Christian Skau: FROM SZEMERÉDI TO GREEN/TAO: COMBINATORICS AND ADDITIVE NUMBER THEORY

 

In 1975 this year's Abel Prize recipient, the Hungarian mathematician Endre Szemerédi, proved a long-standing conjecture of Erdøs: if A is a subset of the natural numbers of positive upper density, then A contains arbitrarily long arithmetic progressions. His proof is purely of combinatorial nature and extremely complicated--only a handful people have read and understood it. In 1977 Furstenberg gave a new proof using ergodic theory. Specifically, he showed that any measure-preserving system has a multiple recurrence property, which is a vast generalization of a classical result of Poincaré. What is amazing is that Szemerédi's theorem and Furstenberg's theorem actually are equivalent!   We will indicate how the Szemeredi theorem -- and Furstenberg's proof of it -- plays a key role in the proof of the now famous Green/Tao result from 2004: within the prime numbers there exist arbitrarily long arithmetic progressions.  
Published Apr. 10, 2012 3:17 PM