
  • Hex FPL - hexagonal fully packed loop-state generated directly from a matching on a hexagon lattice. Simply color in those edges which are not involved in a lattice.
  • proximal pairing - pairing of terminals on an ASM FPL or a hex FPL such that 1-2, 3-4, 5-6, etc OR 2-3, 4-5, 6-7, etc.
  • proper proximal pairing - pairing of terminals on an ASM FPL or a hex FPL such that 1-2, 3-4, 5-6, etc
  • distal pairing - pairing of terminals on an ASM FPL such that one terminal (call it a) goes to the farthest possible terminal (call it b), and a-i pairs to b+i for all i
  • 2-distal pairing - pairing of terminals on an Hex FPL that is equivalent to the definition of distal on an ASM FPL. Has been shown not to exist.
  • 3-distal pairing - The pairing of terminals on an Hex FPL that contains three equal sized "cables."

Boytcho's proof about lack of 2-distal patterns in semi-regular hexagons starts by assuming there is a 2-distal pattern and picking the pair that are next to each other. then following the lines around the edge. One will find that somewhere before getting to the opposite side, you are forced to pair two adjacent terminals.

But Jim pointed out that he had made a "best-case" assumption during the proof. then he offered an example of how such a move could turn bad. He had a bad proof that you can't pick a subsequence of {1,2,3,...20} where no two elements differ by 3 or 5. The naive thought is that the best case is {1,2,3,9,10,11,17,18,19}. But the real answer is {1,3,5,7,9,11,13,15,17,19}.

Jim has been wondering about the FPL/DPFL terminology question. He needs to get that straight for his talk in Paris.

Mike has successfully defended his dissertation. He also has looked at other eigenvalues of the Razamov-Stroganov matrix. He has seen inklings of interesting patters that he doesn't understand yet.

On another note, Rachel will be in charge of the t-shirt over the summer, because Jim will not be around.

I give a talk about the history of Hamiltonians. I'll skip that information right now, except to say that a Hamiltonian can describe a mechanical system, where the matrix elements correspond to some potential energy.

Abe bitched about C++. Somebody mentioned that somebody said that C++ was a practical joke played on programmers. Abe has also been playing with CVS. He expects the up-down checking program to be done over the summer.

Dominic is switching schools. He's looking at interspersed ASMs; maybe he's finding things. Dominic asked Jim about the other way to form pairings from a TOAD.


If you look at the lines we get versus height functions, you see that we get a contour map. The contour shown represents the contour of height 3.5. There Polynomially many pairings for some reason. We noticed interesting things that go on in the frozen regions.

Nick has been playing with Baxter permutations and helping Abe with the Compatibility Graph Program.

Somebody mentions that there are a lot of mathematicians in France.

Geir forgot to bring the drinks he bought.

Dan asks about the compatibility graph. Then he asks about compactness and tilings.

Jim gave a beautiful diagonalization proof of that non-trivial theorem he mentioned before.

Kristin asks about how the Razamov-Stroganov implies something. Then we talked about the eigenvalue problem.

Then we all went and had a picnic under Van Vleck's tower. That was successful.