Approaches to Szemeredi's theorem
Ecole Normale Superieure, France.
Where a long-lasting conjecture of Erdos is solved by the most brilliant mathematical artists, there are interesting mathematics. Where four different approaches shed independently a specific light on a beautiful, elementary-looking theorem, there are very interesting mathematics. Where all this is fullfilled, there are amazing mathematics.
This is the case of Szemeredi's theorem ; it states that a dense subset of the integers necessarily contains long arithmetic progressions.
We will attempt to present various approaches to the theorem (combinatorial, ergodic, Fourier-analytic) and to explain the relationship to other theorems such as Green and Tao's about long arithmetic progressions of prime numbers.
This introductory talk requires no prior knowledge of the subject, so that everybody can join and enjoy.