Three dear friends, colleagues, and teachers Lior Tzafriri, Aryeh Dvoretzky and Michael Maschler passed away last year. I want to tell you a little about their mathematics.
Lior Tzafriri ( 1936-2008 )
Lior Tzafriri worked in functional analysis.
One of the crowning achievements of Banach space theory is the Lindenstrauss-Tzafriri theorem: A Banach space, each of whose closed subspaces is complemented, (that is, is the range of a bounded linear projection) must be isomorphic to a Hilbert space. This was a long-standing conjecture and, as the authors wrote in their paper published in the Israel J. of Mathematics, the proof is surprisingly simple. One of the tools they used was Dvoretzky’s theorem that we will mention below.
Next I want to tell you about a theorem of Bourgain and Tzafriri related to the famous Kadison-Singer conjecture.
Jean Bourgain and Lior Tzafriri considered the following scenario: Let be a real number. Let be a matrix with norm 1 and with zeroes on the diagonal. An by principal minor is “good” if the norm of is less than .
Consider the following hypergraph :
The vertices correspond to indices . A set belongs to if the sub-matrix of is good.
Bourgain and Tzafriri showed that for every there is so that for every matrix we can find so that .
Moreover, they showed that for every nonnegative weights there is so that the sum of the weights in is at least times the total weight. In other words, (by LP duality,) the vertices of the hypergraph can be fractionally covered by edges.
The “big question” is if there a real number so that for every matrix can be covered by good sets. Or, in other words, if the vertices of can be covered by edges. This question is known to be equivalent to an old conjecture by Kadison and Singer (it is also known as the “paving conjecture”). In view of what was already proved by Bourgain and Tzafriri what is needed is to show that the covering number is bounded from above by a function of the fractional covering number.
Michael Maschler ( 1927-2008 )
Michael Maschler was a game theorist. (I mentioned Michael in a post about Rationality.)
A famous theorem by Aumann and Maschler deals with reapeated games with incomplete information. I will describe the simplest possible case.
Start with a two-player game. Each player has a several strategies and any pair of strategies leads to payoffs for the two players. Such a game can be described by a payoff matrix where for each pair of strategies we have pair of payoffs for the two players.
Now consider several twists to this story.
1) The payoffs are unknown: they can be one out of two possible payoff matrices. A priori each of these matrices is equally likely.
2) The game is repeated: it is played infinitely many times (with the same payoff matrix.) The overall payoff for a player is his expected payoff and he gets it only “at the end”. The players do not see the payoffs; they only see how the other players play.
Now suppose one player has secret information and knows which of the two payoff matrices is being played. Aumann and Maschler described situations where the player is better off ignoring his secret information (in order not to expose it) and situations where he is better off using his secret information, and also intermediate scenarios. But a major insight from their theory is that whatever part of your secret information you use is eventually revealed!
Michael Maschler and Micha A. Perles discovered a very interesting solution concept, the subadditive solution, to the Nash bargaining problem. The Nash bargaining problem can be described as follows: Given a compact convex set in the positive orthant you consider the following scenario: If two players can agree on a point then player I will get x dollars and player II will get y dollars. If they fail to agree they both get 0. John Nash posed several axioms for a solution concept which lead to a unique solution: the point in with maximum value of . Another set of Axioms leading to another solution concept was proposed by Ehud Kalai and Meir Smorodinsky. Maschler and Perles introduced the following axiom: The solution for K+L dominates the solution for K + the solution for L. They identified a new and beautiful solution concept.
Aryeh Dvoretzky ( 1916-2008 )
Aryeh Dvoretzky, my academic great-grandfather, worked in Analysis and Probability Theory.
The famous and beautiful Dvoretzky’s theorem asserts that every centrally symmetric convex body in sufficiently high dimension has an almost spherical section. This was a conjecture of Grothendieck (and there is some evidence that the proof by Dvoretzky was one of the reasons Grothendieck moved to other areas.) It is one of the most fundamental and useful results in Banach space theory and convexity. More precisely Dvoretzky’s theorem asserts that a random dimensional centrally symmetric convex body has a -dimensional section whose “distance” from a ball is at most and
As a matter of fact, a random section will have this property with high probability.
Dvoretzky, Erdos, and Kakutani discovered several seminal properties of Brownian motions. A famous result they proved in 1961 asserts that Brownian motions never increase. (or more precisely, almost surely have no point of increase.) In 1996 Yuval Peres found a simple proof for this theorem on Brownian motions and Aryeh was very happy about it.
Some more material: Aryeh Dvoretzky Obituary in Isr J. Math and in Haaretz (Henbrew). Lior Tzafriri obituary by Prof. M. Zippin. More can be found in the “in memoriam” page of the HU Mathematics Institute. A drawing of Lior by a student. Michael Maschler in memoriam.