On-line catalogue and orders (secure paiement, VISA or MASTERCARD only)

Journals available by subscription

Annales scientifiques de l'ENS

Astérisque

Bulletin de la SMF

Mémoires de la SMF

Revue d'Histoire des Mathématiques

Gazette des Mathématiciens

Books

Astérisque

Cours Spécialisés

Documents Mathématiques

Mémoires de la SMF

Panoramas et Synthèses

Séminaires et Congrès

Jean Morlet Chair Series

SMF/AMS Texts and Monographs

La Série T

Volumes "Journée Annuelle"

Other Books

Donald E. Knuth - French translations

Nicolas Bourbaki's seminar new edition

Jean Leray's scientific works new edition

Revue de l'Institut Elie Cartan

Electronic Editions

Annales scientifiques de l'ENS

Bulletin de la SMF

Revue d'Histoire des Mathématiques

Séminaires et Congrès

More information / Subscription

Publications for a general public

L'explosion des mathématiques (smf.emath.fr)

Mathématiques L'explosion continue (smf.emath.fr)

Zoom sur les métiers des maths (smf.emath.fr)

Zoom sur les métiers des mathématiques et de l'informatique (smf.emath.fr)

Où en sont les mathématiques ?

La Série T

For the authors

Submission of manuscripts

Formats and documentation

More info

Electronic distribution list (smf.emath.fr)

Information for bookselers and subscription agencies (smf.emath.fr)

Publications de la SMF
fr en
Your IP number: 54.166.207.223
Access to elec. publ.: SémCong

Astérisque

Presentation of the publication

Titles

Last Titles

Editorial staff committee / Secretary

Year:
Number:

Search


Catalogue & orders

Astérisque - Titles - 390 (2017) 303-334

Titles < 2017 < 390

Séminaire Bourbaki, volume 2015/2016, exposés 1104-1119
Astérisque 390 (2017), xi+533 pages
Download the document
Presentation, Summary

Exposé 1114 : NIP, Keisler measures and combinatorics
Sergei STARCHENKO
Astérisque 390 (2017), 303-334
Download the document

Résumé :
Exposé 1114 : Propriété de non-indépendance (NIP), mesures de Keisler et combinatoire
Keisler measures were introduced by H.J. Keisler in 1987 as finitely additive probability measures on Boolean algebras of definable sets. Almost 20 years later Keisler's work was revisited, significantly improved and deepened in a series of papers by E. Hrushovski, A. Pillay, Y. Peterzil, P. Simon. In this talk I will survey Keisler measures and try to demonstrate that Keisler's measures on so-called distal structures provide a very natural framework for various combinatorial problems.

Mots-clefs : NIP, Keisler measures, distal theories, Erdős-Hajnal Conjecture, Regularity lemma, VC-dimension.

Abstract:
En 1987, H. J. Keisler a introduit l'étude des mesures de probabilités finiment additives sur les algèbres de Boole d'ensembles définissables. Près de 20 ans plus tard, une série d'articles de E. Hrushovski, A. Pillay, Y. Peterzil, P. Simon a revisité le travail de Keisler, en l'approfondissant et l'améliorant de façon significative. Dans cet exposé, j'expliquerai ces mesures de Keisler et essayerai de montrer que les mesures de Keisler sur les structures distales fournissent un cadre très naturel pour l'étude de divers problèmes de combinatoire.

Keywords: Propriété de non-indépendance (NIP), mesures de Keisler, théories distales, conjecture d'Erdős-Hajnal, lemme de régularité, dimension de Vapnik-Chervonenkis.

Class. math. : 03C68, 03C45, 03C98, 05C69, 05D10, 28E05.


ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Alon, Noga and Pach, János and Pinchasi, Rom and Radoičić, Radoš and Sharir, Micha
Crossing patterns of semi-algebraic sets
J. Combin. Theory Ser. A 111 (2005) 310–326
Math Reviews MR2156215
2
Basu, Saugata
Combinatorial complexity in o-minimal geometry
Proc. Lond. Math. Soc. 100 (2010) 405–428
Math Reviews MR2595744
3
Bhaskara Rao, K. P. S. and Bhaskara Rao, M.
Theory of charges
Pure and Applied Mathematics, vol. 109, Academic Press, Inc., New York, 1983
Math Reviews MR751777
4
Chatzidakis, Zoé and Hrushovski, Ehud
Difference fields and descent in algebraic dynamics. I
J. Inst. Math. Jussieu 7 (2008) 653–686
Math Reviews MR2469450
5
Chernikov, Artem and Simon, Pierre
Externally definable sets and dependent pairs II
Trans. Amer. Math. Soc. 367 (2015) 5217–5235
Math Reviews MR3335415
6
Chernikov, Artem and Simon, Pierre
Definably amenable NIP groups
7
8
van den Dries, Lou
Approximate groups [according to Hrushovski and Breuillard, Green, Tao]
Séminaire Bourbaki, vol. 2013/2014, exp. no 1077, Astérisque 367-368 (2015) 79–113
Math Reviews MR3363589
9
Ducros, Antoine
Les espaces de Berkovich sont modérés (d'après Ehud Hrushovski et François Loeser)
Séminaire Bourbaki, vol. 2011/2012, exp. no 1056, Astérisque 352 (2013) 459–507
Math Reviews MR3087354
10
Dunford, N. and Schwartz, J.T.
Linear Operators: General theory
Interscience Publ., 1957
11
Duret, Jean-Louis
Les corps pseudo-finis ont la propriété d'indépendance
C. R. Acad. Sci. Paris Sér. A-B 290 (1980) A981–A983
Math Reviews MR584282
12
Fox, Jacob and Gromov, Mikhail and Lafforgue, Vincent and Naor, Assaf and Pach, János
Overlap properties of geometric expanders
J. reine angew. Math. 671 (2012) 49–83
Math Reviews MR2983197
13
Fox, Jacob and Pach, János and Suk, A.
A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing
SIAM Journal of Computing 45 (2016) 2199–2223
14
Gurevich, Y. and Schmitt, P. H.
The theory of ordered abelian groups does not have the independence property
Trans. Amer. Math. Soc. 284 (1984) 171–182
Math Reviews MR742419
15
Hrushovski, Ehud
Stable group theory and approximate subgroups
J. Amer. Math. Soc. 25 (2012) 189–243
Math Reviews MR2833482
16
Hrushovski, Ehud and Loeser, François
Non-archimedean tame topology and stably dominated types
Annals of Math. Studies, vol. 192, Princeton Univ. Press, Princeton, NJ, 2016
Math Reviews MR3445772
17
Hrushovski, Ehud and Peterzil, Ya'acov and Pillay, Anand
Groups, measures, and the NIP
J. Amer. Math. Soc. 21 (2008) 563–596
Math Reviews MR2373360
18
Hrushovski, Ehud and Pillay, Anand
On NIP and invariant measures
J. Eur. Math. Soc. (JEMS) 13 (2011) 1005–1061
Math Reviews MR2800483
19
Hrushovski, Ehud and Pillay, Anand and Simon, Pierre
Generically stable and smooth measures in NIP theories
Trans. Amer. Math. Soc. 365 (2013) 2341–2366
Math Reviews MR3020101
20
Keisler, H. Jerome
Measures and forking
Ann. Pure Appl. Logic 34 (1987) 119–169
Math Reviews MR890599
21
Laskowski, Michael C.
Vapnik-Chervonenkis classes of definable sets
J. London Math. Soc. 45 (1992) 377–384
Math Reviews MR1171563
22
Loeb, Peter A.
Conversion from nonstandard to standard measure spaces and applications in probability theory
Trans. Amer. Math. Soc. 211 (1975) 113–122
Math Reviews MR0390154
23
oś, Jerzy
Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres
in Mathematical interpretation of formal systems
(1955) 98–113
Math Reviews MR0075156
24
oś, Jerzy and Marczewski, E.
Extensions of measure
Fund. Math. 36 (1949) 267–276
Math Reviews MR0035327
25
Marker, David
Model theory
Graduate Texts in Math., vol. 217, Springer, New York, 2002
Math Reviews MR1924282
26
Medvedev, Alice and Scanlon, Thomas
Invariant varieties for polynomial dynamical systems
Ann. of Math. 179 (2014) 81–177
Math Reviews MR3126567
27
Scanlon, Thomas
Counting special points: logic, Diophantine geometry, and transcendence theory
Bull. Amer. Math. Soc. (N.S.) 49 (2012) 51–71
Math Reviews MR2869007
28
Shelah, Saharon
Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory
Ann. Math. Logic 3 (1971) 271–362
Math Reviews MR0317926
29
Shelah, Saharon
Classification theory and the number of nonisomorphic models
Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam-New York, 1978
Math Reviews MR513226
30
Shelah, Saharon
Classification theory for elementary classes with the dependence property—a modest beginning
Sci. Math. Jpn. 59 (2004) 265–316
Math Reviews MR2062198
31
Shelah, Saharon
Dependent first order theories, continued
Israel J. Math. 173 (2009) 1–60
Math Reviews MR2570659
32
Simon, Pierre
Distal and non-distal NIP theories
Ann. Pure Appl. Logic 164 (2013) 294–318
Math Reviews MR3001548
33
Simon, Pierre
A guide to NIP theories
Lecture Notes in Logic, vol. 44, Cambridge Univ. Press, 2015
34
Simon, Pierre
A note on `Regularity lemma for distal structures'
Proc. Amer. Math. Soc. 144 (2016) 3573–3578
Math Reviews MR3503725
35
Tent, Katrin and Ziegler, Martin
A course in model theory
Lecture Notes in Logic, vol. 40, Association for Symbolic Logic, La Jolla, CA; Cambridge Univ. Press, Cambridge, 2012
Math Reviews MR2908005
36
37
Vapnik, V. and Chervonenkis, A.
On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
Theory of Probability Its Applications 2 (1971) 264–280