Last week I took part in a lovely and impressive conference on the state of geometry and functional analysis, celebrating Vitali Milman’s 70th birthday. The conference started in Tel Aviv, continued at the Dead Sea, and returned to Tel Aviv. I gave a talk on the last day of the conference and many of the nice things I planned to mention about Vitali had already been mentioned by others. (For example, the unusual way he draws convex sets in high dimensions and the philosophy behind it, debating with Vitali, his special language Vitalian…) I did mention briefly the connection between Dvoretzky’s theorem and face numbers of centrally symmetric polytopes that was discovered in a paper by Figiel, Lindenstrauss, and Milman and which inspired my conjecture, and the strong connections between concentration of measure and combinatorics. Then I described how Vitali changed the landscape of Israeli mathematics and the lives of many dozens (perhaps hundreds) of people (all for the better) in his amazing efforts in absorbing mathematicians from the former Soviet Union into Israel.
I also briefly mentioned a lecture by David Milman (Vitali’s father) in the late ’70s that made a strong impression on me. (I was a graduate student at the time). David Milman is the Milman from the famous Krein-Milman theorem, and he talked about mathematics and mathematicians in the Soviet Union. His lecture mentioned areas of mathematics that I knew very little about, and personal descriptions of mathematicians that we only knew by name and saw little chance to ever meet. But things developed very differently from what we expected. A decade later we had opportunities to meet many of the Soviet mathematicians in conferences in the West, and some of the younger people mentioned by David Milman in his talk like Sinai, Novikov and Gromov, were present at my talk.
I also mentioned another surprising mathematical development that I enjoyed observing over the years. Vitali noticed that the well-known Knaster conjecture implies a strong form of Dvoretzky’s theorem (in terms of the asymptotic in ). I remember he used to say “the ball is now in the hands of the topologists.” Then Kashin and Szarek used Banach space techniques to disprove the Knaster conjecture. (Still, related topological results may lead to the hoped-for sharp form of Dvoretzky’s theorem, so the ball is in everybody’s hands.) After all this, I went on with my own lecture which was about quantum computation.
Few among many highlights: Vitaly Bergelson gave a blackboard talk about ergodic theorems and attributed the Poincare Recurrence theorem to King Solomon. He did not have time to describe the 10 major open problems he was planning to, and I will be very happy to host them here over my blog. Mark Rudelson gave a clear and inspiring talk on random matrices. Random matrices were dominant in several other lectures. David Preiss described how certain higher dimensional extensions of the Erdos-Szekeres theorem would imply that all “negilgible” sets are null sets. The hope is that this is not actually the case and that there are rich families of negligible sets which are not null.
On Saturday, while floating near Semyon Alesker in the Dead Sea we saw a large cluster of experts on the Mahler conjecture– Shlomo Reisner, Dimitry Ryabogin and Artem Zvaivich — and we decided to swim towards them and ask them to describe what is going on withthe conjecture. Swimming in the Dead Sea is not easy but when we finally reached the group it was worth the effort. Dimitry already talked at the conference about the stability result by Nazarov, Ryabogin and Zvaivich asserting that the cube is a local minimum for the volume of a centrally symmetric body multiplied by the volume of its polar body, and he and Artem told us a little more about it. This is a difficult result and it is not yet known how to extend it to the other Hammer polytopes. We also heard about a promising new approach by Nazarov.
David Milman (right) with Troyanski.