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H 1 Mutually enriching connections between ergodic theory and combinatorics - part 6

Auteurs : Bergelson, Vitaly (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : * The early results of Ramsey theory :
    Hilbert's irreducibility theorem, Dickson-Schur work on Fermat's equation over finite fields, van der Waerden's theorem, Ramsey's theoremand its rediscovery by Erdos and Szekeres.

    * Three main principles of Ramsey theory :
    First principle: Complete disorder is impossible. Second principle: Behind every 'Partition' result there is a notion of largeness which is responsible for a 'Density' enhancement of this result. Third principle: The sought-after configurations which are always to be found in large sets are abundant.

    * Furstenberg's Dynamical approach :
    Partition Ramsey theory and topological dynamics Dynamical versions of van der Waerden's theorem, Hindman's theorem and Graham-Rothschild-Spencer's geometric Ramsey.
    Density Ramsey theory and Furstenberg's correspondence principle Furstenberg's correspondence principle. Ergodic Szemeredi's theorem. Polynomial Szemeredi theorem. Density version of the Hales-Jewett theorem.

    * Stone-Cech compactifications and Hindman's theorem :
    Topological algebra in Stone-Cech compactifications. Proof of Hind-man's theorem via Poincare recurrence theorem for ultrafilters.

    * IP sets and ergodic Ramsey theory :
    Applications of IP sets and idempotent ultrafilters to ergodic-theoretical multiple recurrence and to density Ramsey theory. IP-polynomial Szemeredi theorem.

    * Open problems and conjectures

    If time permits:
    * The nilpotent connection,
    * Ergodic Ramsey theory and amenable groups

    Codes MSC :
    05D10 - Ramsey theory
    11D41 - "Higher degree equations; Fermat's equation"
    12D10 - Polynomials: location of zeros (algebraic theorems), {For the analytic theory, See 26C10, 30C15}
    37Axx - Ergodic theory
    37B20 - Notions of recurrence
    54D80 - Special constructions of spaces (spaces of ultrafilters, etc.)

      Informations sur la Vidéo

      Réalisateur : Hennenfent, Guillaume
      Langue : Anglais
      Date de publication : 02/11/2016
      Date de captation : 20/10/2016
      Sous collection : Research School
      Format : MP4
      arXiv category : Dynamical Systems ; Combinatorics
      Domaine : Dynamical Systems & ODE ; Combinatorics
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2016-10-20_Bergelson_6.mp4

    Informations sur la rencontre

    Nom de la rencontre : Jean-Morlet Chair - Doctoral school: Applications of Ergodic Theory in Number Theory / Chaire Jean-Morlet - Ecole doctorale : Applications de la théorie ergodique à la théorie des nombres
    Organisateurs de la rencontre : Adamczewski, Boris ; Ferenczi, Sébastien ; Kulaga-Przymus, Joanna ; Lemanczyk, Mariusz ; Troubetzkoy, Serge
    Dates : 17/10/2016 - 21/10/2016
    Année de la rencontre : 2016
    URL Congrès : https://www.chairejeanmorlet.com/1560.html

    Citation Data

    DOI : 10.24350/CIRM.V.19072203
    Cite this video as: Bergelson, Vitaly (2016). Mutually enriching connections between ergodic theory and combinatorics - part 6. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19072203
    URI : http://dx.doi.org/10.24350/CIRM.V.19072203

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