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.