Timothy Chow Launched Polymath12 on Rota Basis Conjecture and Other News

Polymath12

Timothy Chow launched polymath12 devoted to the Rota Basis conjecture on the polymathblog. A classic paper on the subject is the 1989 paper by Rosa Huang and Gian Carlo-Rota.

Let me mention a strong version of Rota’s conjecture (Conjecture 6 in that paper) due to Jeff Kahn that asserts that if you have n^2 bases B_{i,j} 1 \le i \le n,1\le j \le n of an n-dimensional vector space then you can choose b_ {ij} \in B_{i,j} such that for every k,1 \le k\le n, the vectors b_{ik}:1\le i \le n form a basis of V and also the vectors b_{kj}:1\le j \le n form a basis. This conjecture can be regarded as a strengthening of the Dinits Conjecture whose solution by Galvin is described in this post. Rota’s original conjecture is the case where B_{i,j} depends only on i.

A very nice special case of Rota’s conjecture was proposed by Jordan Ellenberg in the polymath12 thread: Given n-1 trees on n vertices, is it always possible to order the edges of each tree so that the kth edges in the trees form a tree for every k?

Two recent papers in the Notices AMS

The February 2017 issue of the Notices of the AMS has two beautiful papers on topics we discussed here. Henry Cohn  wrote an article A conceptual breakthrough in sphere packing about the breakthrough on sphere packings in 8 and 24 dimensions  (see this post) and  Art Duval, Carly Klivans, and Jeremy Martin wrote an article on the the Partitionability Conjecture. (That we mentioned here.)

I have some plans to write about the partitionability conjecture (and an even more general conjecture of mine) soon. But now I would like to draw your attention to a weakening of these conjectures, still implying the “numerical” consequences of the original conjectures, that was proved in 2000 by Art Duval and Ping Zhang in their paper Iterated homology and decompositions of simplicial complexes . The partitionability conjecture is about decompositions into subcubes (or intervals in the Boolean lattice) and the result is about decomposition into subtrees of the Boolean lattice. (See here for the massage “trees not cubes!” in another context.)

Two steps from fame

trampolin

This pictures was taken by Edna Wigderson in a 50th birthday party for Avi Wigderson. Unfortunately the picture shows us while landing from our 3 meter high (3.28 yards) jumps and thus does not fully capture the achievement.

The  guy on the left is Bernard Chazelle, a great computer scientist and geometer,  a long time friend of me and Avi, and the father of Damien Chazelle the director and writer of the movie La La Land now nominated for the record 14 Oscar awards. I wish Damien to win a record number of Oscars and to continue writing,  directing, and producing wonderful movies so as to keep shattering his own records and giving excitement and joy to hundreds of millions, perhaps even billions, of people. (Update: Six Oscars, Damien the youngest ever director to win.)

The Jerusalem Baroque orchestra and 2017 Bach Festival

For those in Israel let me draw your attention to the Jerusalem Baroque orchestra and especially to the 2017 Bach Festival. (I thanks Menachem Magidor for telling me about this wonderful orchestra.)

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