On generically stable types in dependent theories

A note on generically stable measures and fsg groups

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.

Berkovich空间中的Radon测度与ACVF中的一般稳定测度

Radon measures in the Berkovich space and generically stable measures in ACVF

本文研究ACVF模型中的Keisler测度，证明全局M-不变Keisler测度的一般稳定性、正交于值群及稳定受控三者等价，还建立了射影簇的一般稳定测度与Berkovich解析化上Radon测度的一一对应，并给出相关应用。

Generically stable and smooth measures in NIP theories

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.

一般稳定的凯斯勒测度

Generically stable Keisler measures

本文针对一阶理论T的凯斯勒测度，证明了频率解释测度、可定义且其随机扩张r_μ在随机化理论T^R中一般稳定、自平均这三个条件等价，解决了相关长期研究目标，通过AI模型得到了反向蕴含关系。

Remarks on generic stability in independent theories

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…

Local Keisler Measures and NIP Formulas

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.

连续逻辑中的一般稳定测度与远端正则性

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。

Type decomposition in NIP theories

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.

On Stably Pointed Varieties and Generically Stable Groups in ACVF

We give a geometric description of the pair $(V,p)$, where $V$ is an affine algebraic variety over a non-trivially valued algebraically closed field $K$ with valuation ring $\mathcal{O}_K$ and $p$ is a Zariski dense generically stable type concentrated on $V$, by defining a fully faithful functor to the category of schemes over $\mathcal{O}_K$ with residual dominant morphisms over $\mathcal{O}_K$. We also study a maximum modulus principle on schemes over $\mathcal{O}_K$ and show that the schemes obtained by this functor enjoy it.

Keisler 测度与一般稳定随机类型

Keisler Measures and Generically Stable Random Types

本文引入 Keisler 测度的 rgs 和 irgs 概念，通过经典一阶公式在概率划分上的平均刻画这些概念，并证明 irgs 测度是依赖的、对称的，且模型论不稳定事件的测度为零。

Morley sequences in dependent theories

We characterize nonforking (Morley) sequences in dependent theories in terms of a generalization of Poizat's special sequences and show that average types of Morley sequences are stationary over their domains. We characterize generically stable types in terms of the structure of the "eventual" type. We then study basic properties of "strict Morley sequences", based on Shelah's notion of strict nonforking. In particular we prove "Kim's lemma" for such sequences, and a weak version of local character.

Sequential approximations for types and Keisler measures

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…

Classification over a predicate -- the general case. Part I -- structure theory

We begin a systematic development of structure theory for a first order theory, which is stable over a monadic predicate. We show that stability over a predicate implies quantifier free definability of types over stable sets, introduce an independence notion and explore its properties, prove stable amalgamation results, and show that every type over a model, orthogonal to the predicate, is generically stable.

Stable theories and representation over sets

In this paper we explore the representation property over sets. This property generalizes constructibility, however is weak enough to enable us to prove that the class of theories $T$ whose models are representable is exactly the class of stable theories. Stronger results are given for omega-stable.

Karp height of models of stable theories

A trichotomy theorem for countable, stable, unsuperstable theories is offered. We develop the notion of a `regular ideal' of formulas and study types that are minimal with respect to such an ideal.

Minimal types in stable Banach spaces

We prove existence of wide types in a continuous theory expanding a Banach space, and density of minimal wide types among stable types in such a theory. We show that every minimal wide stable type is "generically" isometric to an l_2 space. We conclude with a proof of the following formulation of Henson's Conjecture: every model of an uncountably categorical theory expanding a Banach space is prime over a spreading model, isometric to the standard basis of a Hilbert space.

Generic wide indiscernible sets

We prove that a wide Morley sequence in a wide generically stable type is isometric to the standard basis of an $\ell_p$ space for some $p$.

关于稳定Kim分裂和罗斯理论

On stable Kim-forking and rosy theories

该研究在稳定Kim分裂猜想假设下证明了所有NSOP₁罗斯理论均为简单理论，并展示了弗罗贝尼乌斯域理论具有稳定Kim分裂性质。

Exact saturation in pseudo-elementary classes for simple and stable theories

We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact satura…

On maximal stable quotients of definable groups in NIP theories

For G a group definable in an NIP theory we prove that there is a smallest type-definable subgroup H of G such that the quotient G/H is stable. This generalizes the existence of G^00, the smallest type-definable subgroup of G of bounded index.

Borel completeness of some aleph_0 stable theories

We study aleph_0-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of lambda-Borel completeness and prove that such theories are lambda-Borel complete. Using this, we conclude that an aleph_0-stable theory has 2^lambda pairwise non-L(infinity,aleph_0) equivalent models of size lambda for all infinite cardinals lambda if and only if T either has eni-DOP or is eni-deep.

Type-amalgamation properties and polygroupoids in stable theories

We show that in a stable first-order theory, the failure of higher-dimensional type amalgamation can always be witnessed by algebraic structures which we call n-ary polygroupoids. This generalizes a result of Hrushovski that failures of 4-amalgamation in stable theories are witnessed by definable groupoids (which are 2-ary polygroupoids in our terminology). The n-ary polygroupoids are definable in a mild expansion of the language (adding a unary predicate for an infinite Morley sequence).

Elimination of Hyperimaginaries and Stable Independence in simple CM-trivial theories

In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.

On the class of flat stable theories

A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence relation, flat theories are defined as an appropriate version of superstability. It is shown that in a flat theory every type has finite weight and therefore flat theories are strong. Furthermore, it is shown that under reasonable conditions any type is non-orthogonal to a regular one. Concerning groups in flat theories, it is shown that type-definable groups behave…

Generic expansion and Skolemization in NSOP$_1$ theories

We study expansions of NSOP$_1$ theories that preserve NSOP$_1$. We prove that if $T$ is a model complete NSOP$_1$ theory eliminating the quantifier $\exists^{\infty}$, then the generic expansion of $T$ by arbitrary constant, function, and relation symbols is still NSOP$_1$. We give a detailed analysis of the special case of the theory of the generic $L$-structure, the model companion of the empty theory in an arbitrary language $L$. Under the same hypotheses, we show that $T$ may be generically expanded to an NSOP$_1$ theory with built-in Skolem functions. In order to obtain these results, we…

独立理论中的依赖测度

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。

Superstable theories and representation

In this paper we give characterizations of the super-stable theories, in terms of an external property called representation. In the sense of the representation property, the mentioned class of first-order theories can be regarded as "not very complicated". This was done for "stable" and for "aleph_0-stable". Here we give a complete answer for "superstable".

稳定理论中的非分叉独立性

Non-forking independence in stable theories

本文指出在稳定理论中，利用类型平稳性，通过简单条件即可描述模型及代数闭子集上的非分叉独立性，无需依赖Kim和Pillay工作的全部力量。

Higher amalgamation properties in stable theories

For a complete, stable theory $T$ we construct, in a reasonably canonical way, a related stable theory $T^*$ which has higher independent amalgamation properties over the algebraic closure of the empty-set. The theory $T^*$ is an algebraic cover of $T$ and we give an explicit description of the finite covers involved in the construction of $T^*$ from $T$. This follows an approach of E. Hrushovski. If $T$ is almost strongly minimal with a $0$-definable strongly minimal set, then we show that $T^*$ has higher amalgamation over any algebraically closed subset.

Minimal types in super-dependent theories

We give necessary and sufficient geometric conditions for a theory definable in an o-minimal structure to interpret a real closed field. The proof goes through an analysis of thorn-minimal types in super-rosy dependent theories of finite rank. We prove that such theories are coordinatised by thorn-minimal types and that such a type is unstable if an only if every non-algebraic extension thereof is. We conclude that a type is stable if and only if it admits a coordinatisation in thorn-minimal stable types. We also show that non-trivial thorn-minimal stable types extend stable sets.

泛型稳定性与收敛模式

Generic Stability and Modes of Convergence

该研究在经典与连续逻辑中探究泛型稳定型/测度，分析其与随机化及型/测度收敛模式的联系，推进相关逻辑领域研究。

$\omega$-稳定理论的类型$\omega$-Vaught猜想

The Type $ω$-Vaught's Conjecture for $ω$-stable Theories

本文提出Vaught猜想的一个新变体，基于Martin猜想和$\omega$-Vaught猜想，针对$\omega$-稳定理论证明之，并将Scott句所需量词数从$\omega+\omega$改进为$\omega+5$（可数情形）和$\omega+6$（不可数情形）。

Strict independence

We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular it follows from our investigation that in resilient theories strict non-forking is symmetric. Based on this study, we develop notions of weight which characterize NTP2, dependence and strong dependence. Many of our proofs rely on careful analysis of sequences that witness dividing. We prove simple characterizations of such sequences in resilient theories, as well as of Morley sequences which are witnesses. As a by-product we obtain information …

The universality spectrum of stable unsuperstable theories

It is shown that if T is stable unsuperstable, and aleph_1< lambda =cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)< mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g. aleph_omega < 2^{aleph_0} then T has no universal model in aleph_omega. These results are generalized to kappa =cf(kappa) < kappa (T) in the place of aleph_0. Also: if there is a universal model in lambda >|T|, T stable and kappa < kappa (T) then there is a universal tree of height kappa +1 in cardinality lambda .

Non-commutative groupoids obtained from the failure of $3$-uniqueness in stable theories

We construct a possibly non-commutative groupoid from the failure of $3$-uniqueness of a strong type. The commutative groupoid constructed by John Goodrick and Alexei Kolesnikov in \cite{GK} lives in the center of the groupoid. A certain automorphism group approximated by the vertex groups of the non-commutative groupoids is suggested as a "fundamental group" of the strong type.

Generic stability and stability

We prove two results about generically stable types $p$ in arbitrary theories. The first, on existence of strong germs, generalizes results from D. Haskell, E. Hrushovski and D. Macpherson on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of $p^{(m)}$ for all $m$. We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If $p(x)\in S(B)$ is stable and does not fork over $A$ then $p\restriction A$ is stable. (They had solved some special cases.)

具有 Schröder-Bernstein 性质的一般实值理论是稳定的

General real-valued theories with the Schröder-Bernstein property are stable

本文证明了具有 Schröder-Bernstein 性质的 Keisler 风格一般实值理论是稳定的，推广了经典逻辑中的相应结果，并探讨了该技术在连续理论中的适用性。

Dependence Logic in Pregeometries and $ω$-Stable Theories

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence.

Homology groups of types in stable theories and the Hurewicz correspondence

We give an explicit description of the homomorphism group H_n(p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups H_i(q) are trivial for i at least 2 but less than n. The group H_n(p) turns out to be isomorphic to the automorphism group of a certain piece of the algebraic closure of n independent realizations of p; it was shown earlier by the authors that such a group must be abelian. We call this the "Hurewicz correspondence" in analogy with the Hurewicz Theorem in algebraic topology.

NIP理论中不变类型的极大稳定商

Maximal stable quotients of invariant types in NIP theories

本文证明了NIP理论中不变类型的实现集上存在最精细的相对类型可定义等价关系且具有稳定商，该结论是已有关于类型可定义群极大稳定商结果的对应物。

可定义卷积与幂等Keisler测度III：一般稳定性、一般传递性与修正的Newelski猜想

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。