Sunday, March 1, 2009

Gil Kalai's notes on Günter Ziegler's lecture

on the associahedron and its relatives are posted here. A nice introduction and lots of nice pictures of "participants in the associahedron saga" including, for some reason, two of your humble servant. It is good to see many good friends and colleagues, and in particular, to remember fondly Lou's and Bernd's warm hospitality to Misha Kapranov and me during our first year in the US in 1990-91. And I believe, I have first met both Gil and Günter that year, when they were visiting Cornell.

Among other things, Gil (or should I say Gilushka) demonstrates his command of Russian diminutives although Sergey (Seriushinka) needs some more work, I am afraid. :) In any case, many thanks to Gil for a nice posting.


Anonymous said...

Thanks Andrei,

Indeed Sergey introduced me to the Joy of Russian diminutives. I suppose it should be Seriujinka. I am not sure how to translate the sounds into letters. Certainly I need some remindings of the beautiful theory (of diminutives); I remember there were 4 or 5 forms for each name and also some sporadic diminutives to some special names.

Regarding the conjecture that the associahedron is a polytope. I heard about it from Micha Perles (my thesis advisor) and Yaacov Kupitz (who was also a graduate student of Perles.) They found it in Tamari's works and were not aware of Stasheff's work. (I was quite myopic and did not think this problem is as important as it turned out to be.)

Let me also say that without understanding a single word (I guessed a few) I could still felt the spirit of your recent post on Ilya.

avzel said...

Hi Gil,

Very good to hear from you. It's a shame we didn't have a chance to chat during my visit to Jerusalem last May.

Concerning the diminutives, I believe Seriozhen'ka would be the closest to what you had in mind ("n' " stands for "soft n", and I am not totally sure how it should be depicted). This form is probably a bit too much, it seems to be appropriate for the closest relatives only; most of Sergey's friends (like myself) would use Seriozha when talking to him.

> I was quite myopic and did not think this problem is as important as it turned out to be.

I have checked your post again and haven't found any angry comments, so am not sure what you mean. :) As far as I am concerned, your account is accurate as much as any historical claim can be totally accurate.

By the way, despite all the progress, the problem of finding the diameter of the associahedron still seems to be open, right? I didn't really look into this problem but it seems that the plausible conjecture is that the lower bound 2n-6 proved for large n by Thurston et al, is actually the exact answer for all n > 10. I wonder also if anything at all is known about the diameters of associahedra for other root systems, or Postnikov's generalized permutohedra.

As for my little tribute to Ilya Iosifovich, I am sure you got it right. I apologize that it is in Russian; surely you have plenty of Russian speakers around you who can translate it.

Anonymous said...

Dear Andrei, I do not know any more news regarding the diameter of the associahedra beyond the work of Sleator, Tarjan and Thurston. (I do not even remember if a combinatorial proof for their result was found.)

Also nothing is known to me on the diameter problem for more general secondary polytopes (and fiber polytopes and Postnikov's generalized permutahedra)that you raised. Do the hyperbolic methods that Sleator, Tarjan and Thurston use extend? I think you once proposed that such examples may be relevant in the context of the Hirsch Conjecture.

avzel said...

Gil, I have no idea whether the hyperbolic methods could be extended to other polytopes. As for the Hirsch Conjecture, I remember vaguely our discussion of it, but I don't remember if I had any serious reasons to suspect that associahedra could be used there. If I remember right, I was just trying to make a point that we don't have lots of examples of higher-dimensional polytopes to guide our intuition.

Anonymous said...

"I was just trying to make a point that we don't have lots of examples of higher-dimensional polytopes to guide our intuition."

I agree completely!