Ehud Hrushovski, Anand Pillay, Pierre Simon
We study stable like behaviour in first order theories without the independence property. We introduce generically stable measures, give characterizatiions, and show their ubiquity. We also introduce generic compact domination. We also prove the approximate definability of arbitrary Borel probability measures on definable sets in the real and p-adic fields.
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We study generically stable measures in the local, NIP context. We show that in this setting, a measure is generically stable if and only if it admits a natural finite approximation.
We prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and a distal-like quotient.
We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent (NIP) theories.
We discuss two constructions for obtaining generically stable Keisler measures in an NIP theory. First, we show how to symmetrize an arbitrary invariant measure to obtain a generically stable one from it. Next, we show that suitable sigma-additive probability measures give rise to generically stable measures. Also included is a proof that generically stable measures over o-minimal theories and the p-adics are smooth.
Externally definable fsg groups in NIP theories
本文证明NIP结构中外部可定义的fsg群与可解释群可定义同构，利用诚实定义与群块重构方法，并推广了外部可定义集合与群的相关结果，包括证明Eleftheriou猜想。
Smooth measures and the canonical retraction in NIP theories
本文将Simon关于NIP理论中典范收缩的结果从类型推广到测度，并利用光滑测度构造了测度上的典范收缩，证明其与Simon的构造一致。
Concerning Keisler Measures over ultraproducts
本文研究超积上 Keisler 测度的伪有限性与稳定性，证明有限可逼近测度必为伪有限，并刻画 Morley 积与伪有限积一致的条件，进而构造伪有限 NIP 群上的一般稳定幂等测度。
Generically Stable Measures and Distal Regularity in Continuous Logic
该研究将经典逻辑的相关结论推广至连续逻辑框架，在NIP度量理论中发展生成稳定与光滑Keisler测度理论，给出其等价刻画，并结合弱正交性结果与切割引理，得到远端正则性与强Erdős-Hajnal性质的解析版本。
We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.
We prove that if μis a generically stable stable measure in a first order theory with NIP and mu(ϕ(x,b)) = 0 for all b, then μ^{(n)}(\exists y(ϕ(x_1,y)\wedge ... \wedge ϕ(x_n,y))) = 0. We deduce that if G is an fsg grooup then a definable subset X of G is generic just if every translate of X does not fork over \emptyset.
Generic stability, randomizations, and NIP formulas
本文证明了Keisler测度、一般稳定性、随机化与NIP公式之间的若干关联，引入Keisler-Morley测度，证明相关极限性质，并刻画一般稳定类型在Morley乘积下的封闭性及NIP公式的温和扩张特征。
I give an account of Shelah's notion of strong dependence, or strong NIP, in terms of suitable generically stable measures, forking, and weight.
We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results from [L. Lovász, B. Szegedy, "Regularity partitions and the topology of graphons", An irregular mind, Springer Berlin Heidelberg, 2010, 415-446]. Besides, we revise the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the results from [A. Chernikov, S. Starchenko, "Regularity lemma for distal structures", J. Eur. Math. Soc. 20 (2018), 2437-…
This paper is a modified chapter of the author's Ph.D. thesis. We introduce the notions of sequentially approximated types and sequentially approximated Keisler measures. As the names imply, these are types which can be approximated by a sequence of realized types and measures which can be approximated by a sequence of `averaging measures' on tuples of realized types. We show that both generically stable types (in arbitrary theories) and Keisler measures which are finitely satisfiable over a countable model (in NIP theories) are sequentially approximated. We also introduce the notion of a smoo…
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd(A)$ (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ definably amenable, (iii) definitions, characterizations an…
We initiate a systematic study of the convolution operation on Keisler measures, generalizing the work of Newelski in the case of types. Adapting results of Glicksberg, we show that the supports of generically stable (or just definable, assuming NIP) measures are nice semigroups, and classify idempotent measures in stable groups as invariant measures on type-definable subgroups. We establish left-continuity of the convolution map in NIP theories, and use it to show that the convolution semigroup on finitely satisfiable measures is isomorphic to a particular Ellis semigroup in this context.
Some NIP-like phenomena in NTP$_{2}$
本文研究了NTP₂中的NIP-like现象，提出了NTP₂-光滑度量和distality概念，并探讨了有限交替定理及有限可满足性在NTP₂中的扩展问题。
In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as $ϕ$-types in simple theories that are definable and finitely satisfiable in a small model, but not finitely approximated. Our proofs demonstrate in…
Dependent measures in independent theories
本文提出Keisler测度的依赖性概念，将NIP理论中一般稳定测度的结果推广到任意理论，并证明依赖测度刻画Glivenko-Cantelli函数类。
Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture
本文研究可定义群上的幂等Keisler测度，提出一般传递性并证明其足以发展局部稳定群论，在阿贝尔群及rosy群等新情形确立一般稳定幂等型的一般传递性，分类幂等测度，并构造可数NIP群测度卷积半群的极小左理想，证明修正的Newelski猜想。
The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some basic results regarding the effect of the properties of B on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author's result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transfer…
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