I just heard the terrible news that Oded Schramm was killed in a hiking accident. Oded was hiking on Guye Peak near Snoqualmie Pass near Seattle. This is a terrible loss to Oded’s family, and our hearts and thoughts are with his wife Avivit and children Tselil and Pele. This is a great personal loss to me, to his many friends, and to mathematics.
Yuval Peres, a long time friend and collaborator of Oded, and as of a few months ago the head of the Microsoft theory group wrote a moving account of the sad news, with short remarks on Oded as a mathematician and as a person. Terry Tao wrote a description of some of Oded works concerning two dimensional stochastic processes and SLE. Luca Trevisan also devoted a post to Oded and mentioned connections of his work with computer science. (Luca wrote a later post on Oded’s recent work with Benjamini and Shapira on property testing for planar and minor-free graphs). Yuval Peres mentioned Oded as the best collaborator one can hope for and the story of Lawler, Schramm and Werner amazing discoveries of course comes to mind. Let me mention Oded’s main collaborator for many years Itai Benjamini. Oded and Itai’s beautiful collaboration, imaginative mathematical excursions, and friendship is something to be cherished.
(Sept. 8 ) Russ Lyons and David Wilson set up a memorial blog for Oded. Oded’s memorial page has also a link to a Video of Oded Schramm’s Memorial at Microsoft Research on Saturday September 6, 2008. (Sept. 11 ) An obituary in the NYT. (Sept 14) An obituary in Haaretz (in Hebrew).
(January 22, 2009): There will be a memeorial workshop in honor of oded and his mathematics at Microsoft Research, Redmond August 30-31, 2009.
I looked at Oded’s Master thesis entitled “Borsuk’s Problem and the Set of Middle Points of Diameters,” and I would like to tell you a little about it. It was submitted on 21 of May 1987, and it is carefully hand written 45 pages (in Hebrew) with many drawings and footnotes. The main theorem (perhaps Oded’s first theorem) reads as follows:
Theorem 7.1: For every and every large enough, , where
Here, is the smallest number of sets of diameter smaller than needed to cover a set of diameter in .
To prove Theorem 7.1 Oded studies the set of middle points of all diameters in .
In a footnote on page 8 Oded explains: “We use the term ‘diameter of a set ‘ for two meanings. On the one hand, the diameter of is the number , and on the other hand a diameter of is a line segment whose end points are points in whose distance equals . We hope that this will not cause confusions.”
And here is the first Lemma that Oded regards as trivial but supplies a proof nevertheless.
Lemma 3.2: Let be a set of diameter in and let be a compact convex subset of disjoint from then the diameter of is smaller than .
The rough strategy to cover a set with sets of smaller diameter is now clear. We need to find “large” (compact convex) subsets of that are disjoint from and for this, writes Oded, we need to show that is “small” for various meaning of the word “small”.
In the main Chapter 3 of the thesis Oded studies carefully properties of the set when is a set of constant width in . A few of these properties are needed for the main theorem and some others are of independent interest. For example, Corollary 3.9 asserts that in dimension three or more every set of constant width contains a rectangle with verticeson the boundary whose diagonals are diameters. (Or, in other words, contains two diameters with the same middle point.) Oded proves this claim using the fact that the -dimensional real projective space cannot be embedded into and asks if there is an elementary proof. Oded studies in the thesis the Lipschitzmap mapping a direction in to the middle point of the diameter in this direction. He shows that the set must have “cusps” and he studies these cusps.
Oded did not publish his Master thesis as a paper since shortly after he submitted it he discovered a different strategy that gave better bounds for giving . More than twenty years later this is still the bound to beat. A different proof yielding precisely the same value of was found by Bourgain and Lindenstrauss. A fundamental open question posed by Oded at that time was: how small can the quantity be for a set of constant width 1 in , as tends to infinity. Fixing the diameter, the maximum volume is attained by balls. But perhaps, when the dimension tends to infinity, cannot be any smaller than what you get for balls.