Appalachian Set Theory workshop: Slawomir Solecki

November 22, 2014, at the University of Illinois at Chicago

Slawomir Solecki

Ultrafilter space methods in Infinite Ramsey Theory

Professor Solecki will present a part of Infinite Ramsey Theory by explaining the methods and results stemming from Ellis’ idempotent lemma. These methods rely heavily on ultrafilters and their spaces and are far reaching generalizations of the initial argument due to Galvin and Glazer. The lectures will include proofs of older and more recent results, including the Furstenberg-Katznelson and Gowers theorems. The lectures will present a a new unified treatment of this part of Infinite Ramsey Theory, including some applications if time allows.

Here is a reading list containing some background material for the workshop:

 

  • Basics:
    • The ultrafilter entry on Wikipedia.
    • Pages 409-413 in W. Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409-419.
  • Intermediate:
    • Sections 2 and 3 in V. Bergelson, A. Blass, N. Hindman, Partition theorems for spaces of variable  words, Proc. London Math. Soc. 68 (1994), 449-476.
  • Advanced:
    • H. Furstenberg, Y. Katznelson, Idempotents in compact semigroups and Ramsey theory,   Israel J. Math. 68 (1989), 257-270.
    • Chapter 2 in S. Todorčević, Introduction to Ramsey Spaces, Annals of Mathematics Studies, 174. Princeton University Press, 2010.

 

  • Supplementary:
    • Chapters 1-5 in  N. Hindman, D. Strauss, Algebra in the Stone-Čech Compactification, de Gruyter  Expositions in Mathematics 27, Walter de Gruyter & Co., 2012.

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