Posted by: matheuscmss | February 15, 2010

## Ergodic Ramsey Theory: where Combinatorics and Number Theory meet Dynamics

Hello! Since everybody (including myself) in Rio de Janeiro is in Carnival mood, this post will be very brief. Two weeks ago, Vitaly Bergelson and I shared a mini-course entitled Ergodic Ramsey Theory: where Combinatorics and Number Theory meet        Dynamics. This mini-course was a part of actvities of the first Brazilian School of Dynamical Systems held at Maceió from February 1 to Februrary 11 (my lectures were in the first week and Vitaly´s lecture were in the second week). During my lectures, I focused on the basic aspects of Ergodic Ramsey Theory, so that Vitaly could touch upon the more sophisticated (and recent) results. In particular, I talked about Ramsey Theorem, van der Waerden theorem, Szemeredi theorem, Furstenberg´s proof of Szemeredi theorem via his correspondence principle and multiple recurrence theorem, multiple recurrence for weak-mixing and compact systems, weak-mixing and compact extensions and Furstenberg-Zimmer structural theorem. Of course, the material of my part of the mini-course is largely covered by Furstenberg´s book and Terence Tao´s lecture notes, but I thought it could be useful for any interested student to take a look at the following set of notes prepared jointly with my friend Yuri Lima:

• First Lecture: pigeonhole principle, friendship theorem and Ramsey theorem; van der Waerden theorem and Hindman theorem; Szemeredi theorem; basic principles in Ramsey theory and Ergodic Ramsey theory; dynamical proofs of van der Wander and Szemeredi theorem modulo appropriate multiple recurrence results; Furstenberg´s multiple recurrence theorem; (there is also three appendices about the combinatorial proof of van der Waerden theorem, proof of topological multiple recurrence theorem of Furstenberg and Weiss and Furstenberg´s correspondence principle)
• Second Lecture: weak-mixing systems; characterizations of weak-mixing (inclunding its stability under products with ergodic systems and its spectral counterpart); proof of Furstenberg´s multiple recurrence theorem for weak-mixng systems; van der Corput trick;
• Third Lecture: compact systems; proof of Furstenberg´s multiple recurrence theorem for compact systems; Kronecker systems, topological classification of minimal equicontinuous systems, metrical classification of ergodic compact systems; spectral characterization of compact systems; compact and weak-mixing extensions; multiple recurrence results for weak-mixing and compact extensions; dichotomy between structure and ramdomness; Furstenberg-Zimmer structural theorem and “completion´´ of the proof of Furstenberg´s multiple recurrence theorem;
• Fourth Lecture: Harmonic Analysis proof of Roth theorem and its relationship with Furstenberg-Zimmer structural theorem.
• Fifth Lecture: Ultrafilters and Hindman theorem.