What is... Szemerédi’s Theorem?
Vortrag von Andrea Ulliana
Datum: 12.11.24 Zeit: 16.30 - 17.30 Raum:
As firstly conjectured by Erdös and Turán in 1936, in 1972 Szemerédi proved that any positive density subset of \(N\) contains arbitrary long arithmetic progressions. Determinant contributions came from very different fields: harmonic analysis, graph theory and ergodic theory. This theorem uncovered deep connections between these fields and sits at the foundation of the celebrated Green-Tao theorem about arithmetic progressions of prime numbers. During the talk we will introduce the notion of density of a subset of \(N\) and we will motivate the statement of the theorem. We will then turn our attention to Roth’s theorem (Szemerédi’s theorem for arithmetic progressions of length 3): we will sketch the harmonic analysis proof by Roth (1953) and we will mention Szemerédi’s alternative one, that exploits the ‘subgraph removal lemma’ and opened the way to a proof of Szemerédi’s theorem. Finally we will discuss Furstenberg alternative proof (1977) of Szemerédi’s theorem, based on his ‘Correspondence principle’ between subsets of \(N\) and measure preserving dynamical systems.