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Topological dynamics and definable groups

Anand Pillay

Journal of Symbolic Logic 78 (2):657-666 (2013)

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Type-definability, compact lie groups, and o-minimality.Anand Pillay - 2004 - Journal of Mathematical Logic 4 (02):147-162.details We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures (...) Download Export citation Bookmark 23 citations
Forking and dividing in NTP₂ theories.Artem Chernikov & Itay Kaplan - 2012 - Journal of Symbolic Logic 77 (1):1-20.details We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2. Download Export citation Bookmark 27 citations
A Note on Generically Stable Measures and fsg Groups.Ehud Hrushovski, Anand Pillay & Pierre Simon - 2012 - Notre Dame Journal of Formal Logic 53 (4):599-605.details We prove (Proposition 2.1) that if $\mu$ is a generically stable measure in an NIP (no independence property) theory, and $\mu(\phi(x,b))=0$ for all $b$ , then for some $n$ , $\mu^{(n)}(\exists y(\phi(x_{1},y)\wedge \cdots \wedge\phi(x_{n},y)))=0$ . As a consequence we show (Proposition 3.2) that if $G$ is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and $X$ is a definable subset of $G$ , then $X$ is generic if and only if every translate of $X$ does not (...) Download Export citation Bookmark 3 citations
(1 other version)[Omnibus Review].Anand Pillay - 1984 - Journal of Symbolic Logic 49 (1):317-321.details Download Export citation Bookmark 28 citations

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