50th
Annual Meeting of the Australian
Mathematical Society
Plenary
Talk in Macquarie Theatre
Tuesday
26 September 2006 at 16:30
Terence
Tao (University of California,
Los Angeles)
Long Arithmetic
Progressions in the Primes
A famous and difficult theorem
of Szemeredi asserts that every subset of the integers of positive
density will contain arbitrarily long arithmetic progressions;
this theorem has had four different proofs (graph-theoretic, ergodic,
Fourier analytic, and hypergraph-theoretic), each of which has
been enormously influential, important, and deep. It had been
conjectured for some time that the same result held for the primes
(which of course have zero density). I shall discuss recent work
with Ben Green obtaining this conjecture, by viewing the primes
as a subset of the almost primes (numbers with few prime factors)
of positive relative density. The point is that the almost primes
are much easier to control than the primes themselves, thanks
to sieve theory techniques such as the recent work of Goldston
and Yildirim. To "transfer" Szemeredi's theorem to this
relative setting requires that one borrow techniques from all
four known proofs of Szemeredi's theorem, and especially from
the ergodic theory proof.
