Some Themes Around First Order Theories Without the Independence Property

  • Anand PillayEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2111)


The aim of these notes (as well as the course of lectures they are based on) is to describe some current work around theories with NIP (not the independence property). This is a broad class of first order theories, including natural examples such as algebraically closed fields, differentially closed fields (both of which are stable) as well as real closed fields, p-adically closed fields and algebraically closed valued fields (which are unstable).This is really a paper on “pure” model theory, but I will comment here and there on applications and connections.



Supported by EPSRC grant EP/I002294/1.


  1. 1.
    N. Alon, D.J. Kleitman, Piercing convex sets and the Hadwiger-Debrunner (p, q)- problem. Adv. Math. 96, 103–112 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Chernikov, I. Kaplan, Forking and dividing in NTP 2 theories. J. Symb. Log. 77, 1–20 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Chernikov, P. Simon, Externally definable sets and dependent pairs. Isr. J. Math. 194, 409–425 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Chernikov, P. Simon, Externally definable sets and dependent pairs II. Trans. AMS (to appear)Google Scholar
  5. 5.
    A. Chernikov, I. Kaplan, S. Shelah, On nonforking spectra (preprint, 2012)Google Scholar
  6. 6.
    E. Hrushovski, F. Loeser, Non-archimedean Tame Topology, and Stably Dominated Types. Princeton Monograph Series (to appear)Google Scholar
  7. 7.
    E. Hrushovski, Y. Peterzil, A. Pillay, Groups, measures and the NIP. J. AMS 21, 563–596 (2008).Google Scholar
  8. 8.
    E. Hrushovski, A. Pillay, On NIP and invariant measures. J. Eur. Math. Soc. 13, 1005–1061 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    E. Hrushovski, A. Pillay, P. Simon, On generically stable and smooth measures in NIP theories. Trans. AMS 365, 2341–2366 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    H.K. Keisler, Measures and forking. Ann. Pure Appl. Log. 45, 119–169 (1987)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. Matousek, Bounded VC-dimension implies a fractional Helly theorem. Discrete Comput. Geom. 31, 251–255 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Pillay, Geometric Stability Theory (Oxford University Press, Oxford, 1996)zbMATHGoogle Scholar
  13. 13.
    A. Pillay, Externally definable sets and a theorem of Shelah, in Felgner Festchrift (College Publications, London, 2007)Google Scholar
  14. 14.
    A. Pillay, On weight and measure in NIP theories. Notre Dame J. Formal Log. (to appear)Google Scholar
  15. 15.
    S. Shelah, Dependent first order theories, continued. Isr. J. Math. 173, 1–60 (2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    S. Shelah, Dependent dreams and counting types (preprint 2012)Google Scholar
  17. 17.
    S. Shelah, Strongly dependent theories. Isr. J. Math. (to appear)Google Scholar
  18. 18.
    V. N. Vapnik, A.Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities. Theory Prob. Appl. 16, 264–280 (1971)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.University of LeedsLeedsUK
  2. 2.University of Notre DameNotre DameUSA

Personalised recommendations