Professor Terance Tao

University of California, Los Angeles

The proof of the Poincaré conjecture
In a series of three terse papers in 2003 and 2004, Grisha Perelman made spectacular advances in the theory of the Ricci flow on 3-manifolds, leading in particular to his celebrated proof of the Poincare conjecture (and most of the proof of the more general geometrization conjecture). Remarkably, while the Poincare conjecture is a purely topological statement, the proof is almost entirely analytic in nature, in particular relying on nonlinear PDE tools together with estimates from Riemannian geometry to establish the result. In this talk we discuss some of the ingredients used in the proof, and sketch a high-level outline of the argument.

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.